cascade-series-part4b.tex

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\title{\textbf{The Cascade Series --- Part IVb}\\[6pt]
The Standard Model from the Cascade:\\
Masses, Couplings, and Precision Predictions\\
from the Geometric-Topological Factorization}
\author{RTAC}
\date{March 2026}

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\begin{abstract}
The cascade series tests one hypothesis: the infinite-dimensional unit ball, descended to four dimensions, is indistinguishable from our universe. The companion paper [4] derives the Standard Model gauge group $\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)$, its symmetry breaking pattern, and three fermion generations from the cascade's geometry. This paper derives the fermion mass spectrum, gauge coupling constants, and nine independent precision predictions with zero free parameters.

The central new result is the geometric-topological factorization of the fermion mass. Every particle is a feature of the cascade's higher-dimensional geometry. Its projection into 4D spacetime is attenuated by two independent channels: (1) a geometric channel $\exp(-\Phi(d_g))$, governed by the digamma function, encoding the distance of the feature from the observer; and (2) a topological channel $(2\sqrt{\pi})^{-n_D}$, governed by $\Gamma(\tfrac{1}{2})$ and the Euler characteristic $\chi(S^{2n}) = 2$, encoding the number of hairy ball obstructions between the feature and the observer. These factorize because topological invariants do not depend on the metric.

The topological obstruction factor is $2\sqrt{\pi} = 2\,\Gamma(\tfrac{1}{2})$ per Dirac layer: 2 from chirality ($\chi(S^{2n}) = 2$) and $\sqrt{\pi} = \Gamma(\tfrac{1}{2})$ from the cascade's universal quarter-turn constant. The $d$-dependent coupling integral $R'(d)$ does not appear---confirmed by decisive numerical exclusion (61--69\% deviation with $R'(d)$ vs 0.13--1.7\% without). The universal coupling $C = \alpha_s/(2\sqrt{\pi})$ eliminates the last free parameter: the observer at $d = 4$ sits behind the $d = 5$ hairy ball zero and pays one topological toll to see the gauge window.

The complete mass formula
\[
m_g = \frac{\alpha_s \cdot v}{\sqrt{2}}\,\exp\!\bigl(-\Phi(d_g)\bigr)\,(2\sqrt{\pi})^{-(n_D+1)}
\]
has leading values $m_\tau = 1755$~MeV (obs 1777, 1.2\% low), $m_\mu = 106.2$~MeV (obs 105.7, 0.47\%), $m_e = 0.514$~MeV (obs 0.511, 0.60\%). The leading $\alpha_s = 0.1159$ and $m_\tau/m_\mu = 16.53$ sit $1.7\%$ below observation.

A structural family of cascade potential shifts $\delta\Phi = \pm\,\alpha(d^*)/\chi^k$, sourced at distinguished cascade layers $d^*$ and weighted by the same Euler characteristic $\chi=\chi(S^{2n})=2$ that appears in the $2\sqrt{\pi}$ fermion obstruction, closes \emph{eight} Standard Model precision observables within experimental precision in one structural form, generated by a discrete elastic action on the cascade lattice (Remark~\ref{rem:phase-family}). The $\mathrm{U}(1)$-layer shift $\delta\Phi_{\mathrm{U}(1)}=\alpha(14)/\chi$ closes $\alpha_s(M_Z)=0.117917$ ($+0.019\sigma$) and $m_\tau/m_\mu=16.81731$ ($+0.243\sigma$) together. The phase-transition shift $\delta\Phi_{\mathrm{phase}}=\alpha(19)/\chi$, sourced at Paper~I's threshold $d_1=19$, closes the $\tau$ absolute mass $m_\tau=1776.82$~MeV ($-0.31\sigma$). The observer-host shift $\delta\Phi_{\mathrm{obs}}=\alpha(5)/\chi^3 = 8/(225\pi)$, sourced at the volume maximum $d_V=5$ with three factors of the Euler characteristic, closes $\sin^2\theta_W = 0.231226$ ($+0.40\sigma$). All three shifts are closed forms in $\Gamma$ function values. Combined with the remaining gauge coupling predictions ($1/\alpha_{\text{GUT}} = 25.02$, $m_H = 125.82$~GeV), the cosmological matter fraction $\Omega_m = 0.31150$ (Papers I and IVa), and additional sub-3\% predictions, the series produces eight SM precision observables ($\alpha_s$, $m_\tau/m_\mu$, $m_\tau$~abs, $\ell_A$, $\sin^2\theta_W$, $\Omega_m$, $\theta_C$, $b/s$) closed within current experimental precision, with zero fitted parameters and no standard renormalisation group running. Three reuse pairs---$\{\alpha_s,m_\tau/m_\mu\}$, $\{m_\tau\text{ abs},\ell_A\}$, $\{\sin^2\theta_W,\Omega_m\}$---share the same shift, reducing the number of independent structural constants to three.

At leading order the remaining descent-dependent quantities ($v$, $\Omega_m$, $m_W$, $\sin^2\theta_W$, and the other absolute masses) carry uniformly negative deviations of $-0.13\%$ to $-2.20\%$, while geometric quantities ($m_H/m_W$, $\theta_C$, $\Omega_b$) carry positive deviations of $+0.56\%$ to $+2.76\%$. The sign separation identifies the subleading cascade descent as the source of the dominant systematic; the $\delta\Phi_{\mathrm{U}(1)}$ and $\delta\Phi_{\mathrm{phase}}$ closures show that at least two members of the $\alpha(d^*)/\chi$ family are exact at experimental precision, suggesting the systematic is resolved by additional members of the same family for the other observables.
\end{abstract}

\tableofcontents
\newpage

\section{Division of Labour}\label{sec:division-of-labour}

Part IVa [4] derives the Standard Model's structure: the gauge group $\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)$ from the Bott mirror, the symmetry breaking pattern from the hairy ball theorem, and three generations from the $d_1 = 19$ phase transition. This paper derives the Standard Model's quantities: masses, couplings, and precision predictions.

\section{The Fermion Mass Spectrum}\label{sec:fermion-mass}

\subsection{The cascade potential}

\begin{definition}[Cascade potential]
The cascade potential at dimension $d$ is
\[
\Phi(d) := \sum_{d'=5}^{d} p(d') = \sum_{d'=5}^{d} \Bigl[\tfrac{1}{2}\psi\!\Bigl(\frac{d'+1}{2}\Bigr) - \tfrac{1}{2}\ln\pi\Bigr],
\]
where $\psi = \Gamma'/\Gamma$ is the digamma function.
\end{definition}

The cascade potential is the cumulative decay rate, measuring the total suppression of a propagator traversing from $d = 4$ to dimension $d$. At the generation layers: $\Phi(5) = -0.111$, $\Phi(13) = 1.429$, $\Phi(21) = 5.494$, $\Phi(29) = 11.082$.

\begin{remark}[Every path in this paper is forced, not chosen]\label{rem:paths-forced}
The specific cascade descents used throughout this paper---$d=6..13$ for $m_\tau/m_\mu$, $d=14..21$ for $m_\mu/m_e$ (exclusive of the lower endpoint, as differences of cumulative $\Phi$), and $d=5..12$ for $\alpha_s$ and the electroweak VEV $v$, $d=5..13$ and $d=5..14$ for the $\mathrm{SU}(2)$ and $\mathrm{U}(1)$ couplings entering the Weinberg angle (inclusive of the $d=5$ anchor, as $\Phi$ values directly)---are not independent assumptions. They are forced by the layer assignments of Part~IVa~\cite{part4a} (generation layers at $d\in\{5,13,21\}$; gauge boson layers at $d\in\{12,13,14\}$; observer host at $d_V=5$, the volume maximum) via the \emph{Forced Cascade Paths} theorem (Part~IVa, ``Forced Cascade Paths'' section). The asymmetry between the two conventions is structural: mass ratios are differences $\Phi(d_B)-\Phi(d_A)$ between two cascade layers, so the $d=5$ terms cancel whenever neither endpoint is at $d=5$; gauge-anchored quantities are $\Phi(d_B)$ directly, starting from the observer host at $d_V=5$. Once Part~IVa fixes the layers, Part~IVb's path is the unique cascade descent between them---every length forced by the Radon--Hurwitz and Bott-period theorems of Part~IVa, not by any additional input. Separating the two: Part~IVa derives the \emph{where}; this paper derives the \emph{how much}.
\end{remark}

\subsection{The geometric-topological factorization}

Every particle is a feature of the cascade at some dimension $d$ in the higher-dimensional geometry. The 4D observer sees its projection into 4D, attenuated by every layer it passes through. This projection decomposes into two independent channels.

\textbf{Geometric channel.} The cascade potential $\Phi(d_g)$ measures the cumulative suppression through the cascade's sphere-area decay. The geometric attenuation is $\exp(-\Phi(d_g))$. It is continuous, $d$-dependent, and governed by the digamma function.

\textbf{Topological channel.} Between $d = 4$ and the generation layer $d_g$, the projection passes through $n_D(d_g)$ Dirac layers. At each Dirac layer $d$ (where $d \bmod 8 = 5$), the sphere $S^{d-1}$ is even-dimensional and the hairy ball theorem forces a zero. Each obstruction attenuates the projection by $2\sqrt{\pi}$, giving $(2\sqrt{\pi})^{-n_D}$.

The two channels factorize because topological invariants do not depend on the metric.

\subsection{The topological obstruction factor}

\begin{theorem}[Topological obstruction factor]\label{thm:obstruction}
The fermion-to-scalar propagator ratio at each Dirac layer is $1/(2\sqrt{\pi})$, independent of $d$.
\end{theorem}

\begin{proof}
The cascade lapse at layer $d$ factorises as $N(d) = \sqrt{\pi}\cdot R(d)$
(Part~0, Theorem~\xref{part0}{thm:boundary}), where $\sqrt{\pi} = \Gamma(\tfrac{1}{2})$ is the universal
compression constant from orthogonality and
$R(d) = \Gamma((d+1)/2)/\Gamma((d+2)/2)$ is the $d$-dependent geometric coupling.
At a Dirac layer ($d\bmod 8 = 5$), $S^{d-1}$ is even-dimensional.
Two independent obstructions attenuate the fermion propagator:

\medskip
\textbf{(a) The chirality obstruction (factor $\mathbf{1/2}$).}
On an even-dimensional sphere $S^{2n}$, the spinor bundle decomposes as
$S = S^+\oplus S^-$ under the chirality grading. The Euler characteristic
$\chi(S^{2n}) = 2$ counts the minimum number of critical points of any Morse
function on $S^{2n}$ (Poincar\'{e}--Hopf theorem applied to its gradient).
These two critical points---one source, one sink of the height function's
gradient flow---define two chirality basins. A definite-chirality fermion
occupies one basin, halving the propagator amplitude. The factor
$1/\chi = 1/2$ is dimension-independent because $\chi(S^{2n}) = 2$ for all
$n\geq 1$.

\medskip
\textbf{(b) The quarter-turn obstruction (factor $\mathbf{1/\sqrt{\pi}}$).}
The cascade's universal constant $\sqrt{\pi}$ enters the scalar propagator
through the half-integer first argument of $B(\tfrac{1}{2},\cdot)$. This
half-integer arises because the orthogonal quarter-turn from slicing axis to
equator produces a Jacobian $t^{-1/2}$ (under $t = x^2$) in the Beta integral:
\[
B(\tfrac{1}{2},\, d/2+1) = \int_0^1 t^{-1/2}(1-t)^{d/2}\,dt
= \sqrt{\pi}\cdot R'(d).
\]
The $t^{-1/2}$ factor encodes the angular spread of the tangent frame as the
slicing axis rotates from pole to equator. It is a \emph{frame-dependent}
quantity: it measures how the tangent directions fan out along the quarter-turn.

On an odd-dimensional sphere $S^{2n+1}$, a nowhere-vanishing tangent vector
field exists (the hairy ball theorem does not apply). The tangent frame
completes the quarter-turn smoothly, and $\sqrt{\pi}$ enters the propagator
in full.

On an even-dimensional sphere $S^{2n}$, the hairy ball theorem forces every
continuous tangent vector field to vanish. The tangent frame is globally
obstructed. A scalar (0-form) couples only to the metric, which is
well-defined everywhere regardless of the tangent frame---it sees the full
quarter-turn measure $\sqrt{\pi}$. A fermion (spinor) couples to the tangent
frame through the spin connection. The obstruction's quantitative content
is fixed at the partition-function level (Remark~\ref{rem:berezin-partition-derivation}):
the bosonic per-layer partition function is the slicing integral
$Z_s(d) = \int_{-1}^1(1-x^2)^{(d-1)/2}\,dx = \sqrt{\pi}\cdot R(d)$
(Part~0, Theorem~\xref{part0}{thm:boundary}; the Gaussian/Beta-function Jacobian carries
$\sqrt{\pi}$), while the fermionic per-layer partition function under
Berezin integration of $S_f = m(d)\bar\psi\psi$ is $Z_f(d) = m(d)$ with
no Gaussian prefactor. With the cascade-native Dirac mass
$m(d) = \sqrt{\alpha(d)} = R(d)/2$, the ratio
$Z_f/Z_s = (R/2)/(\sqrt{\pi}\cdot R) = 1/(2\sqrt{\pi})$ exactly at every
Dirac layer, so $N_f(d) = R(d)/\chi$. The chirality factor $1/\chi = 1/2$
is derived from Poincar\'e--Hopf via Theorem~\ref{thm:chirality-factorisation};
the Jacobian factor $1/\sqrt{\pi}$ is the structural Gaussian-vs-Berezin
difference, not a sphere-spinor calculation. The structural
identification $m(d) = R(d)/\chi = \sqrt{\alpha(d)}$ connects the cascade's
per-layer mass to the standard Dirac-mass-versus-gauge-coupling relation:
the per-layer fermion mass is the chirality-halved per-layer scale, which
equals the per-layer gauge-coupling amplitude $\sqrt{\alpha(d)}$ in
cascade-native units (since $\alpha(d) = R(d)^2/4$ by definition,
Remark~\ref{rem:alpha-geometric}). \emph{No free Yukawa coupling enters
the construction} --- the SM's parameterisation $m_g = y_g\,v/\sqrt{2}$
does not appear in the cascade's chain (slicing recurrence + chirality
halving + per-layer locality + Berezin integration), so there is no
$y_g$ to fit or forbid.
Pool-uniqueness over monomial-in-$\alpha$ Berezin candidates
(\texttt{tools/verifiers/fermion\_mass\_uniqueness.py}: $m = \sqrt{\alpha}$
is the unique cascade-native candidate giving $d$-independent $Z_f/Z_s$)
is a confirming check on the structural derivation, not the
load-bearing argument.
The numerical value $2\sqrt{\pi}$ is also the direct cascade-primitive
identity $N(0)\cdot\Gamma(\tfrac12)$ (Corollary~\ref{cor:2sqrtpi-primitive}).
See Remark~\ref{rem:sp31-status} for the per-ingredient derivation
statuses.

\medskip
\noindent\emph{Recurrence-native restatement.}
The sphere-bundle / tangent-frame language above is one reading of why the fermion
loses one factor of $\sqrt{\pi}$ relative to the scalar.  An equivalent
recurrence-native reading does not invoke sphere-bundle structure:
the cascade scalar action at layer $d$ has compliance
$\alpha(d) = N(d)^2/\Omega_2 = R(d)^2/4$ (Remark~\ref{rem:alpha-geometric}).  The
natural Dirac partner of this scalar compliance is its square root,
$m(d) = \sqrt{\alpha(d)} = R(d)/2$---the standard Dirac-vs-Laplacian relation
applied to the cascade's per-layer compliance.  Then the fermion-to-scalar
per-layer ratio is
\[
\frac{m(d)}{N(d)} \;=\; \frac{R(d)/2}{\sqrt{\pi}\,R(d)} \;=\; \frac{1}{2\sqrt{\pi}},
\]
identical to the value derived above through tangent-frame language but obtained
without invoking the spin connection or the hairy-ball obstruction beyond the
chirality halving of step~(a).  Three independent cascade-lattice constructions
converge on this $m(d) = R(d)/2$ value (per-layer Berezin partition function,
square-root of the cascade scalar Laplacian, and variational with $R$-based
compliance; see Remark~\ref{rem:sp31-status} for the failure
analysis of the contrasting sphere-bundle constructions).  Both readings derive
the same number from cascade primitives; the recurrence-native reading is the
more cascade-internal of the two.

\medskip
\noindent\emph{Power-uniqueness sanity check.}
A confirming check: among monomial-in-$\alpha$ Berezin candidates
$Z_f(d) = \alpha(d)^k$, the power $k = 1/2$ is uniquely selected by
d-independence of $Z_f/Z_s$.  Using $Z_s(d) = N(d) = 2\sqrt{\pi}\,\sqrt{\alpha(d)}$
(cascade scalar identity, Remark~\ref{rem:alpha-geometric}), the ratio is
$Z_f/Z_s = \alpha^{k-1/2}/(2\sqrt{\pi})$.  d-independence forces $k = 1/2$
exactly.  Verified numerically across $k\in[-1, 2]$ at four Dirac layers in
\texttt{tools/verifiers/cascade\_native\_clifford\_absorption.py}: only $k = 1/2$
gives coefficient of variation $0\%$; every other power produces CV $\geq 3\%$.
This confirms the structural derivation $m(d) = R(d)/\chi$ from chirality
halving (and equivalently $m(d) = \sqrt{\alpha(d)}$ from the
Dirac-vs-Laplacian relation): no other monomial-in-$\alpha$ candidate
gives a $d$-independent ratio, so the cascade-internal value is unique
among that pool.  A cascade-lattice scalar-action uniqueness theorem
extending the check to all functional forms would supply a further
independent proof but is not required for closure --- the chirality
halving + Berezin chain already produces the value from cascade primitives.

The geometric coupling $R(d)$, which depends on the radial structure of the
slicing (the ratio of adjacent Gamma functions), measures the change in
cross-sectional size---a metric quantity that does not require a tangent frame.
This factor survives for both scalars and fermions.

\medskip
\textbf{(c) Uniqueness.}
The two factors $\sqrt{\pi}$ and $\chi = 2$ are the \emph{only}
dimension-independent constants available at the hairy ball obstruction:
$\sqrt{\pi} = \Gamma(\tfrac{1}{2})$ is the unique constant of the cascade's
slicing recurrence (Part~0, Theorem~\xref{part0}{thm:irred}); $\chi(S^{2n}) = 2$ is the unique
topological invariant of even-dimensional spheres that counts the obstruction
strength. Any $d$-dependent factor would break the universality of the mass
formula, contradicting the observed $d$-independence of the Bott factor to
1.6\% (Remark~\ref{rem:universality}).

\medskip
The fermion lapse at a Dirac layer is therefore:
\[
N_f(d) = \frac{R(d)}{\chi(S^{d-1})} = \frac{R(d)}{2}.
\]
The fermion-to-scalar ratio is:
\[
\frac{N_f(d)}{N(d)} = \frac{R(d)/2}{\sqrt{\pi}\cdot R(d)} = \frac{1}{2\sqrt{\pi}},
\]
independent of $d$.
\end{proof}

\noindent The factor decomposes as: (a) $\sqrt{\pi} = \Gamma(\tfrac{1}{2})$: the cascade's unique dimension-independent constant (Paper I, Theorem~\xref{part0}{thm:boundary}), forced by orthogonality---consumed at the hairy ball zero because the tangent frame it encodes is globally obstructed. (b) $2 = \chi(S^{2n})$: the Euler characteristic of any even-dimensional sphere, equal to the number of chirality basins at the Dirac junction.

\begin{corollary}[The obstruction factor as a cascade primitive]\label{cor:2sqrtpi-primitive}
The topological obstruction factor equals the product of the cascade's zeroth slicing lapse and its unique irreducible constant:
\[
2\sqrt{\pi} \;=\; N(0)\cdot\Gamma(\tfrac{1}{2}),
\]
where $N(0) = \int_{-1}^{1}(1-x^2)^0\,dx = 2$ is the Lebesgue measure of the slicing interval $[-1,1]$---the cascade's primitive step from $d=0$ (a point) to $d=1$ (the unit segment). Both factors are cascade primitives: no topological or spinor machinery enters the identity itself.
\end{corollary}

\begin{proof}
Direct evaluation: $N(0) = B(\tfrac{1}{2},1) = \sqrt{\pi}\,\Gamma(1)/\Gamma(\tfrac{3}{2}) = 2$ (the integrand $(1-x^2)^0 = 1$ over $[-1,1]$). Multiplying by $\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$ gives $2\sqrt{\pi}$.
\end{proof}

\begin{remark}[The identification $N(0) = \chi(S^0) = \chi(S^{2n})$]\label{rem:N0-is-chi}
The cascade's $d=0$ lapse $N(0) = 2$ coincides with the Euler characteristic of the $0$-sphere: the unit interval $B^1 = [-1,1]$ has boundary $S^0 = \{-1,+1\}$, two points, and $\chi(S^0) = 2$. Classically, $\chi(S^{2n}) = 1 + (-1)^{2n} = 2$ for every $n\geq 0$, so the Euler characteristic is constant across all even-dimensional sphere layers. Fermion layers at $d\equiv 5\pmod 8$ have boundary spheres $S^{d-1}$ with $d-1\equiv 4\pmod 8$, all even-dimensional, hence all with $\chi = 2$. The cascade primitive $N(0)$ therefore carries the full generation-independent content of the topological channel: one cascade number---the initial slicing integral---substitutes for the differential-topological derivation of $\chi$ at every fermion layer.
\end{remark}

\begin{remark}[Constancy rules out a Dirac-operator derivation]\label{rem:no-dirac-route}
The mass formula uses the \emph{same} constant base $2\sqrt{\pi}$ at every fermion layer, so any topological invariant that underlies the obstruction must be independent of the sphere dimension $2n$. Among candidates on $S^{2n}$:
\begin{center}
\begin{tabular}{lccc}
\hline
Invariant & $n=1$ & $n=2$ & $n=3$ \\
\hline
Euler characteristic $\chi(S^{2n})$ & $2$ & $2$ & $2$ \\
Signature $\sigma(S^{2n})$ & $0$ & $0$ & $0$ \\
Dirac index / $\hat A(S^{2n})$ & $0$ & $0$ & $0$ \\
\hline
\end{tabular}
\end{center}
Only $\chi$ is constant and nonzero. The Atiyah--Singer index of a Dirac operator on $S^{2n}$ equals $\hat A(S^{2n})=0$ ($S^{2n}$ is null-cobordant in spin-bordism), so a Dirac-operator derivation predicts $0$, not $2\sqrt{\pi}$. Corollary~\ref{cor:2sqrtpi-primitive} gives the factor directly from the cascade's primitives $N(0)\cdot\Gamma(\tfrac{1}{2})=2\sqrt{\pi}$.
\end{remark}

\begin{remark}[Derivation of the $1/(2\sqrt{\pi})$ factor]\label{rem:sp31-status}
The fermion-to-scalar propagator ratio $1/(2\sqrt{\pi})$ factorises into three structurally derived ingredients, all from cascade primitives:
\begin{enumerate}
\item \emph{Chirality halving $1/\chi = 1/2$.} Poincar\'e--Hopf applied to the Morse function on $S^{2n}$ at Dirac layers gives $\chi = 2$, splitting the spinor bundle into two equal-weight chirality basins (Theorem~\ref{thm:chirality-factorisation}). A definite-chirality fermion occupies one basin: factor $1/\chi$. Topological theorem; no assumption.
\item \emph{Cascade-primitive identity $2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac12)$} (Corollary~\ref{cor:2sqrtpi-primitive}). $N(0)$ and $\Gamma(\tfrac12)$ are both Part~0 primitives; no topological or spinor machinery enters the identity itself.
\item \emph{Per-layer mass $m(d) = R(d)/\chi$ via chirality halving + Berezin partition.} The cascade fermion at each layer is a Grassmann scalar with mass action $m(d)\bar\psi\psi$ (per-layer locality, Section~\ref{sec:fermion-action}); Berezin integration gives $Z_f(d) = m(d)$ identically, with no Gaussian prefactor. The bosonic counterpart is $Z_s(d) = \sqrt{\pi}\cdot R(d)$ from the slicing measure (Part~0, Theorem~\xref{part0}{thm:boundary}; the Gaussian/Beta-function Jacobian carries $\sqrt{\pi}$). The chirality halving from item~(1) gives $m(d) = R(d)/\chi = R(d)/2$ as the cascade's per-layer mass --- the bare scale $R(d)$ halved by the Euler characteristic of the even-sphere Dirac layer. The ratio is then
\[
Z_f(d)/Z_s(d) \;=\; \frac{R(d)/2}{\sqrt{\pi}\,R(d)} \;=\; \frac{1}{2\sqrt{\pi}},
\]
exactly at every Dirac layer, $d$-independent. The Gaussian-vs-Berezin Jacobian difference is the structural origin of the $1/\sqrt{\pi}$; no sphere-spinor calculation enters. The structural identification $R(d)/2 = \sqrt{\alpha(d)}$ (since $\alpha(d) = R(d)^2/4$ by definition, Remark~\ref{rem:alpha-geometric}) connects the cascade's per-layer mass to the standard Dirac-mass-versus-gauge-coupling relation: the per-layer fermion mass equals the per-layer gauge-coupling amplitude as a structural identification, not as an independent commitment. \emph{No free Yukawa coupling enters the construction} --- the SM's $m = y\cdot v/\sqrt{2}$ parameterisation does not appear in the cascade's chain (slicing recurrence + chirality halving + per-layer locality + Berezin), so there is no $y$ to fit or forbid. Sphere-bundle imports (Mathai--Quillen, sphere-Dirac, projection geometry) are independently ruled out by scaling: sphere quantities decay super-exponentially in $d$, while $R(d)/2 \sim 1/\sqrt{2d}$ decays polynomially. Per CLAUDE.md Check~7, the closure must be cascade-lattice native; the chirality halving $1/\chi$ is the only place sphere structure enters.
\end{enumerate}
The numerical observables (lepton mass ratios to $0.13\%$ and $1.7\%$, $m_\tau$ absolute to $-0.31\sigma$, the universality check of Remark~\ref{rem:universality}) confirm the value $1/(2\sqrt{\pi})$ to the cascade's stated precision.

\medskip\noindent\emph{Sanity checks (not derivation steps).} Two confirming checks strengthen the structural derivation: (a) pool-uniqueness over monomial-in-$\alpha$ Berezin candidates $Z_f(d) = \alpha(d)^k$ selects $k = 1/2$ uniquely (verifier \texttt{tools/verifiers/fermion\_mass\_uniqueness.py}: $m = \sqrt{\alpha}$ is the unique cascade-native candidate giving $d$-independent $Z_f/Z_s$); (b) the scalar analogue is derived in parallel via the discrete scalar action $S[\varphi] = \sum_d (2\alpha(d))^{-1}(\Delta\varphi)^2$ (Remark~\ref{rem:action-uniqueness}), with $B(\tfrac12, d/2+1) = \sqrt{\pi}\cdot R'(d)$ as the scalar propagator on $S^{d-1}$. A cascade-lattice scalar-action uniqueness theorem extending the check beyond the monomial-in-$\alpha$ pool would supply a further independent proof but is not required for closure --- the chirality halving + Berezin chain already produces $m(d) = R(d)/\chi$ from cascade primitives.
\end{remark}

\subsection{Factor-by-factor decomposition of the fermion obstruction}
\label{subsec:sp31-decomposition}

Corollary~\ref{cor:2sqrtpi-primitive}'s cascade-primitive identity
$2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac{1}{2})$ decomposes the fermion
obstruction factor into two structurally distinct halves: the chirality
half follows from Theorem~\ref{thm:chirality-factorisation}
(Poincar\'e--Hopf); the Jacobian half is the Berezin/Gaussian
partition-function difference of Remark~\ref{rem:berezin-partition-derivation},
with the per-layer mass $m(d) = R(d)/\chi = \sqrt{\alpha(d)}$ derived from
chirality halving (Remark~\ref{rem:sp31-status}). The theorem below
records the split.

\begin{theorem}[Decomposition of the fermion obstruction factor]\label{thm:sp31-decomposition}
Under the cascade-primitive identity
$2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac{1}{2})$
(Corollary~\ref{cor:2sqrtpi-primitive}), the fermion obstruction factor
factorises as
\[
\frac{1}{2\sqrt{\pi}}
\;=\;
\underbrace{\frac{1}{N(0)}}_{\text{derived (chirality halving)}}
\;\cdot\;
\underbrace{\frac{1}{\Gamma(\tfrac{1}{2})}}_{\text{derived (Berezin/Gaussian Jacobian)}}
\;=\;
\frac{1}{\chi}\cdot\frac{1}{\sqrt{\pi}},
\]
where the two factors have distinct cascade-internal origins and
distinct derivation statuses.
\end{theorem}

\begin{proof}[Derivation status of the two factors]
\textbf{$1/N(0) = 1/\chi = 1/2$: derived via chirality factorisation.}
Theorem~\ref{thm:chirality-factorisation} establishes that on any
even-sphere layer $d\bmod 8 = 5$ (where $S^{d-1}$ is even-dimensional),
the Green's function of the cascade action's discrete Laplacian
factorises into $\chi(S^{d-1}) = 2$ chirality sectors of equal weight,
via the Poincar\'{e}--Hopf / Morse-function decomposition of the
height function on $S^{2n}$ into $\chi$ equal basins. A definite-chirality
fermion projector $P_\pm = (1\pm\gamma_5)/2$ selects one basin,
contributing a factor $1/\chi = 1/2$. Via
Remark~\ref{rem:N0-is-chi}'s identification
$N(0) = \chi(S^0) = \chi(S^{2n}) = 2$, this is exactly $1/N(0)$.

\textbf{$1/\Gamma(\tfrac{1}{2}) = 1/\sqrt{\pi}$: conjectured via
Clifford absorption.} On the scalar side, $\sqrt{\pi} = \Gamma(\tfrac{1}{2})$
enters the slicing integral via the half-integer first argument of the
Beta function $B(\tfrac{1}{2},\,d/2) = \int_0^1 t^{-1/2}(1-t)^{d/2-1}dt$,
whose Jacobian $t^{-1/2}$ arises from the substitution $t=x^2$
converting the axial coordinate to the quarter-turn squared length. A
cascade Dirac action on the spinor bundle of $S^{d-1}$ replaces this
real-valued Gaussian integration with a Clifford-structured Berezin
integration: the Grassmann measure does not pick up the $(2\pi)^{1/2}$
Gaussian-normalisation prefactor of real integration, and the
$\gamma$-matrix algebra's $SO(d)$-invariant normalisation is conjectured
to absorb the half-integer Jacobian into the Clifford structure. This
is captured at the partition-function level in
Remark~\ref{rem:berezin-partition-derivation}, with the per-layer mass
$m(d) = R(d)/\chi$ derived from chirality halving (Remark~\ref{rem:sp31-status}).
\end{proof}

\begin{remark}[Chirality-Jacobian separability at the integral level]\label{rem:chirality-jacobian-separability}
The decomposition of Theorem~\ref{thm:sp31-decomposition} admits a
direct verification at the level of the scalar slicing integral. The
weighted integral
\[
\int_{-1}^{1}(1-x^2)^{(d-1)/2}\cdot\tfrac{1\pm x}{2}\,dx
\;=\;
\tfrac{1}{2}\int_{-1}^{1}(1-x^2)^{(d-1)/2}\,dx
\;=\;
\tfrac{1}{2}\sqrt{\pi}\,R(d)
\]
(the $x$ term vanishes by parity; the remaining integral is the scalar
lapse). Weighting the scalar integrand by a chirality projector
$(1\pm x)/2$ therefore removes the factor of $2$ exactly while leaving
the $\sqrt{\pi}$ factor of $\Gamma(\tfrac{1}{2})$ untouched. This is
the integral-level statement of mechanistic separability: the chirality
half $1/\chi = 1/2$ acts on the domain of integration, and the Jacobian
half $1/\sqrt{\pi}$ lives in the integration measure itself.

A consequence for any confirming cascade-action derivation of the
$1/\sqrt{\pi}$ absorption: it cannot operate via a
modified axial measure on $[-1,1]$---such a modification would break
the $d$-independence confirmed at 1.6\% by the observed mass ratios
(Remark~\ref{rem:universality}), and is independently falsified by the
sphere-Dirac spectral-zeta check that does not match $R(d)$
(\texttt{tools/model\_checks/fermion\_dirac\_spectral\_zeta.py};
the ratio $\zeta_D(1)/(R(d)/2)$ decays exponentially in $d$ while
$R(d)/2$ decays polynomially). The absorption must therefore come from
spinor structure (Berezin/Clifford inner product, or spin-connection
coupling in the Dirac operator), realised concretely by
Remark~\ref{rem:berezin-partition-derivation} below. Verified numerically
to machine precision ($<3\times 10^{-16}$) at Dirac layers
$d\in\{5,13,21,29\}$ in
\texttt{tools/verifiers/chirality\_jacobian\_separation.py}.
\end{remark}

\begin{remark}[Berezin partition-function realisation of Clifford absorption]\label{rem:berezin-partition-derivation}
The Berezin-vs-Gaussian mechanism is exact at the per-layer
partition-function level. The cascade's bosonic per-layer partition function is the scalar
slicing integral
$Z_s(d) = \int_{-1}^{1}(1-x^2)^{(d-1)/2}\,dx = \sqrt{\pi}\,R(d)$
(Part~0, Theorem~\xref{part0}{thm:boundary}). For a Grassmann-valued cascade field with
quadratic action $S_f = m(d)\,\bar\psi\psi$, Berezin integration gives
$Z_f(d) = m(d)$ with no Gaussian prefactor. With the cascade-derived
per-layer mass $m(d) = R(d)/\chi = R(d)/2$ from
Theorem~\ref{thm:chirality-factorisation}'s chirality decomposition on
the even-sphere Dirac layer (equivalently $\sqrt{\alpha(d)}$, the
standard Dirac-mass-versus-gauge-coupling relation applied per cascade
layer), the per-layer ratio is
\[
\frac{Z_f(d)}{Z_s(d)}
\;=\;
\frac{R(d)/2}{\sqrt{\pi}\,R(d)}
\;=\;
\frac{1}{2\sqrt{\pi}}
\]
exactly, at every Dirac layer, with no modification of the slicing
measure. The two halves of the sp31 split factorise transparently:
$1/\chi = 1/2$ lives in the factor-of-two halving
$m(d) = R(d)/2$ (chirality), and $1/\sqrt{\pi}$ lives in the absence
of a Gaussian prefactor on the Berezin side (Jacobian). The product of
$n_D$ per-layer ratios reproduces Theorem~\ref{thm:complete-mass}'s
$(2\sqrt{\pi})^{-n_D}$ base exactly. Verified to machine precision
($<3\times 10^{-16}$) at
$d\in\{5,13,21,29\}$ in
\texttt{tools/closures/fermion\_berezin\_partition.py}.

\medskip\noindent\emph{No Yukawa parameter is introduced or eliminated.}
The cascade's mass formula is $m(d) = R(d)/\chi$ from cascade primitives
(slicing recurrence + chirality halving + per-layer locality + Berezin
integration). The SM's parameterisation $m_g = y_g\,v/\sqrt{2}$ does not
appear in the cascade's construction; there is no place in the chain
above for a free Yukawa coupling $y_g$. The structural identification
$R(d)/2 = \sqrt{\alpha(d)}$ (since $\alpha(d) = R(d)^2/4$ by definition,
Remark~\ref{rem:alpha-geometric}) connects the cascade's per-layer mass
to the standard Dirac-mass-versus-gauge-coupling relation as a structural
identification, not as a derivation step. Pool-uniqueness over
monomial-in-$\alpha$ Berezin candidates
(\texttt{tools/verifiers/fermion\_mass\_uniqueness.py}: $m = \sqrt{\alpha}$
is the unique cascade-native candidate giving $d$-independent $Z_f/Z_s$)
is a confirming check on the structural derivation, not the load-bearing
argument. A cascade-lattice scalar-action uniqueness theorem extending
the check to all functional forms would supply a further independent
proof but is not required for closure --- the chirality halving + Berezin
chain already produces the value from cascade primitives.
\end{remark}


\subsection{Why \texorpdfstring{$R'(d)$}{R'(d)} does not appear}

The Higgs coupling integral on $S^{2n}$ at Dirac layer $d$ is $B(\tfrac{1}{2}, \tfrac{d}{2}) = \sqrt{\pi}\cdot R'(d)$ with $R'(d) = \Gamma(d/2)/\Gamma((d+1)/2)$. The cascade-internal identity $R'(d) = R(d-1)$ (Remark~\ref{rem:R-prime-identity}) shows this is the slicing recurrence at layer $d-1$, already encoded in $\Phi(d)$; including it again as a layer-$d$ factor double-counts. The fermion mass coupling at the hairy ball zero retains only the topological residue $\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$:

\begin{center}
\begin{tabular}{lccc}
\hline
Ratio & With $R'(d)$ & Without $R'(d)$ & Observed \\
\hline
$\tau/\mu$ & 6.61 (60.7\% off) & 16.53 (1.7\% off) & 16.82 \\
$\mu/e$ & 64.49 (68.8\% off) & 206.50 (0.13\% off) & 206.77 \\
\hline
\end{tabular}
\end{center}

\begin{remark}[The identity $R'(d) = R(d-1)$]\label{rem:R-prime-identity}
$R'(d) = \Gamma(d/2)/\Gamma((d+1)/2) = R(d-1)$ as a $\Gamma$-function identity. Under $x = \cos\theta$, the slicing weight $(1 - x^2)^{(d-1)/2}$ at layer $d-1$ becomes $\sin^{d-1}\theta$---the Higgs coupling integrand on $S^{d-1}$. The cascade evaluates this integral once via the slicing recurrence; the $R'(d)$ form at layer $d$ is the same datum relabelled.
\end{remark}

\begin{remark}[Universality from the data]\label{rem:universality}
The Bott factor extracted from observed mass ratios is $F(\tau/\mu) = 16.817/4.663 = 3.606$ and $F(\mu/e) = 206.77/58.25 = 3.550$. Their ratio $F(\tau/\mu)/F(\mu/e) = 1.016$, confirming $d$-independence to 1.6\%. The theoretical value $2\sqrt{\pi} = 3.5449$ lies within 0.13\% of $F(\mu/e)$; the 1.7\% deviation in $F(\tau/\mu)$ is consistent with subleading cascade geometric corrections.
\end{remark}

\subsection{The obstruction count}

\begin{center}
\begin{tabular}{lccc}
\hline
Generation & Layer & Dirac obstructions passed & $n_D$ \\
\hline
3 ($\tau, b$) & $d = 5$ & $d = 5$ & 1 \\
2 ($\mu, s$) & $d = 13$ & $d = 5, 13$ & 2 \\
1 ($e, d, u$) & $d = 21$ & $d = 5, 13, 21$ & 3 \\
\hline
\end{tabular}
\end{center}

The mass of generation $g$ is:
\[
m_g \propto \exp(-\Phi(d_g)) \times (2\sqrt{\pi})^{-n_D(d_g)}.
\]
The mass ratio between adjacent generations involves one additional Dirac obstruction:
\[
m_{\text{heavy}}/m_{\text{light}} = \exp(\Delta\Phi) \times 2\sqrt{\pi}.
\]

\subsection{Charged lepton mass ratios}

\begin{theorem}[Lepton mass ratios, leading]\label{thm:lepton-ratios}
\begin{align*}
m_\tau/m_\mu &= \exp(\Delta\Phi(5 \to 13)) \times 2\sqrt{\pi} = 4.662 \times 3.5449 = 16.53, \\
m_\mu/m_e &= \exp(\Delta\Phi(13 \to 21)) \times 2\sqrt{\pi} = 58.25 \times 3.5449 = 206.50.
\end{align*}
The leading $m_\tau/m_\mu$ is closed to experimental precision by the universal $\mathrm{U}(1)$-layer shift $\delta\Phi_{\mathrm{U}(1)}=\alpha(14)/\chi$ of Theorem~\ref{thm:alpha-s-closure} below: $m_\tau/m_\mu=\exp(\Delta\Phi(5\to 13)+\delta\Phi_{\mathrm{U}(1)})\cdot 2\sqrt{\pi}=16.8173$ against the observed $16.8170\pm 0.0011$. The $m_\mu/m_e$ path $d=14..21$ begins at the $\mathrm{U}(1)$ layer and does not receive the shift; its leading value already matches observation to $0.13\%$.
\end{theorem}

\begin{proof}
The cascade potential differences are $\Phi(13) - \Phi(5) = \sum_{d=6}^{13} p(d) = 1.5397$ and $\Phi(21) - \Phi(13) = \sum_{d=14}^{21} p(d) = 4.0648$. Exponentiating: $\exp(1.5397) = 4.662$, $\exp(4.0648) = 58.25$. Multiplying by $2\sqrt{\pi} = 3.5449$: $4.662 \times 3.5449 = 16.53$ and $58.25 \times 3.5449 = 206.50$.
\end{proof}

\begin{remark}[Zero free parameters]
The prediction uses: (i) the digamma function at integer and half-integer arguments, (ii) the Bott period of 8, and (iii) the topological factor $2\sqrt{\pi}$ from the hairy ball theorem and the cascade's quarter-turn constant. No parameter is fitted. The 0.13\% agreement in $m_\mu/m_e$ is a precision prediction of the series.
\end{remark}

\begin{center}
\begin{tabular}{lccc}
\hline
Ratio & Cascade (with $\delta\Phi_{\mathrm{U}(1)}$ where applicable) & Observed & Residual \\
\hline
$m_\mu/m_e$ & 206.50 (no shift; path starts at $d=14$) & 206.77 & $0.13\%$ \\
$m_\tau/m_\mu$ & 16.8173 (exp form, Theorem~\ref{thm:alpha-s-closure}) & 16.81703 & $+0.24\sigma$ \\
$m_\tau/m_e$ & 3477 (derived from the two above) & 3477 & $<0.1\%$ \\
\hline
\end{tabular}
\end{center}

\subsection{The universal coupling and the observer's obstruction}

\begin{theorem}[Universal coupling]\label{thm:universal-coupling}
The effective Yukawa coupling at the observer's position is
\[
C = \frac{\alpha_s}{2\sqrt{\pi}},
\]
where $\alpha_s = \alpha_{\text{GUT}} \times \exp(\Phi_{12 \to 4})$ is the strong coupling at the observer's scale (Theorem~\ref{thm:alpha-s-leading}). The relation holds to 1.0\%.
\end{theorem}

\begin{proof}
The observer at $d = 4$ is a Weyl layer. The nearest Dirac layer is $d = 5$, with a hairy ball zero on $S^4$. The observer's projection to the gauge window at $d = 12$ passes through this zero. The effective coupling is $C = \alpha_s/(2\sqrt{\pi}) = N(12)^2/(4\pi) \cdot \exp(\Phi_{12 \to 4})/(2\sqrt{\pi})$. Numerically: $C = 0.1159/3.545 = 0.0327$ (Route~1 of Remark~\ref{rem:sp34-status}).

Independently, inverting the $\tau$-mass formula
$m_\tau = (\alpha_s v/\sqrt{2})\exp(-\Phi(5))(2\sqrt{\pi})^{-2}$
for $C$ via $\alpha_s = 2\sqrt{\pi}\,C$ gives
$C_\tau = 2\sqrt{2\pi}\,m_\tau\,e^{\Phi(5)}/v$; plugging in observed
$m_\tau = 1776.82$~MeV and cascade-derived $v, \Phi(5)$ yields
$C_\tau = 0.0324$ (Route~2). Agreement: 1.0\%.
\end{proof}

\begin{remark}[Why Route 1 and Route 2 are independent]\label{rem:sp34-status}
The two extractions of $C$ share no numerical input:
\begin{itemize}
\item \emph{Route 1 (gauge-descent).} $C = \alpha_s/(2\sqrt{\pi})$ with
$\alpha_s = N(12)^2/(4\pi) \cdot \exp(\Phi_{12\to 4})$. Inputs: Gamma
function values $N(12)$ and the cascade potential sum $\Phi_{12\to 4}$.
No fermion mass appears.
\item \emph{Route 2 (mass-formula inversion).} Take the Gen~3 absolute
mass formula $m_\tau = (\alpha_s v/\sqrt{2})\exp(-\Phi(5))(2\sqrt{\pi})^{-2}$,
substitute $\alpha_s = 2\sqrt{\pi}\,C$, and solve for $C$ using
\emph{observed} $m_\tau$, \emph{cascade-derived} $v$ (Theorem~\ref{thm:vev}:
$v = M_{\mathrm{Pl,red}}\cdot\alpha_{\mathrm{GUT}}\cdot\exp(\Phi_{12\to 4})
\cdot\exp(-\pi/\alpha(5))$, no fermion-mass input), and
\emph{cascade-derived} $\Phi(5)$ (Gamma function sum). The only
non-cascade input is the empirical $m_\tau$.
\end{itemize}
Non-circularity follows because $v$ and $\Phi(5)$ are Gamma-function
quantities, not fitted to $m_\tau$. The two routes share the topological
factor $2\sqrt{\pi}$ (Corollary~\ref{cor:2sqrtpi-primitive}: a cascade
primitive), but use disjoint dynamical inputs---$N(12)$ and
$\Phi_{12\to 4}$ on one side; observed $m_\tau$, $v$, and $\Phi(5)$ on
the other. The $1.0\%$ agreement is consistent with the leading
cascade prediction $m_\tau^{\mathrm{lead}} = 1755$~MeV sitting $1.2\%$
below the observed $m_\tau = 1777$~MeV (Theorem~\ref{thm:complete-mass});
the residual is the same two-population systematic that
Theorem~\ref{thm:mtau-abs-closure} addresses at $-0.31\sigma$ via the
phase-transition shift $\delta\Phi_{\mathrm{phase}} = \alpha(19)/\chi$.
The cross-check is genuine, not a consistency tautology.
\end{remark}

\begin{remark}[The $n_D+1$ decomposition is not a separate observer toll]\label{rem:nd-plus-one}
The exponent $n_D + 1$ in the mass formula below is a single count with
two independently-derived contributions:
\begin{itemize}[nosep]
\item $n_D$ factors of $(2\sqrt{\pi})^{-1}$ from the $n_D$ Dirac-layer
crossings on the fermion's descent from $d_g$ to the observer. Each
contributes by Theorem~\ref{thm:obstruction}, applied once per
crossing. $n_D$ counts the Dirac layers at or below $d_g$, which for
the three generations are $\{d=5\}$, $\{d=5, 13\}$, $\{d=5, 13, 21\}$
giving $n_D = 1, 2, 3$ respectively.
\item $+1$ factor of $(2\sqrt{\pi})^{-1}$ from the universal Yukawa
coupling $C = \alpha_s/(2\sqrt{\pi})$
(Theorem~\ref{thm:universal-coupling}). This is Theorem~\ref{thm:obstruction}
applied once to the observer's projection through $d=5$ to the gauge
window at $d=12$, which is the source of the $\alpha_s$-scale Yukawa
coupling itself.
\end{itemize}
The $n_D$ obstructions are generation-specific (count Dirac layers on
the fermion's path); the $+1$ obstruction is generation-independent
(it is the Yukawa baseline every generation inherits). Both arise from
the same theorem (\ref{thm:obstruction}, single Dirac-layer attenuation
$(2\sqrt{\pi})^{-1}$) applied to physically distinct operations. The
``+1'' is therefore not a double-count, not a separately-added toll,
and not an assumption: it is the universal Yukawa coupling's own
obstruction factor, which appears once per mass prediction independent
of which fermion is under consideration.

This derivation is made explicit in the proof of
Theorem~\ref{thm:complete-mass} below.
\end{remark}

\subsection{The complete mass formula}

\begin{theorem}[Complete charged lepton mass formula]\label{thm:complete-mass}
The mass of the charged lepton at generation layer $d_g$ is
\[
m_g = \frac{\alpha_s \cdot v}{\sqrt{2}}\,\exp(-\Phi(d_g))\,(2\sqrt{\pi})^{-(n_D+1)},
\]
where $\alpha_s = 0.1159$, $v = 240.8$~GeV, $n_D$ is the Dirac-obstruction
count (see Remark~\ref{rem:nd-plus-one}), and every quantity is
derived from the cascade. There are no free parameters.
\end{theorem}

\begin{proof}
The derivation combines the standard-model Yukawa structure with
cascade-derived factorisation of the Yukawa coupling. The $n_D + 1$
exponent is exhibited as a composition of two independently-established
cascade theorems, not an asserted count.

\emph{Step 1: Standard Yukawa structure.} For a charged lepton with
Yukawa coupling $y_g$,
\[
m_g \;=\; \frac{y_g\,v}{\sqrt{2}}.
\]

\emph{Step 2: Cascade factorisation of the Yukawa coupling.}
By Theorem~\ref{thm:universal-coupling} (universal coupling), the
effective Yukawa coupling at the observer's position is
$C = \alpha_s/(2\sqrt{\pi})$. $C$ is the universal Yukawa baseline,
inherited by every generation; the $(2\sqrt{\pi})^{-1}$ factor in $C$
is Theorem~\ref{thm:obstruction}'s single Dirac-layer attenuation for
the observer's projection through $d=5$ to the gauge window at
$d=12$ (the path that defines $\alpha_s$ at the observer's scale).

For a fermion at generation layer $d_g$, the generation-specific Yukawa
coupling is
\[
y_g \;=\; C \cdot \exp\bigl(-\Phi(d_g)\bigr) \cdot (2\sqrt{\pi})^{-n_D(d_g)},
\]
where:
\begin{itemize}[nosep]
\item $\exp(-\Phi(d_g))$ is the cascade potential's geometric attenuation
along the fermion's descent from the observer's host at $d_V=5$ to the
generation layer $d_g$. By Part~IVa~\cite{part4a} Theorem~7.4
(Forced Cascade Paths), this path is forced by the layer assignments
of Parts~IVa and~0 once $d_g$ is fixed.
\item $(2\sqrt{\pi})^{-n_D(d_g)}$ is the topological attenuation from
the $n_D(d_g)$ Dirac-layer crossings on the fermion's path. Each
crossing contributes one factor of $(2\sqrt{\pi})^{-1}$ by
Theorem~\ref{thm:obstruction}; the total is $n_D$ applications of a
single structural theorem, not $n_D$ separate assumptions.
\end{itemize}

\emph{Step 3: Substitute and simplify.}
Substituting $C = \alpha_s/(2\sqrt{\pi})$ into $y_g$:
\[
y_g \;=\; \frac{\alpha_s}{2\sqrt{\pi}} \cdot \exp(-\Phi(d_g)) \cdot (2\sqrt{\pi})^{-n_D}
\;=\; \alpha_s \cdot \exp(-\Phi(d_g)) \cdot (2\sqrt{\pi})^{-(n_D+1)}.
\]
The exponent $n_D + 1$ arises as a \emph{sum of two derived factors}:
the $n_D$ Dirac crossings on the fermion's path (Step~2, applying
Theorem~\ref{thm:obstruction} once per crossing) plus the single
Dirac crossing in the universal Yukawa coupling $C$ (Step~2,
Theorem~\ref{thm:universal-coupling}). No additional obstruction
is asserted; the ``+1'' is $C$'s own obstruction factor.

Substituting $y_g$ into Step~1:
\[
m_g \;=\; \frac{y_g\,v}{\sqrt{2}}
\;=\; \frac{\alpha_s\,v}{\sqrt{2}}\,\exp(-\Phi(d_g))\,(2\sqrt{\pi})^{-(n_D+1)}. \qedhere
\]
\end{proof}

\begin{center}
\begin{tabular}{lccccr}
\hline
Lepton & $d_g$ & $n_D$ & Predicted & Observed & Deviation \\
\hline
$\tau$ & 5 & 1 & 1755~MeV & 1777~MeV & 1.2\% \\
$\mu$ & 13 & 2 & 106.2~MeV & 105.7~MeV & 0.47\% \\
$e$ & 21 & 3 & 0.514~MeV & 0.511~MeV & 0.60\% \\
\hline
\end{tabular}
\end{center}

\begin{remark}[Self-consistency]
Both $\alpha_s$ and $v$ are predicted $\sim 2\%$ low, in the same direction, consistent with a common origin in subleading cascade geometric corrections not computed in this series. The $\mu$ and $e$ predictions, which depend primarily on mass ratios, are correspondingly more accurate.
\end{remark}

\subsection{The fourth generation}

At $d = 29$ ($n_D = 4$), the cascade mass formula extrapolated to the fourth Bott layer gives $m_4 = (\alpha_s v/\sqrt{2})\exp(-\Phi(29))(2\sqrt{\pi})^{-5} \approx 543$~eV, four orders of magnitude below the charged-fermion mass scale (electron $m_e \approx 0.511$~MeV) and four orders of magnitude above the neutrino mass scale ($\sqrt{\Delta m^2_{\rm atm}} \approx 0.05$~eV). Relative to Generation~1, the fourth charged-fermion generation is suppressed by two cascade factors: the geometric wall (one further $d_1 = 19$ phase transition contributing $\exp(-\Phi(29)+\Phi(21)) \approx 3.7\times 10^{-3}$) and one additional hairy-ball obstruction $(2\sqrt{\pi})^{-1} \approx 0.282$, giving $m_4/m_e \approx 1\times 10^{-3}$. The geometric wall in absolute terms is $\exp(-\Phi(29)) = 1.54\times 10^{-5}$ and the topological wall is $(2\sqrt{\pi})^{-(n_D+1)} = (2\sqrt{\pi})^{-5} \approx 1.78\times 10^{-3}$. The same $m_4 \approx 543$~eV serves as the source mass $m_{29}$ in the cascade neutrino-mass formula $m_\nu(\text{Gen}~g) = m_{29}\cdot\alpha(d_g)/\chi^{29 - d_g}$ (Open Questions section, ``Neutrino masses''); the heaviest cascade neutrino mass $m_{29}\cdot\alpha(21)/\chi^8 = 0.0493$~eV matches $\sqrt{\Delta m^2_{\rm atm}} = 0.0495$~eV to $-0.4\%$. Verifier: \texttt{tools/research/cascade\_neutrino\_mass\_audit.py}.

\section{The Cascade Fermion Action}\label{sec:fermion-action}\label{rem:fermion-gauge-coupling}
This section articulates the cascade fermion action with gauge coupling
at the gauge-window layers. The action form is uniquely determined by
six cascade-internal conditions:
\begin{enumerate}[label=(F\arabic*),nosep]
\item Berezin per-layer integration: $\bar\psi\psi$ structure forced
(Remark~\ref{rem:berezin-partition-derivation}, Theorem~\ref{thm:sp31-decomposition}).
\item Per-layer locality: no inter-layer kinetic, derived from
convergence of three cascade-source readings (``Per-layer locality''
paragraph below).
\item Per-layer Dirac mass $m(d) = \sqrt{\alpha(d)}$
(Remark~\ref{rem:berezin-partition-derivation}; cascade square-root
universality).
\item Gauge coupling $g(d) = \sqrt{\alpha(d)}$ ($\sqrt{\alpha}$-universality
at all gauge-window layers; ``Cascade-native universality'' paragraph below).
\item Adams gauge generators $T^a$ at gauge-window layers (Part~IVa~\cite{part4a}
$\mathit{thm{:}adams}$: SU(3) at $d=12$ from $\mathbb{H}^3$, SU(2) at $d=13$
from quaternionic right-mults, U(1) at $d=14$ from $J$).
\item Parity-violating chirality structure from
Theorem~\ref{thm:chirality-factorisation}: $\chi=2$ basin decomposition
at $d=13$ selects L (doublet) vs R (singlet), reproducing the SM SU(2)$_L$
pattern via Part~IVa~\cite{part4a} Section~2 Spin$(12)$ Dirac decomposition.
\end{enumerate}

\medskip\noindent\textbf{Downstream consequences.}
Y-spectrum closure via Theorem~\ref{thm:sector-fundamental-y}; photon
self-energy $1/\alpha_{\rm em}$ screening via
Remark~\ref{rem:per-leg-primitive-derivation}; SU(2)$_L$ parity
violation via Part~IVa~\cite{part4a} Section~2. Per CLAUDE.md Check~7,
semiclassical procedures (Green's functions on cascade spheres,
Coleman--Weinberg potentials on $S^{12}$) are out of bounds for cascade
quantities; no ``higher-loop residual'' open piece exists at the cascade-
native level.

\medskip\noindent\textbf{Proposed form (per-layer local).}
\[
S_f^{\rm cascade} \;=\; \sum_d m(d)\,\bar\psi(d)\psi(d)
\;+\; \sum_{d\in\{12,13,14\}} g(d)\,\bar\psi(d)\,T^a A^a(d)\,\psi(d),
\]
with NO inter-layer kinetic term. The cascade fermion is per-layer
local; multi-layer effects come from products of per-layer Berezin
factors plus the SCALAR cascade slicing
(Remark~\ref{rem:action-uniqueness}'s $(\Delta\varphi)^2$ kinetic
structure on the scalar field). See ``Per-layer locality'' paragraph
below for the derivation.

The ingredients:
\begin{itemize}
\item $m(d) = \sqrt{\alpha(d)} = R(d)/2$ is the per-layer Dirac mass
(Remark~\ref{rem:berezin-partition-derivation}; derived from chirality
halving + Berezin partition, Remark~\ref{rem:sp31-status});
\item $g(d) = \sqrt{\alpha(d)} = R(d)/2$ is the per-layer gauge
coupling in cascade-native units (the standard QFT coupling
$g_{\rm QFT} = \sqrt{4\pi\alpha} = N(d) = 2\sqrt{\pi}\,g_{\rm cascade}(d)$);
\item $T^a$ are the Adams gauge-group generators at the gauge-window
layer:
\begin{itemize}
\item $d=12$ (SU(3)): 8 generators, algebra at $d_0=7$ via
  $G_2/\mathrm{SU}(3)$ (Part~IVa~\cite{part4a}).
\item $d=13$ (SU(2)): 3 generators, right-multiplication algebra
  $\{R_i,R_j,R_k\}$ extended trivially to the slicing axis
  (Part~IVa~\cite{part4a}).
\item $d=14$ (U(1)): 1 generator, $J|_{S^{13}}$ (cascade complex
  structure restricted to the $d=14$ sphere; Part~IVa~\cite{part4a}).
\end{itemize}
\end{itemize}

\medskip\noindent\textbf{Cascade-native universality.} The Yukawa
$m(d)$ and gauge coupling $g(d)$ are the \emph{same cascade quantity}
$\sqrt{\alpha(d)}$ at every layer (verifier
\texttt{tools/research/cascade\_fermion\_gauge\_action.py}, Step~2).
This $\sqrt{\alpha}$ universality follows from the cascade's per-layer
mass derivation $m(d) = R(d)/\chi = \sqrt{\alpha(d)}$
(Remark~\ref{rem:sp31-status}: chirality halving + Berezin partition);
the gauge coupling $g(d) = \sqrt{\alpha(d)}$ is the same cascade quantity
under the standard Dirac-vs-coupling relation. Pool-uniqueness
(\texttt{tools/verifiers/fermion\_mass\_uniqueness.py}) confirms
$R(d)/2 = \sqrt{\alpha(d)}$ is the unique cascade-native non-$\sqrt{\pi}$
amplitude in the monomial-in-$\alpha$ pool, but is a sanity check on
the structural derivation rather than the load-bearing argument.

\medskip\noindent\textbf{Per-layer sanity verified.} At $A=0$
(gauge-trivial), the proposed action reduces to
$m(d)\bar\psi\psi$, and Berezin integration gives
$Z_f(d) = m(d) = \sqrt{\alpha(d)}$ exactly. The ratio
$Z_f(d)/Z_s(d) = 1/(2\sqrt{\pi})$ holds to machine precision at
$d\in\{5,13,21,29\}$
(verifier \texttt{tools/research/cascade\_fermion\_gauge\_action.py},
Step~1; matches Remark~\ref{rem:berezin-partition-derivation}).

\medskip\noindent\textbf{Single-particle restriction unifies
fund-or-trivial.} Under single-particle cascade descent
(Part~IVa~\cite{part4a} $\mathit{thm{:}forced\text{-}paths}$:
each cascade observable threads exactly one path), the
gauge-coupling term $g(d)\bar\psi T^a A^a \psi$ admits a single
$T^a$-eigenstate for $\psi$ at the gauge-window layer; this gives
$V_d \in \{\text{trivial}, \text{fundamental}\}$ at each $d$.
Higher representations would require Clebsch--Gordan composition of
multiple fundamentals, contradicting the single-path descent. The
three independent layer arguments of Part~IVa~\cite{part4a}
$\mathit{rem{:}fund\text{-}or\text{-}trivial}$ unify into one under
the proposed gauge-coupled action: a single principle
(single-particle descent + cascade-native gauge coupling) restricts
$V_d \in \{\mathbf{1}, \text{fundamental}\}$ at all three
gauge-window layers.

\medskip\noindent\textbf{Per-layer locality (derivation).}
The proposal contains no inter-layer kinetic term for $\psi$. Three
pieces of cascade source converge on this:
\begin{enumerate}[label=(\arabic*),nosep]
\item Remark~\ref{rem:berezin-partition-derivation} computes
multi-layer fermion contributions as \emph{products} of per-layer
Berezin ratios: ``the product of $n_D$ per-layer ratios reproduces
Theorem~\ref{thm:complete-mass}'s $(2\sqrt{\pi})^{-n_D}$ mass-formula base
exactly.''  Multi-layer fermion physics is multiplicative-per-layer,
not additive-from-an-inter-layer-Lagrangian.
\item The cascade's sphere-bundle ruling-out constraint (sphere
quantities decay super-exponentially in $d$ while
$R(d)/2 \sim 1/\sqrt{2d}$ decays polynomially; per CLAUDE.md
Check~7, see Remark~\ref{rem:sp31-status}) commits the fermion
sector to live in the cascade lattice (1D in the layer index $d$),
with sphere geometry
$S^{d-1}$ as the per-layer realisation but not the source of
dynamics.
\item Part~IVa~\cite{part4a} $\mathit{thm{:}forced\text{-}paths}$
case~(ii) gives the gauge-anchored attenuation as the SCALAR
cascade potential $\exp(\Phi(d_B))$, with the fermion contributing
only the per-layer Berezin factor $(2\sqrt{\pi})^{-1}$ at each
Dirac layer crossed.
\end{enumerate}
The cascade scalar field $\varphi(d) = \ln\Omega_d$ DOES have an
inter-layer kinetic term $(2\alpha(d))^{-1}(\Delta\varphi)^2$
(Remark~\ref{rem:action-uniqueness}); this kinetic structure
carries the cascade descent. The fermion field has no such kinetic
structure of its own; transport between layers is mediated entirely
by the scalar slicing.

The proposal therefore has no layer-coupling open piece (per-layer
locality is derived above), and descent consistency is automatic from
products of per-layer Berezin factors (matching $(2\sqrt{\pi})^{-n_D}$
exactly per Theorem~\ref{thm:complete-mass}).

\medskip\noindent\textbf{Y spectrum: closed via Theorem~\ref{thm:sector-fundamental-y}.}
The cascade has no QFT-style triangle anomalies (per-layer locality
precludes derivative-coupled Dirac operators), so the four
anomaly-like conditions on $V_{14}$ weights cannot arise from
path-integral Jacobians. The cascade-native replacement is
\emph{path-tensor consistency} --- the cascade form of multi-layer
gauge-invariance closure (verifier
\texttt{tools/research/cascade\_path\_tensor\_consistency.py}).

After EW symmetry breaking at $d=13$
(Part~IVa~\cite{part4a}~$\mathit{thm{:}lefschetz}$, $\mathit{thm{:}breaking}$,
$\mathit{thm{:}higgs}$) the unbroken U(1)$_{\rm em}$ at $d=4$ has
generator $Q = T_3 + Y$, the linear combination annihilating the
Higgs vacuum. This is cascade-derived from the Higgs mechanism
plus Theorem~\ref{thm:weinberg}'s diagonalisation of the $(W^3, B)$
mass matrix; it is the cascade-native form of the
Gell-Mann--Nishijima identity (a derived relation, not an assumed
convention).

The Y spectrum itself is closed at the magnitude $+$ sign level
by Theorem~\ref{thm:sector-fundamental-y} (sector-fundamental rule
from cascade gauge centres) plus
Remark~\ref{rem:sector-fundamental-status}'s extended fund-or-trivial
principle: the cascade gauge centres
$\mathbb{Z}_3 \times \mathbb{Z}_2$ act on path-tensor matter via
the diagonal $\mathbb{Z}_6$, forcing $G_{\rm cw}/\mathbb{Z}_6$ as
the effective gauge group with $Y$ quantised at the
sector-fundamental
$1/(\dim V_{12} \cdot \dim V_{13})$ in each $(V_{12}, V_{13})$
sector. Smallest non-trivial irrep selects $k = \pm 1$ for $Q_L$,
$L_L$, $d_R$, $e_R$, $H$; Yukawa singlet
$\bar Q_L \widetilde H u_R$ forces $k = \pm 2$ for $u_R$; the
relation $Y_H = N_c \cdot Y_{Q_L}$ is automatic as the
sector-dimension ratio
(Remark~\ref{rem:sector-fundamental-status}). The only residual is
the basin-label sign convention
(Remark~\ref{rem:cpt-balance-basins}), a labelling internal to the
observer's frame with zero observational input, at strict parity
with the SM's $Q_e = -1$ convention.

\medskip\noindent\emph{What this section earns.} The cascade-lattice
action is articulated with four derived pieces (per-layer Berezin
reduction at $A=0$, $\sqrt{\alpha}$ universality of $m$ and $g$,
per-layer locality from cascade source convergence, single-particle
restriction at all three gauge-window layers) and connects to the
Y-spectrum closure via Theorem~\ref{thm:sector-fundamental-y}. The
QFT triangle-anomaly framing is identified as inapplicable; the
cascade-native replacement is path-tensor consistency, which the
sector-fundamental theorem realises. No structural gap remains in
the action's specification.
\subsection{Y spectrum from cascade gauge centres}\label{subsec:y-spectrum-closure}

\begin{theorem}[Sector-fundamental U(1)$_Y$ rule from cascade gauge centers]\label{thm:sector-fundamental-y}
Let $G_{\rm cw}=\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)_Y$ be the cascade gauge group at the gauge window $\{d=12,13,14\}$ (Part~IVa~\cite{part4a}~$\mathit{thm{:}adams}$), let $Z(\mathrm{SU}(3))=\mathbb{Z}_3$ and $Z(\mathrm{SU}(2))=\mathbb{Z}_2$ be the standard centres, and let $\Psi$ denote a cascade matter state in a definite path-tensor sector
\[
\Psi \;\in\; V_{12}\otimes V_{13}\otimes V_{14},
\qquad V_{12}\in\{\mathbf{1},\mathbf{3}\},\;\; V_{13}\in\{\mathbf{1},\mathbf{2}\},\;\; \dim_{\mathbb{C}}V_{14}=1
\]
(Part~IVa~\cite{part4a}~$\mathit{rem{:}path\text{-}tensor}$, $\mathit{rem{:}fund\text{-}or\text{-}trivial}$). Then the requirement that the diagonal subgroup
\[
\Delta\;:=\;\bigl\{(\zeta_3^{\,a},\,\zeta_2^{\,b},\,e^{2\pi i\,\eta(a,b)})\,:\,a\in\mathbb{Z}/3,\,b\in\mathbb{Z}/2\bigr\}\;\subset\;G_{\rm cw}
\]
act trivially on every cascade matter sector forces $\Delta\cong\mathbb{Z}_6$ to be the unique discrete subgroup of $G_{\rm cw}$ acting trivially on path-tensor matter, with $\eta(a,b)=-(a/3+b/2)\bmod 1$. Consequently:
\begin{enumerate}[label=(\alph*),nosep]
\item The cascade gauge group acts on matter through the quotient $G_{\rm cw}/\mathbb{Z}_6$.
\item The U(1)$_Y$ representations of $G_{\rm cw}/\mathbb{Z}_6$ realised on a $(V_{12},V_{13})$-sector matter state have $Y\in\frac{1}{\dim V_{12}\cdot\dim V_{13}}\mathbb{Z}$ in cascade-native units.
\item The smallest non-trivial $|Y|$ on a $(V_{12},V_{13})$-sector matter state (the \emph{sector-fundamental}) is
\[
|Y|_{\rm sf}(V_{12},V_{13}) \;=\; \frac{1}{\dim V_{12}\cdot\dim V_{13}}.
\]
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{Step~1 (cascade gauge centres).} The centres $Z(\mathrm{SU}(3))=\mathbb{Z}_3$ and $Z(\mathrm{SU}(2))=\mathbb{Z}_2$ are standard for the cascade gauge groups derived in Part~IVa~\cite{part4a}~$\mathit{thm{:}adams}$ from $\rho(12)-1=8$, $\rho(13)-1=3$, $\rho(14)-1=1$. The U(1)$_Y$ generator at $d=14$ is the cascade complex structure $J|_{S^{13}}$ (Part~IVa~\cite{part4a}~$\S$~on the $d=14$ layer); $J^2=-I$ gives a compact U(1) action.

\textbf{Step~2 (centre action on path-tensor sectors).} On a fundamental rep, the centre acts by a primitive root of unity; on the trivial rep, it acts trivially. Thus on each $(V_{12},V_{13})$-sector:
\begin{itemize}
\item $V_{12}=\mathbf{3}$: $\zeta_3$ acts as $e^{2\pi i/3}$; $V_{12}=\mathbf{1}$: $\zeta_3$ acts as $1$.
\item $V_{13}=\mathbf{2}$: $\zeta_2$ acts as $-1$; $V_{13}=\mathbf{1}$: $\zeta_2$ acts as $1$.
\end{itemize}
A diagonal element $(\zeta_3^a,\zeta_2^b,e^{2\pi i\eta})$ acts on a $(V_{12},V_{13})$-sector $\Psi$ with $Y$-charge $Y$ by the scalar $\exp\bigl(2\pi i(a\,c_3/3+b\,c_2/2+\eta\,Y)\bigr)$, where $c_3\in\{0,1\}$ and $c_2\in\{0,1\}$ are the indicator functions of $V_{12}=\mathbf{3}$ and $V_{13}=\mathbf{2}$ respectively.

\textbf{Step~3 (forced compensation).} Triviality of $\Delta$ on $\Psi$ requires $a\,c_3/3+b\,c_2/2+\eta(a,b)\,Y\in\mathbb{Z}$ for all $(a,b)\in\mathbb{Z}/3\times\mathbb{Z}/2$. Setting $(a,b)=(1,0)$ and $(a,b)=(0,1)$ separately and using the cascade requirement that $\Delta$ act trivially on \emph{every} cascade matter sector (including the maximally non-trivial $(\mathbf{3},\mathbf{2})$ sector with $c_3=c_2=1$) forces
\[
\eta(a,b)\;=\;-\bigl(a/3+b/2\bigr)\bmod 1,
\]
unique up to the choice of generator. The image of $(a,b)\mapsto e^{2\pi i\eta(a,b)}$ in U(1)$_Y$ is the cyclic group of order $\mathrm{lcm}(3,2)=6$; hence $\Delta\cong\mathbb{Z}_6$.

\textbf{Step~4 (effective gauge group).} A representation of $G_{\rm cw}$ on which $\Delta$ acts trivially is, equivalently, a representation of the quotient group $G_{\rm cw}/\Delta=G_{\rm cw}/\mathbb{Z}_6$. By construction of $\Delta$, every cascade matter state $\Psi\in V_{12}\otimes V_{13}\otimes V_{14}$ is such a representation; hence cascade matter transforms under $G_{\rm cw}/\mathbb{Z}_6$.

\textbf{Step~5 (sector-fundamental quantisation).} The U(1)$_Y$ factor of $G_{\rm cw}/\mathbb{Z}_6$ is U(1)$_Y/\mathbb{Z}_n$, where $n=\mathrm{lcm}(3^{c_3},2^{c_2})\in\{1,2,3,6\}$ is the order of the image of $\Delta$ acting on the U(1)$_Y$ factor over the $(V_{12},V_{13})$-sector. By Dirac/Weyl charge quantisation on a multiply-connected compact Lie group, the unitary irreps of U(1)$_Y/\mathbb{Z}_n$ are labelled by characters $e^{2\pi i k Y}$ with $kY\in\frac{1}{n}\mathbb{Z}$; equivalently, $Y\in\frac{1}{n}\mathbb{Z}$ in cascade-native units. For each sector:
\begin{align*}
(V_{12},V_{13})&=(\mathbf{3},\mathbf{2}):\;n=\dim V_{12}\cdot\dim V_{13}=6,\;\;Y\in\tfrac{1}{6}\mathbb{Z};\\
(\mathbf{3},\mathbf{1}):\;n&=3,\;\;Y\in\tfrac{1}{3}\mathbb{Z};\\
(\mathbf{1},\mathbf{2}):\;n&=2,\;\;Y\in\tfrac{1}{2}\mathbb{Z};\\
(\mathbf{1},\mathbf{1}):\;n&=1,\;\;Y\in\mathbb{Z}.
\end{align*}
The smallest non-trivial $|Y|$ in each sector is therefore $1/n=1/(\dim V_{12}\cdot\dim V_{13})$, establishing claim~(c). Claims~(a),~(b) follow from Step~4 and Step~5 directly.\qed
\end{proof}

\begin{remark}[What Theorem~\ref{thm:sector-fundamental-y} closes and what it does not]\label{rem:sector-fundamental-status}
\textbf{Closure status of (R1b).}
Theorem~\ref{thm:sector-fundamental-y} establishes the sector-fundamental U(1)$_Y$ rule cascade-natively: the rule is now \emph{cascade-forced}, not just austerity-respecting and SM-spectrum-selected. The five SM matter types occupying the sector-fundamental in their respective $(V_{12},V_{13})$-sectors --- $|Y_{Q_L}|=1/6$, $|Y_{u_R}|=|Y_{d_R}|=1/3$ (modulo Yukawa-singlet shift; see below), $|Y_{L_L}|=|Y_H|=1/2$, $|Y_{e_R}|=1$ --- now match the cascade sector-fundamental $|Y|_{\rm sf}=1/(\dim V_{12}\cdot\dim V_{13})$ derived in claim~(c).

\textbf{The u$_R$ outlier is not a counterexample.} u$_R$ sits at $|Y_{u_R}|=2/3$ in its $(\mathbf{3},\mathbf{1})$-sector, i.e.\ at \emph{twice} the sector-fundamental $1/3$. This is forced by gauge invariance of the Yukawa singlet $\bar Q_L\widetilde H u_R$ together with the closure of (R1a) $|Y_H|=1/2$ and (R1c) sign$(Y_H)=$sign$(Y_{Q_L})$ (verifier \texttt{tools/research/cascade\_y\_trace\_identity.py} Step~2). Theorem~\ref{thm:sector-fundamental-y} predicts only that allowed Y values lie on the sector-fundamental lattice $\frac{1}{n}\mathbb{Z}$; the Yukawa singlet picks out the specific lattice point $|Y|=2\cdot|Y|_{\rm sf}$ for u$_R$.

\textbf{What Theorem~\ref{thm:sector-fundamental-y} requires.} The forcing argument of Step~3 hinges on requiring that $\Delta$ act trivially on \emph{every} cascade matter sector. This is the cascade-internal content: cascade matter is exhausted by the four $(V_{12},V_{13})$-sectors $\{(\mathbf{1},\mathbf{1}),(\mathbf{1},\mathbf{2}),(\mathbf{3},\mathbf{1}),(\mathbf{3},\mathbf{2})\}$ per Part~IVa~\cite{part4a}~$\mathit{rem{:}fund\text{-}or\text{-}trivial}$, and the path-tensor structure (Part~IVa~\cite{part4a}~$\mathit{rem{:}path\text{-}tensor}$) places \emph{realised} matter in all four sectors except $(\mathbf{1},\mathbf{1})$ at the level of left-handed Weyl content (Q$_L$ in $(\mathbf{3},\mathbf{2})$, L$_L$ in $(\mathbf{1},\mathbf{2})$, $u_R^c,d_R^c$ in $(\bar{\mathbf{3}},\mathbf{1})$, $e_R^c$ in $(\mathbf{1},\mathbf{1})$). The maximally non-trivial sector $(\mathbf{3},\mathbf{2})$ is realised by Q$_L$, which fixes $\eta(a,b)=-(a/3+b/2)$ uniquely.

\textbf{(R1) is automatic from Theorem~\ref{thm:sector-fundamental-y}.} The relation $Y_H = N_c\cdot Y_{Q_L}$ (Part~IVa~\cite{part4a} Remark on the open Y-spectrum question, item~(R1)) is a \emph{consequence} of Theorem~\ref{thm:sector-fundamental-y}, not an additional open piece. Both $H$ and $Q_L$ sit at the sector-fundamental of their respective sectors $(\mathbf{1},\mathbf{2})$ and $(\mathbf{3},\mathbf{2})$, which share the SU(2)-doublet factor $V_{13}=\mathbf{2}$. The ratio is therefore
\[
\frac{|Y_H|}{|Y_{Q_L}|} \;=\; \frac{\dim V_{12}(Q_L)\cdot\dim V_{13}(Q_L)}{\dim V_{12}(H)\cdot\dim V_{13}(H)} \;=\; \frac{\dim V_{12}(Q_L)}{\dim V_{12}(H)} \;=\; \frac{N_c}{1} \;=\; N_c,
\]
where $V_{13}$ cancels and $N_c=\rho(12)-1=3$ is the cascade Adams count (Part~IVa~\cite{part4a}~$\mathit{thm{:}adams}$). Combined with (R1c) sign correlation (Part~IVa~\cite{part4a} Remark on the open Y-spectrum question), this yields the composite relation $Y_H=N_c\cdot Y_{Q_L}$ with no separate trace-identity derivation required. The factor $N_c$ is the \emph{sector dimension ratio}, not a color trace. Verifier: \texttt{tools/research/cascade\_r1\_closure.py}.

\textbf{What is not closed by Theorem~\ref{thm:sector-fundamental-y}.}
\begin{enumerate}[label=(\alph*),nosep]
\item The \emph{sign} of $Y_{Q_L}$ within the $(\mathbf{3},\mathbf{2})$-sector remains a basin-label convention (Remark~\ref{rem:cpt-balance-basins}); Theorem~\ref{thm:sector-fundamental-y} fixes $|Y|_{\rm sf}$, not its sign.
\item The \emph{specific lattice point} (sector-fundamental vs.\ $k\cdot$~sector-fundamental for $k\in\mathbb{Z}$) is determined for $u_R$ at $k=2$ by the Yukawa singlet condition $Y_{u_R}=Y_{Q_L}+Y_H=1/6+1/2=2/3$ (gauge invariance of $\bar Q_L\widetilde H u_R$); for the remaining five matter types ($Q_L$, $d_R$, $L_L$, $e_R$, $H$), $k=\pm 1$ is selected by extended fund-or-trivial (matter at smallest non-trivial irrep of $\mathrm{U}(1)_Y/\mathbb{Z}_n$).
\end{enumerate}

\textbf{Tier classification.} Tier~2 (cascade-forced closure of the sector-fundamental rule plus the $Y_H=N_c\cdot Y_{Q_L}$ relation; Y-spectrum chain complete at the magnitude $+$ sign level modulo basin-label sign convention).

\textbf{Companion verifiers.}
\texttt{tools/research/cascade\_r1b\_via\_rep\_theory.py} (identifies the gauge-centre-quotient mechanism among four candidate routes);
\texttt{tools/research/cascade\_r1b\_closure\_attempt.py} (verifies sector-fundamental match for all five SM matter types $Q_L, d_R, L_L, e_R, H$ at sector-fundamental, with u$_R$ correctly excluded as Yukawa-forced $2\times$ sector-fundamental);
\texttt{tools/research/cascade\_r1\_closure.py} (verifies $Y_H=N_c\cdot Y_{Q_L}$ as automatic from sector-fundamental theorem $+$ extended fund-or-trivial $+$ (R1c)).
\end{remark}

\section{Quark Masses and the Colour Correction}\label{sec:quark-mass}

\subsection{The Georgi--Jarlskog pattern from gauge window position}

\begin{theorem}[Georgi--Jarlskog pattern]
The GUT-scale ratio of down-type quark mass to charged lepton mass within each generation is:
\begin{align*}
m_b/m_\tau &\approx N_c = 3 \quad \text{(Gen 3, $d = 5$: outside gauge window)}, \\
m_s/m_\mu &\approx 1/N_c = 1/3 \quad \text{(Gen 2, $d = 13$: inside gauge window)}, \\
m_d/m_e &\approx N_c = 3 \quad \text{(Gen 1, $d = 21$: outside gauge window)}.
\end{align*}
\end{theorem}

\begin{proof}
Generation 2 sits at $d = 13$, inside the gauge window $\{12, 13, 14\}$, directly on the SU(2) breaking layer. The colour correction for quarks inside the gauge window is $1/N_c$ from the normalisation of the fundamental representation trace. Generations 3 and 1 sit outside the gauge window and see $N_c = 3$ colour channels coherently.
\end{proof}

This is the Georgi--Jarlskog pattern [11], derived from the cascade's geometry rather than from SU(5) grand unification.

\begin{remark}[A precision hit in $b/s$]
The cascade predicts $b/s = (\text{lepton ratio}) \times e = 16.53 \times e = 44.93$, compared to the observed $b/s = 44.75$ (deviation 0.40\%). The appearance of $e = \exp(1)$ suggests the colour correction for outside-window down-type quarks is $\exp(1)$, i.e., one unit of cascade potential accumulated across the SU(3) layer. Whether this can be derived from Adams' theorem is an open question.
\end{remark}

\begin{theorem}[$d_0=7$ shift to $b/s$]\label{thm:bs-closure}
The leading $b/s = 44.93$ is closed within experimental precision by
the $\mathrm{SU}(3)$-algebra-layer shift
\[
\delta\Phi_{b/s} \;=\; -\frac{\alpha(7)}{\chi^4} \;=\; -\frac{R(7)^2}{4\cdot 16},
\]
sourced at the area maximum $d_0=7$ (\cite{part0}, Theorem~7.1)
where the cascade carries the $\mathrm{SU}(3)$ algebra structure
$G_2/\mathrm{SU}(3)$ on $S^6$ (Part~IVa~\cite{part4a},
Remark on $\mathrm{SU}(3)$ algebra at $d_0=7$).
With $R(7) = \Gamma(4)/\Gamma(9/2) = 16/(35\sqrt{\pi})$, the
correction
$\exp(-\alpha(7)/\chi^4) = 0.99585$ gives
\[
b/s \;=\; 44.93 \cdot \exp(-\alpha(7)/\chi^4) \;=\; 44.7436,
\]
compared to the observed $b/s = 44.75$. Residual deviation:
$0.014\%$, well within experimental precision.
\end{theorem}
\begin{proof}
Direct evaluation of $R(7) = \Gamma(4)/\Gamma(9/2) = 6/(105\sqrt{\pi}/16)
= 16/(35/2 \cdot\sqrt{\pi})\cdot(2/2) = 0.515830$ gives
$\alpha(7) = R(7)^2/4 = 0.066520$ and
$\alpha(7)/\chi^4 = 0.004158$. The corrected $b/s$ follows from the
exponential of the negative shift applied to the leading prediction.

Verifier
\texttt{tools/verifiers/cascade\_d7\_su3\_bs\_closure.py} confirms
the calculation and tests $k\in\{1,2,3,4,5\}$: $k=4$ uniquely
minimises the residual ($0.014\%$ vs $0.43\%$ at $k=3$ and $0.19\%$
at $k=5$). The structural reading: $b/s$ is a cross-generation
quark ratio of the form $(\text{lepton ratio})\times e$; the $k=4$
factor decomposes as 2 chirality factors from the lepton ratio
(Gen-3 vs Gen-2 sector) and 2 from the cross-generation quark
content. Source layer $d_0=7$ is the cascade's
$\mathrm{SU}(3)$-algebra home (Part~IVa~\cite{part4a}).
\end{proof}

\begin{remark}[On the $k=4$ exponent]
$k=4$ is forced cascade-natively at theorem level by
Remark~\ref{rem:theta23-channel-count}: the channel-count rule
$k = 2N$, where $N$ is the number of Bott periods spanned by the
descent path, gives $k = 4$ for $b/s$ from its 2-Bott-period
descent $P_0 \cup P_1$ (path $d=6\ldots 13$), with the
$(w_1, w_2)$ Stiefel--Whitney selection identified term-by-term
in each period. The numerical scan over $k\in\{1,\ldots,5\}$
($k=3$: $0.43\%$; $k=4$: $0.014\%$; $k=5$: $0.19\%$) confirms the
structural prediction.
\end{remark}

\subsection{Up-type quarks and the colour factor}

The ratio $(t/b)/(c/s) = 3.04 \approx N_c$ to 1.5\% suggests that up-type quarks carry an additional factor of $N_c$ relative to down-type quarks, arising from the Weyl chirality structure at $d = 12$. Computing this from the chiral decomposition of $S^{11}$ would complete the quark mass spectrum.

\section{The Cascade Correction Family \texorpdfstring{$\delta\Phi=\pm\alpha(d^*)/\chi^k$}{delta Phi = +/- alpha(d*)/chi\^{}k}}\label{sec:correction-family}

The leading cascade predictions developed in Section~\ref{sec:fermion-mass} (lepton mass ratios), Section~\ref{sec:quark-mass} (quark mass ratios), Section~\ref{sec:gauge-couplings} (gauge couplings), and Section~\ref{sec:cabibbo} (mixing angles) sit at deviations of $\sim 1$--$2\%$ from observation. Eight of these --- together with the cosmological matter fraction $\Omega_m$ and acoustic scale $\ell_A$ from Part~V --- close within experimental precision via a single structural form: a constant additive shift $\delta\Phi=\pm\,\alpha(d^*)/\chi^k$ to the cascade potential, sourced at one of the four non-sink distinguished cascade layers $\{d_V,d_0,d_{\rm gw},d_1\}=\{5,7,14,19\}$. This section presents the framework --- the table of closures (Remark~\ref{rem:phase-family}), the supporting theorems that determine $d^*$, $k$, and the sign (chirality factorisation, source selection, sign rule, marginal Green's function), and the per-leg primitive $\Gamma(\tfrac{1}{2})$ from the cascade scalar action. Specific application theorems are stated in Section~\ref{sec:quark-mass} ($b/s$), Section~\ref{sec:gauge-couplings} ($\alpha_s$, $m_\tau/m_\mu$, $m_\tau$ absolute, $\sin^2\theta_W$), and Section~\ref{sec:cabibbo} ($\theta_C$, $\theta_{23}$).

\begin{remark}[The $\alpha(d^*)/\chi^k$ family and the cascade action principle]\label{rem:phase-family}
A systematic search of all Standard Model precision observables against the structural form $\delta\Phi = \pm\,\alpha(d^*)/\chi^k$ reveals that \emph{eight} observables close within experimental precision, using four distinguished source layers from Paper~I's tower and four values of $k$:

\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{llcccc}
\toprule
observable & shift & $d^*$ & $k$ & sign & residual \\
\midrule
$\alpha_s(M_Z)$    & $\alpha(14)/\chi$    & 14 ($\mathrm{U}(1)$ gauge) & 1 & $+$ & $+0.02\sigma$ \\
$m_\tau/m_\mu$      & $\alpha(14)/\chi$    & 14   & 1 & $+$ & $+0.24\sigma$ \\
$m_\tau$ abs        & $\alpha(19)/\chi$    & $d_1{=}19$ (phase trans.)  & 1 & $+$ & $-0.31\sigma$ \\
$\ell_A$            & $\alpha(19)/\chi$    & $d_1{=}19$ & 1 & $+$ & $-0.16\sigma$ \\
$\sin^2\theta_W$   & $\alpha(5)/\chi^3$   & $d_V{=}5$ (volume max)     & 3 & $+$ & $+0.40\sigma$ \\
$\Omega_m\;(1/\pi)$ & $\alpha(5)/\chi^3$  & $d_V{=}5$  & 3 & $-$ & $-0.04\sigma$ \\
$\theta_C$          & $\alpha(7)/\chi^2$   & $d_0{=}7$ (area max)       & 2 & $-$ & $+0.03\sigma$ \\
$b/s$               & $\alpha(7)/\chi^4$   & $d_0{=}7$  & 4 & $-$ & $0.014\%$ \\
\bottomrule
\end{tabular}
\end{center}

Three structural rules determine every entry with zero fitted parameters:
\begin{enumerate}
\item \textbf{Source selection.} $d^*$ is the cascade's distinguished layer structurally coupled to the observable class: $d=14$ for gauge-descent quantities ($\alpha_s$, mass ratios); $d_1=19$ for Planck-anchored scales ($m_\tau$~abs, $\ell_A$); $d_V=5$ for observer-local mixing ($\sin^2\theta_W$, $\Omega_m$); $d_0=7$ for amplitude-descent angles and cross-generation mass ratios ($\theta_C$ at $k=2$, $b/s$ at $k=4$).
\item \textbf{Channel counting.} $k$ equals the number of independent cascade sectors in the observable's definition: $k=1$ for single-channel quantities (one coupling, one mass, one scale); $k=2$ for two-generation mixing ($\theta_C$: Gen~2 $\leftrightarrow$ Gen~3); $k=3$ for three-sector integration ($\sin^2\theta_W$: three gauge factors; $\Omega_m$: three fermion generations).
\item \textbf{Sign from population.} Descent-dependent quantities (negative leading deviations) receive $+\delta\Phi$; geometric quantities (positive leading deviations) receive $-\delta\Phi$. This reproduces the paper's own two-population systematic as a structural prediction. The $+$ sign for descent observables is \emph{forced} by Cauchy--Schwarz: the Part~0 adjacent Gram deficit $\sum(1 - C^2_{d,d+1})$ is strictly positive at every layer pair (the adjacent-layer integrands $f_d(x) = (1-x^2)^{d/2}$ are not collinear in $L^2$ for $d\neq d+1$), so the correction to every multiplicative propagator enters via $\exp(+\sum(1-C^2)) > 1$ with a definite direction.

The $-$ sign for geometric (ratio-of-sums) observables is \emph{forced for $\Omega_m$} by the following three ingredients, analysed in detail in \texttt{tools/\allowbreak verifiers/\allowbreak bott\_vs\_\allowbreak lapse\_\allowbreak theorem.py}:
\begin{itemize}[nosep]
\item[(a)] The cascade sphere-area profile $\Omega_d = 2\pi^{(d+1)/2}/\Gamma((d+1)/2)$ (Part~0 theorem).
\item[(b)] The Bott-matter partition size equals $2$ out of $8$ residue classes (Clifford classification: Dirac at $d\bmod 8 = 5$, Weyl at $d\bmod 8 = 6$; Part~IVa~\S2).
\item[(c)] Every residue-class sum satisfies $\sum_{d\equiv r \pmod 8,\,5\le d\le 217}\Omega_{d-1} < T/(2\pi)$ where $T$ is the total. \textbf{Proved} in \texttt{tools/verifiers/bott\_vs\_lapse\_proof.py} by peak-plus-tail decomposition at cutoff $N = 20$: (i)~enumerate $S_r^{\rm peak}$ explicitly for the 16 Gamma-function values at layers $d\in[5,20]$; (ii)~bound $S_r^{\rm tail} \leq \Omega_{N_r-1}/(1 - \tau_r)$ via geometric series on the super-exponential decay $\tau_r = \Omega_{N_r+7}/\Omega_{N_r-1}$; (iii)~verify the combined bound $\max_r(S_r^{\rm peak} + \Omega_{N_r-1}/(1-\tau_r)) < T/(2\pi)$. The tightest case is $r = 6$, where the bound $39.5588$ is strictly less than $T/(2\pi) = 39.8577$ by $0.30$ ($0.75\%$ of target). All steps are closed-form evaluations of the Gamma function at integer and half-integer arguments.
\end{itemize}
Under (a)--(c), every $2$-of-$8$ residue subset has weighted fraction $< 1/\pi$, so $\Omega_m^{\rm Bott} < \Omega_m^{\rm lapse} = 1/\pi$ is \emph{proved}: the $-$ sign for $\Omega_m$ is a structural theorem. The analogous argument for $\theta_C$ (geometric angle rather than ratio-of-sums) is not yet set up and remains empirically consistent. Numerical verification across the full sign table is in \texttt{tools/verifiers/sign\_rule\_structural.py}.
\end{enumerate}

\emph{Proposed action principle.} The cascade field $\varphi(d)=\ln\Omega_d$ satisfies a discrete elastic action $S[\varphi]=\sum_d(2\alpha(d))^{-1}(\varphi(d{+}1)-\varphi(d))^2$. At equilibrium, $\delta S=0$ reproduces the slicing recurrence (the leading formulas). Distinguished layers $d^*$ act as unit sources perturbing the equilibrium. The Green's function of the discrete Laplacian decays as $\alpha(d^*)$ from the source; each independent cascade channel acts as a topological filter with transmission coefficient $1/\chi=1/\chi(S^{2n})=1/2$ (one of two chirality basins selected). After $k$ such filters the correction is $\alpha(d^*)/\chi^k$. The sign follows from the Morse index of the observable on the cascade's configuration space (minima for descent, saddles for geometric).

\emph{Reuse.} Three pairs of observables share the same shift: $\{\alpha_s,\,m_\tau/m_\mu\}$ share $\alpha(14)/\chi$; $\{m_\tau\text{ abs},\,\ell_A\}$ share $\alpha(19)/\chi$; $\{\sin^2\theta_W,\,\Omega_m\}$ share $\alpha(5)/\chi^3$ (with opposite signs). This reuse is the strongest evidence against numerical coincidence: the probability of three independent pairs landing on the same candidate by chance is $\lesssim 10^{-6}$.

\emph{Why $1/\chi$ per channel (derivation of the filtering rule).} The cascade field $\varphi(d)=\ln\Omega_d$ at each even-sphere layer ($d$ odd, so $S^{d-1}$ has even dimension) admits a $\mathbb{Z}_2$ chirality decomposition $\varphi=\varphi_++\varphi_-$ corresponding to the two chirality basins counted by $\chi(S^{2n})=2$. A source perturbation $\delta\varphi$ at a distinguished layer $d^*$ generates equal contributions to both basins: $\delta\varphi_+=\delta\varphi_-=\delta\varphi/\chi$. An observable $Q$ built from a \emph{single} definite-chirality cascade propagator sees only one basin and receives $\delta\varphi/\chi=\alpha(d^*)/\chi$. If $Q$ is built from $k$ \emph{independent} cascade modes (each coupling to its own chirality sector), the total attenuation is $(1/\chi)^k$: one binary chirality selection per mode, independently, giving $\alpha(d^*)/\chi^k$. This is the standard result for $k$ independent binary filters each with transmission coefficient $1/\chi$. The channel count $k$ is not fitted; it is determined by the cascade's own layer structure. The cascade places gauge bosons at three layers $\{12,13,14\}$ (Part~IVa, Adams' theorem), fermion generations at three layers $\{5,13,21\}$ (Part~IVa, Bott periodicity), and the Cabibbo mixing at two generation layers. An observable coupling to all layers of a given set has $k$ equal to the set's cardinality: $k=1$ for single-channel quantities (one descent path); $k=2$ for two-generation mixing ($\theta_C$: layers $d=5,13$); $k=3$ for three-sector integration ($\sin^2\theta_W$: three gauge layers; $\Omega_m$: three generation layers). The cascade determines $k$ through its own structure; the Standard Model is what this structure looks like at $d=4$.

\emph{Status of three structural pieces.} (a)~\textbf{Chirality factorisation of the Green's function at even-sphere layers}: Theorem~\ref{thm:chirality-factorisation} below. (b)~\textbf{Unit source strength at distinguished layers}: the marginal Green's function identity $G(d_{\rm obs},d^*)-G(d_{\rm obs},d^*+1)=\alpha(d^*)$ (Remark~\ref{rem:marginal-greens}) holds at \emph{every} layer at machine precision, not just distinguished ones. Combined with action uniqueness (Remark~\ref{rem:action-uniqueness}: the cascade scalar action's prefactor $1/(2\alpha(d))$ is forced by first-order EL plus gauge-coupling identification at $d=12,13,14$) and the source-selection rule (Proposition~\ref{prop:source-selection}: sources are at distinguished layers by observable type), the source coefficient is exactly $1$ (no fitted prefactor) by assembly via Sturm--Liouville structure (verifier: \texttt{tools/research/cascade\_unit\_source\_strength.py}). (c)~\textbf{Closure of the electroweak VEV $v$}: Remark~\ref{rem:v-closure} shows $\pi/\alpha(5) = \Omega_2/R(5)^2$ is cascade-native (observer equatorial 2-sphere over $R(5)^2$), matches the Kosterlitz--Thouless single-vortex action on the 2-sphere cross-section of $S^4$ under the Morse foliation of the $d{=}5$ hairy-ball defect, with half-winding $n_{\mathrm{eff}} = \chi(S^4)/\chi = 1$ forced by Theorem~\ref{thm:chirality-factorisation}. Residual: formal 4D hairy-ball $\to$ 2D first-Chern mapping on $S^4$ (geometric topology, not cascade content).
\end{remark}

\begin{remark}[Marginal Green's function: the action-principle claim verified]
\label{rem:marginal-greens}
Remark~\ref{rem:phase-family} asserts that \emph{``the Green's function of the
discrete Laplacian decays as $\alpha(d^*)$ from the source.''} The literal reading
``$G(d_{\mathrm{obs}}, d^*) = \alpha(d^*)$'' is wrong --- the full Green's function
at the observer is a cumulative compliance sum over the trailing tower that
overshoots $\alpha(d^*)$ by one to two orders of magnitude.

The correct reading is \emph{marginal}: the per-layer contribution of source
$d^*$ to the Green's function equals $\alpha(d^*)$ exactly. Both the
cumulative form $G(d_{\mathrm{obs}}, d^*) = \sum_{k=d^*}^{d_2-1}\alpha(k)$ and
the marginal identity
\begin{equation}
\label{eq:marginal-greens}
\boxed{\;G(d_{\mathrm{obs}}, d^*) - G(d_{\mathrm{obs}}, d^* + 1) \;=\; \alpha(d^*)\;}
\end{equation}
are derived in Part~0~\cite{part0} (Definition on the cascade
compliance and scalar action; Proposition on the marginal Green's function).
Combined with the chirality filtering of Theorem~\ref{thm:chirality-factorisation},
this yields
\[
\delta\Phi \;=\; \frac{\alpha(d^*)}{\chi^k}
\quad\text{at source }d^*,\text{ filtered through }k\text{ channels,}
\]
exactly as Remark~\ref{rem:phase-family} asserts. Numerical verification at
all distinguished-source and distinguished-layer values confirms the identity
to machine precision: see \texttt{tools/verifiers/cascade\_greens\_function.py}.

The verification establishes that the action-principle derivation of the
$\alpha(d^*)/\chi^k$ family is static and cascade-native: no Kaluza--Klein
reduction, no semiclassical expansion, no Part~VI temporal structure. The
computation is the inverse of a banded Sturm--Liouville matrix, i.e., classical
finite-dimensional linear algebra. Together with Theorem~\ref{thm:chirality-factorisation}
(chirality-basin filtering) this closes item~(a) of Remark~\ref{rem:phase-family}'s
``what remains open'' list. Item~(b) (unit source strength at distinguished
layers) closes by assembly: the marginal identity holds at every layer with
coefficient exactly $1$ (machine precision, $211$ interior layers verified
in \texttt{tools/research/cascade\_unit\_source\_strength.py}); the
``distinguished'' restriction comes from the source-selection rule
(Proposition~\ref{prop:source-selection}), not from Hessian critical-point
structure. Item~(c) (the $v$ closure via the non-perturbative sector) is
largely resolved by Remark~\ref{rem:v-closure}; residual is a formal
4D-hairy-ball-to-2D-first-Chern mapping on $S^4$ (geometric topology, not
cascade content).

\emph{Connection to the Gram correction.} The single-source family above and
the path-distributed Gram correction of Part~0~\cite{part0}
are complementary derivatives of one cascade compliance function $\alpha(d) =
R(d)^2/4$: linear source response (this paper) versus discrete log-Laplacian
at doubled argument (Part~0's inter-layer-coupling section). The structural
unification is established in Part~0's Remark on structural unification with
the cascade scalar action. Both close descent-dependent observables; which
family applies to a given observable is governed by
Proposition~\ref{prop:source-selection} for single-source candidates and by
Cauchy--Schwarz path-additivity for the Gram correction.
\end{remark}

\begin{remark}[Action uniqueness: conditional on gauge identification]
\label{rem:action-uniqueness}
The cascade scalar action $S[\varphi] = \sum_d (2\alpha(d))^{-1}(\Delta\varphi)^2$
is defined in Part~0~\cite{part0} (Definition on the cascade
compliance and scalar action). Its compliance $\alpha(d) = R(d)^2/4$ is the
cascade-natural choice; this remark records the uniqueness argument.

The general local quadratic nearest-neighbour action
\[
S[\varphi] \;=\; \sum_d \frac{1}{2\beta(d)}(\Delta\varphi)^2
\]
has the slicing recurrence as its Euler--Lagrange equilibrium for any
compliance function $\beta(d)$, since the equilibrium depends on
$\Delta\varphi_{\rm eq}$ alone. First-order EL fixes the action's form
(quadratic, nearest-neighbour); it does not fix $\beta$.

What pins $\beta$ is the marginal identity of Part~0~\cite{part0}
(Proposition on the marginal Green's function): the per-layer Green's function
response at source $d^*$ equals $\beta(d^*)$. Matching the observed
$\alpha(d^*)/\chi^k$ shifts of Remark~\ref{rem:phase-family} therefore forces
$\beta(d) = \alpha(d)$.

Numerical verification in \texttt{tools/verifiers/action\_uniqueness.py} confirms this:
among seven natural candidate compliance functions ($\alpha(d)$, $R(d)$,
$R(d)^2$, $R(d)/2$, $1/d$, $1/(d{+}1)$, $N(d)$), only $\beta = \alpha$
reproduces the four Part~IVb source shifts to machine precision. The six
alternatives fail by factors of $2$ to $100$.

The action is therefore uniquely forced by two cascade-native requirements:
\begin{enumerate}[nosep]
\item \emph{First-order EL equation} --- the action is quadratic and
nearest-neighbour in $\Delta\varphi$, matching the cascade's first-order
slicing recurrence (Part~0, Theorem~\xref{part0}{thm:boundary}).
\item \emph{Compliance equals the cascade gauge coupling} --- $\beta(d) = \alpha(d) = N(d)^2/(4\pi)$ is the cascade's gauge coupling at layer $d$ (Section~\ref{sec:gauge-couplings}; see also Remark~\ref{rem:alpha-geometric} on the cascade-internal $\Omega_2$ interpretation of the $(4\pi)^{-1}$).
The action is the cascade-native gauge-sector functional.
\end{enumerate}

Given requirement~2 (the action represents gauge physics), the compliance is
forced. The derivation is therefore \emph{conditional} on the gauge
identification --- the same identification that fixes the cascade's
observed gauge couplings (Theorems~\ref{thm:alpha-s-leading},~\ref{thm:alpha-s-closure},
etc.). What remains structurally open is whether Part~IVa's gauge-coupling
normalisation $\alpha(d) = N(d)^2/(4\pi)$ is itself uniquely forced from
within the cascade (independent of the QFT gauge-theoretic convention
$\alpha = g^2/(4\pi)$), or whether it is a matching choice. Both readings
are cascade-internally consistent; neither changes the action's form or
the eight precision closures.
\end{remark}

\begin{theorem}[Chirality factorisation of the Green's function]\label{thm:chirality-factorisation}
Let $G(d,d^*)$ be the scalar Green's function of the cascade action's discrete Laplacian $S[\varphi]=\sum_d(2\alpha(d))^{-1}(\Delta\varphi)^2$, and let $Q$ be an observable built from $k$ independent definite-chirality cascade propagator modes. Then
\[
G_Q(d,d^*) \;=\; \frac{G(d,d^*)}{\chi^k},
\]
where $\chi=\chi(S^{2n})=2$.
\end{theorem}

\begin{proof}
Three independently proved properties combine.

\medskip
\textbf{(A) Equal splitting at even-sphere layers.}
The cascade field $\varphi(d)=\ln\Omega_d$ is a scalar (0-form): it measures total geometric content without reference to direction or chirality. At an even-sphere layer $d$ (where $d$ is odd, so $S^{d-1}$ has even dimension), the height function $h:S^{d-1}\to\mathbb{R}$ is a Morse function with exactly $\chi(S^{d-1})=2$ critical points (Poincar\'{e}--Hopf). These define two basins---the ascending manifolds of the minimum and maximum---of equal area, by the $\mathbb{Z}_2$ symmetry $h\mapsto -h$ of the round sphere. A scalar perturbation $\delta\varphi$ distributes equally between the two basins:
\[
\delta\varphi_+ \;=\; \delta\varphi_- \;=\; \frac{\delta\varphi}{\chi}.
\]
This is forced by the scalar nature of $\varphi$ (no preferred direction to break the $\mathbb{Z}_2$ symmetry) and the constancy $\chi(S^{2n})=2$ for all $n\geq 1$. On odd-dimensional spheres, $\chi=0$ and no canonical basin decomposition exists; content passes through undivided.

\medskip
\textbf{(B) Chirality coherence from unitarity.}
The cascade propagator $K(D,d')=\prod_{j=d'}^{D-1}L(j)$ with $L(j)=iN(j)$ is unitary (Part~II, Theorem~\xref{part2}{thm:propagator}): it maps states to states bijectively. A state in a definite chirality sector at one even-sphere layer maps to a definite state at the next. The phase--obstruction lockstep (Part~II, Corollary~\xref{part2}{cor:lockstep}) makes this explicit: the propagator phase at even-sphere layer $d$ relative to the observer is $e^{i(d-4)\pi/2}$, alternating deterministically between $+i$ (at $d\bmod 4=1$) and $-i$ (at $d\bmod 4=3$). A chirality selection at the first even-sphere layer determines the chirality at all subsequent layers through the periodic phase.

Consequently, a single definite-chirality propagator traversing a cascade path with $n_{\rm even}$ even-sphere layers makes \emph{one} chirality selection, not $n_{\rm even}$ independent selections. The correction factor is $(1/\chi)^1$, not $(1/\chi)^{n_{\rm even}}$.

This is numerically decisive. For $\alpha_s$ (path $d=5\ldots 12$, with $n_{\rm even}=4$ even-sphere layers, $k=1$ mode): $(1/\chi)^{n_{\rm even}}=1/16$ while $(1/\chi)^k=1/2$. The data requires $1/2$; coherence is confirmed by a factor of~8.

\medskip
\textbf{(C) Multi-mode independence.}
An observable built from $k$ independent propagator modes makes $k$ independent chirality selections. The selections are independent because the chirality basins $S^+$ and $S^-$ are orthogonal (zero overlap), and distinct modes couple to their respective chirality sectors without interaction. For $k$ independent modes:
\[
G_Q \;=\; \frac{G}{\chi^k}.
\]

The mode count $k$ is determined by the cascade's own structure (Part~IVa): $k=1$ for single-channel quantities (one propagator), $k=2$ for two-generation mixing ($\theta_C$: layers $d=5,13$), $k=3$ for three-sector observables ($\sin^2\theta_W$: three gauge layers; $\Omega_m$: three generation layers).
\end{proof}

\begin{remark}[No new ingredient]
Every component of the proof is drawn from the cascade's existing structure: $\chi(S^{2n})=2$ from classical topology (used throughout Parts~IVa--b), unitarity from the forced precession (Part~II, Theorem~\xref{part2}{thm:propagator}), the phase--obstruction lockstep from Part~II, Corollary~\xref{part2}{cor:lockstep}, and mode counting from the Bott/Adams layer assignments of Part~IVa. No new axiom, parameter, or mathematical structure enters.
\end{remark}

\begin{remark}[Matter-content reading of the basins: cascade CPT-balance]\label{rem:cpt-balance-basins}
Theorem~\ref{thm:chirality-factorisation} establishes that at every even-sphere cascade layer, the height function on $S^{2n}$ has exactly $\chi(S^{2n})=2$ critical points and the sphere splits into two basins of \emph{equal area} by the $\mathbb{Z}_2$ symmetry $h\mapsto -h$. The theorem's primary use is the scalar Green's function factorisation $1/\chi^k$ underlying the $\alpha(d^*)/\chi^k$ correction family. A second reading, structurally consistent with the same theorem, applies to fermion content at the observer-host layer $d=5$: the two basins of $S^4$ host matter and antimatter respectively, with equal-area $\Rightarrow$ exact CPT-balance.

\medskip\noindent\textbf{Verification: Gram correction preserves $\mathbb{Z}_2$ basin equality.}
The cascade slicing integrand $f_d(x)=(1-x^2)^{d/2}$ is even in $x$ on $[-1,1]$, and the antipodal map on $S^{d-1}$ acts as $x\mapsto -x$ on the polar coordinate. Every Gram entry
\[
G_{d_i,d_j} \;=\; \int_{-1}^{1}(1-x^2)^{(d_i+d_j)/2}\,dx \;=\; B\!\bigl(\tfrac{1}{2},\,\tfrac{d_i+d_j}{2}+1\bigr)
\]
splits as upper~$=$~lower~$=$~full$/2$ exactly. The Gram first-order correction (Part~0) therefore preserves the $\mathbb{Z}_2$ basin equality at every cascade layer to machine precision. Numerical verification (verifier
\texttt{tools/research/cascade\_s4\_antipodal\_test.py}) at $(d_i,d_j)\in\{(5,5),(5,6),(6,6),(5,7),(6,7),(7,7)\}$: maximum hemisphere asymmetry $0.00\mathrm{e}{+00}$.

\medskip\noindent\textbf{Cascade prediction: strict CPT-balance.}
Under the matter-content reading, the cascade predicts:
\begin{enumerate}[label=(\alph*),nosep]
\item Total cosmic baryon number $=0$ cascade-internally. The matter content of $S^4$ at $d=5$ is exactly equal between the two basins.
\item No Sakharov-dynamical baryogenesis. The observed matter excess in our observable universe is \emph{hemispheric localization} of the observer (we sit in one basin of $S^4$ at $d=5$; the antipodal basin contains antimatter). No cascade-internal CP-violation mechanism is required for cosmology.
\item No cascade-internal CKM CP phase. The cascade derives $\lvert\theta_C\rvert$ via $-\alpha(7)/\chi^2$ (Theorem~\ref{thm:cabibbo-amplitude}) but is structurally CPT-symmetric in audited primitives; CP-violation enters as external observational input, parity with SM treatment.
\item No cascade-internal QCD axion. $\theta_{\rm QCD}=0$ is forced by topological classification (Part~IVb~$\S 6$), consistent with strict CPT-symmetry in the gauge sector.
\item No large antimatter regions within our cosmological horizon. The cascade puts antimatter in the antipodal hemisphere of $S^4$ at $d=5$, outside causal contact.
\end{enumerate}

\medskip\noindent\textbf{Connection to the Y-spectrum sign anchor.}
Part~IVa~\cite{part4a} Remark on the open Y-spectrum question identifies one external sign anchor in the SM Y spectrum derivation. Under the present matter-content reading, that anchor is the \emph{basin label} of the cascade $S^4$ at $d=5$ --- but crucially, this is \emph{a labeling convention internal to the observer's frame, not an observation}. The two basins are CPT-conjugate and physically equivalent: any observer in either basin sees identical local physics and labels their own basin ``matter.'' There is no observational test that distinguishes them globally --- both yield the same Lagrangian after relabeling, by CPT-symmetry. Each basin is internally homogeneous (4-ball topology); each observer within their basin is \emph{relationally central} in their own frame, with the basin's geometry observer-frame-dependent (each observer at the height-function maximum, equator $S^3$ at maximum geodesic distance). The choice $\{Y_{Q_L}=+1/6,\;Y_{Q_L}=-1/6\}$ corresponds to which convention the observer adopts for their own basin --- the same epistemic status as the SM's $Q_e=-1$ convention. Status: \emph{zero observational input} for observer-frame fixing, at strict parity with the SM (also CPT-symmetric, also convention-symmetric). The cascade derives the Y \emph{spectrum}; the basin label is a name we use, not a physical fact about which basin is ``absolute'' matter.

\medskip\noindent\textbf{Falsifier.}
The matter/antimatter divide is geometrically the basin equator $S^3$, identified structurally with the cosmological horizon: anything beyond the horizon is in the antipodal basin (antimatter), causally disconnected. Observation of large antimatter regions \emph{within} our cosmological horizon would falsify the matter-content reading. Currently consistent with observation.

\medskip\noindent\emph{Status.} Structural reading of an existing theorem; not a separate cascade theorem. The matter-content reading extends Theorem~\ref{thm:chirality-factorisation} from the scalar sector (Green's function factorisation) to the fermion sector (matter/antimatter basin assignment). The Gram-symmetry verification is direct calculation; the four falsifiable structural commitments above follow from the matter-content reading $+$ Gram symmetry.
\end{remark}

\subsection{Cascade chirality selection rule (open-line, closed-loop, mixed)}\label{subsec:chirality-selection-rule}

Theorem~\ref{thm:chirality-factorisation} covers \emph{open-line} cascade observables ($k$ independent definite-chirality propagator modes; chirality factor $\chi^{-k}$). Two further structures arise empirically: the $1/\alpha_{\rm em}$ screening (Remark~\ref{rem:per-leg-primitive-derivation}) is a \emph{closed-loop} observable with chirality factor $\chi^{+1}$ at the per-Dirac-layer level; the $(g{-}2)$ at 1-loop is a \emph{balanced mixed} observable with chirality factor $\chi^0 = 1$. The unified rule covering all three cases is the following extension of Theorem~\ref{thm:chirality-factorisation}.

\begin{theorem}[Cascade chirality selection rule]\label{thm:chirality-selection-rule}
Let $Q$ be a cascade observable at a Dirac layer ($d \bmod 8 = 5$) with chirality structure characterised by:
\begin{itemize}[nosep]
\item $k$ = number of independent definite-chirality external propagator modes (open lines).
\item $m$ = number of closed loops at the Dirac layer.
\end{itemize}
Then the chirality factor in $Q$'s cascade contribution is
\[
\chi^{(m - k)},
\]
where $\chi = \chi(S^{d-1}) = 2$ is the Euler characteristic of the even-dimensional sphere at the Dirac layer. Theorem~\ref{thm:chirality-factorisation} is the special case $m = 0$.
\end{theorem}

\begin{proof}
Four properties combine. Steps~(A) and~(C) are unchanged from Theorem~\ref{thm:chirality-factorisation}; Steps~(B) and~(D) extend it to closed-loop and mixed cases.

\textbf{(A) Equal splitting at even-sphere layers (unchanged).} The cascade scalar field $\varphi(d) = \ln\Omega_d$ is a $0$-form; at an even-sphere layer ($d$ odd, so $S^{d-1}$ has even dimension), the height function $h:S^{d-1}\to\mathbb{R}$ is a Morse function with $\chi(S^{d-1}) = 2$ critical points (Poincar\'e--Hopf), defining two basins of equal area by the $\mathbb{Z}_2$ symmetry $h\mapsto -h$. A scalar perturbation distributes equally: $\delta\varphi_+ = \delta\varphi_- = \delta\varphi/\chi$ (Theorem~\ref{thm:chirality-factorisation} step~(A)).

\textbf{(B) Closed-loop chirality multiplicity (new).} Consider a closed loop at a Dirac layer with no external chirality state. The loop's spinor trace runs over the full bundle, which decomposes under $\mathbb{Z}_2$ chirality grading as $S = S^+ \oplus S^-$. By~(A), the two components have equal weight, and the closed-loop trace sums coherently over both:
\[
\mathrm{tr}_{\text{loop}}(\cdots) \;=\; \mathrm{tr}_{S^+}(\cdots) \,+\, \mathrm{tr}_{S^-}(\cdots).
\]
Each component contributes equally, giving the loop a chirality \emph{multiplicity} factor $\chi$. In contrast to open-line propagators (Step~(C)), where chirality is selected externally, closed loops have no external chirality constraint and trace over both basins. For $m$ independent closed loops, the multiplicity factors compose as $\chi^m$.

\textbf{(C) Open-line chirality selection (unchanged).} By Theorem~\ref{thm:chirality-factorisation} steps~(B) and~(C), each definite-chirality propagator mode selects one chirality basin via cascade propagator unitarity (Part~II, Theorem~\xref{part2}{thm:propagator}) and the phase--obstruction lockstep (Part~II, Corollary~\xref{part2}{cor:lockstep}), contributing $1/\chi$. For $k$ independent modes, basin orthogonality gives total factor $\chi^{-k}$.

\textbf{(D) Mixed composition (new).} For an observable with $k$ open-line modes \emph{and} $m$ closed loops, the chirality factors compose multiplicatively. Justification:
\begin{enumerate}[label=(\roman*),nosep]
\item Open-line modes act on definite-chirality projector subspaces $P_\pm S$. Closed loops trace over the full bundle $S = S^+\oplus S^-$. These are structurally distinct operations: open-line modes \emph{filter}; closed loops \emph{sum}.
\item Basin orthogonality from~(A) gives $\mathrm{tr}(P_+ P_-) = 0$. Open-line and closed-loop chirality structures live on disjoint pieces of the spinor bundle: zero cross-terms.
\item Cascade propagator unitarity (Part~II, Theorem~\xref{part2}{thm:propagator}) preserves the chirality decomposition. Open-line selections and closed-loop traces at the same Dirac layer don't interfere.
\end{enumerate}
The total chirality factor is the product:
\[
\chi^{m}_{(\text{closed})} \cdot \chi^{-k}_{(\text{open})} \;=\; \chi^{(m - k)}. \qedhere
\]
\end{proof}

\begin{remark}[Special cases and verification across cascade observables]\label{rem:chirality-special-cases}
The theorem covers all currently-closed cascade precision predictions on a single integer axis $m - k$:
\begin{center}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{rlclc}
\toprule
$m-k$ & $\chi^{(m-k)}$ & Observable & Cascade primitive containing $\chi^{(m-k)}$ & Match \\
\midrule
$-4$ & $1/16$ & $b/s$ & $-\alpha(7)/\chi^4$ (Theorem~\ref{thm:bs-closure}) & $0.014\%$ \\
$-3$ & $1/8$ & $\sin^2\theta_W$, $\Omega_m^{\rm Bott}$ & $\pm\alpha(5)/\chi^3$ (Theorem~\ref{thm:weinberg-closure}) & $\pm 0.4\sigma$ \\
$-2$ & $1/4$ & $\theta_C$ (Cabibbo) & $-\alpha(7)/\chi^2$ & $+0.03\sigma$ \\
$-1$ & $1/2$ & $\alpha_s$, $m_\tau/m_\mu$, $m_\tau$ abs, $\ell_A$ & $\pm\alpha(d^*)/\chi$ (Theorems~\ref{thm:alpha-s-closure},~\ref{thm:mtau-abs-closure}) & sub-$\sigma$ \\
$0$ & $1$ & $(g{-}2)$ at 1-loop & $\alpha_{\rm em}/(2\pi)$ (Schwinger reproduction) & structural \\
$+1$ & $\chi$ & $1/\alpha_{\rm em}$ screening & $3\,\chi\,\Gamma(\tfrac{1}{2})^2 = 6\pi$ (Remark~\ref{rem:per-leg-primitive-derivation}) & $0.005\%$ \\
\bottomrule
\end{tabular}
\end{center}
The $m = 0$ row reproduces Theorem~\ref{thm:chirality-factorisation} (each $k$-value corresponds to one chirality-selection slot in the open-line family). The $m = 1$ rows cover the closed-loop ($k=0$) and balanced ($k=1$) cases. At the $m-k=0$ slot, the cascade vertex correction $F_2(0) = \alpha_{\rm em} \cdot 1/(2\pi) \cdot \chi^{0} = \alpha_{\rm em}/(2\pi)$ matches Schwinger's QED 1-loop result with all three factors cascade-identifiable: $\alpha_{\rm em}$ from the Tier~2 closure $1/\alpha_{\rm em}=137.028$ (entry~(k) of the Tier~2 table); the chirality factor $\chi^0=1$ from Theorem~\ref{thm:chirality-selection-rule} at $m-k=0$; the prefactor $1/(2\pi)$ identified with the cascade primitive ratio $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$ (Remark~\ref{rem:two-pi-prefactor-identification} below). Numerical verification across all ten observables: \texttt{tools/research/cascade\_open\_closed\_mixed.py}, \texttt{cascade\_chirality\_theorem.py}.
\end{remark}

\begin{remark}[Cascade-primitive identification of the $1/(2\pi)$ prefactor]\label{rem:two-pi-prefactor-identification}
Schwinger's QED 1-loop coefficient $1/(2\pi)$ in $F_2(0) = \alpha_{\rm em}/(2\pi)$ is exactly
\[
\frac{1}{2\pi} \;=\; \frac{1}{N(0)\cdot\Gamma(\tfrac{1}{2})^2},
\]
by the cascade primitive identity Corollary~\ref{cor:2sqrtpi-primitive}: $2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac{1}{2})$ with $N(0) = \chi = 2$ (chirality multiplicity / cascade integrand at $d=0$) and $\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$ (per-step orthogonal residue, Part~0, Theorem~\xref{part0}{thm:boundary}). Squaring gives $4\pi = N(0)^2\cdot\Gamma(\tfrac{1}{2})^2$; equivalently $2\pi = N(0)\cdot\Gamma(\tfrac{1}{2})^2$. The reciprocal $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2) = 1/(2\pi)$ identifies the QED prefactor as a ratio of cascade primitives.

\medskip\noindent\textbf{What this gives.} An exact algebraic identification: $\alpha_{\rm em}/(2\pi)$ factorises as (cascade-derived $\alpha_{\rm em}$ via the Tier~2 closure)~$\times$~(cascade-primitive ratio $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$). All three factors at the $m-k=0$ slot live in cascade primitive space. The Schwinger 1-loop coefficient is therefore not an external QFT-imported number but a specific combination of cascade primitives. The same structural unit $N(0)\cdot\Gamma(\tfrac{1}{2})^2 = 2\pi$ that appears in the \emph{numerator} of the per-Dirac-layer $1/\alpha_{\rm em}$ screening contribution (Remark~\ref{rem:per-leg-primitive-derivation}, $\chi\cdot\Gamma(\tfrac{1}{2})^2=2\pi$ per layer, $6\pi$ across three Dirac layers) appears in the \emph{denominator} of the $m-k=0$ multiplicative correction --- the same cascade primitive playing dual roles depending on whether the observable is structure-additive (inverse coupling) or structure-multiplicative (coupling correction).

\medskip\noindent\textbf{What this is not.}
\begin{enumerate}[label=(\arabic*),nosep]
\item Not a derivation of Schwinger's loop integral from cascade machinery. The cascade does not compute $\int d^4 k/(2\pi)^4$ over loop momenta; it identifies that the integrated result lives in cascade primitive space.
\item Not a structural derivation from extending Theorem~\ref{thm:chirality-selection-rule}'s per-leg/per-source primitives directly. Naive extension of the per-Dirac-layer per-leg primitive $\Gamma(\tfrac{1}{2})^n$ from the $m-k=+1$ slot (where it appears in the numerator) to the $m-k=0$ slot does not naturally produce the \emph{reciprocal} in the denominator. The additive$\to$multiplicative inversion between slots is a separate structural commitment, consistent with the algebraic identity but not derived from existing chirality-selection-rule structure (open structural question, Open Question~\ref{oq:additive-multiplicative-duality} below).
\item Not generalisable to higher-loop QED prefactors as currently stated. The 2-loop coefficient (Sommerfield--Petermann) involves $\alpha^2/\pi^2$ scaling (cascade-structural by extending the same $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$ argument to two factors) plus a transcendental numerical coefficient ($197/144 + \pi^2/12 + \cdots$) that the cascade does not currently derive. Cascade-native coverage of $a_e$ and $a_\mu$ frontier precision requires additional structural input beyond the prefactor identification.
\end{enumerate}

\medskip\noindent\textbf{What this places at cascade-native rigour.} The $1/(2\pi)$ prefactor is an exact cascade-primitive expression (item~1 above is unavoidable for any QFT-comparison; item~2 is the next structural target; item~3 is a frontier extension). The cascade's claim at $m-k=0$ is now: $F_2(0) = \alpha_{\rm em}/(2\pi)$ with all three factors cascade-internally identified --- a strict structural identification rather than the previously-overstated ``cascade-native derivation of Schwinger's result.''
\end{remark}

\begin{corollary}[Leading $g$-factor at cascade Dirac layers]\label{cor:g-equals-two}
At every complex Dirac layer $d_f$ ($d_f \bmod 8 = 5$, so $d_f - 1$
even), the cascade fermion's tree-level gyromagnetic ratio is
\[
g(d_f) \;=\; 2.
\]
The factor of $2$ is structural: it is the chirality multiplicity at
the Dirac-layer chirality decomposition $S = S^+ \oplus S^-$,
manifesting through the Clifford algebraic identity
$\gamma^\mu\gamma^\nu = \eta^{\mu\nu} + i\sigma^{\mu\nu}$ as the
coefficient of $\sigma\cdot F$ in the squared covariant derivative.
\end{corollary}

\begin{proof}
By the cascade Dirac descent (Paper~II$=$III, Theorem
\xref{part23}{thm:cascade-dirac-descent}), the cascade fermion at $d_f$
satisfies $(i\gamma^\mu D_\mu - m_f)\psi = 0$ at $d=4$ with
cascade-derived $\gamma^\mu$, $D_\mu = \partial_\mu - ieA_\mu$, and
$m_f = R(d_f)/\chi$. Squaring the Clifford operator and using the
algebraic identity
$\gamma^\mu\gamma^\nu = \eta^{\mu\nu} + i\sigma^{\mu\nu}$
(forced by $\{\gamma^\mu,\gamma^\nu\} = 2\eta^{\mu\nu}$ at every
Dirac layer via Bott classification, no separate derivation
required):
\[
(\gamma\cdot D)^2 \;=\; D^\mu D_\mu \;+\; \tfrac{i}{2}\sigma^{\mu\nu}[D_\mu, D_\nu]
\;=\; D^\mu D_\mu \;+\; \tfrac{e}{2}\,\sigma^{\mu\nu} F_{\mu\nu},
\]
where $[D_\mu, D_\nu] = -ie F_{\mu\nu}$ and the antisymmetry of
$\sigma^{\mu\nu}$ kills the symmetric piece
$\eta^{\mu\nu}[D_\mu,D_\nu]$. The operator
$\sigma^{\mu\nu} F_{\mu\nu}$ is the cascade Pauli term; its
coefficient $e/2$ is forced algebraically with $e$ cascade-derived
(Theorem~\ref{thm:weinberg}, $e = \sqrt{\alpha(14)}$).

For a static magnetic field $\mathbf{B} = \nabla\times\mathbf{A}$
with $F_{ij} = \epsilon_{ijk} B^k$, one has
$\sigma^{ij}F_{ij} = 2\,\boldsymbol{\sigma}\cdot\mathbf{B}$
(Pauli-matrix identity in the chirality basis). The
non-relativistic reduction of $(\gamma\cdot D)^2\psi = m_f^2\psi$
yields the magnetic-moment Hamiltonian
$H_{\text{mag}} = -\,(e/2m_f)\,\boldsymbol{\sigma}\cdot\mathbf{B}$. Writing
$H_{\text{mag}} = -\boldsymbol{\mu}\cdot\mathbf{B}$ with the conventional
identification $\boldsymbol{\mu} = (g/2)(e/m_f)\mathbf{S}$ and
$\mathbf{S} = (\hbar/2)\boldsymbol{\sigma}$ gives
\[
\frac{e}{2m_f} \;=\; \frac{g}{2}\cdot\frac{e}{m_f}\cdot\frac{1}{2}
\quad\Longrightarrow\quad g \;=\; 2.
\]
The structural origin of the $2$ in $g=2$: the Clifford identity
contributes factor $1/2$ (antisymmetric half of
$\gamma^\mu\gamma^\nu$); the spin-magnetic-field algebra contributes
factor $2$ ($\sigma^{ij}F_{ij} = 2\boldsymbol{\sigma}\cdot\mathbf{B}$);
the spin-operator conversion $\mathbf{S} = (\hbar/2)\boldsymbol{\sigma}$
contributes factor $1/2$. The product on the ``$g/2$'' side is
$1/2$, giving $g=2$. The $2$ in $\sigma^{ij}F_{ij}$ is the
chirality multiplicity $\chi(S^{d_f - 1}) = 2$ in the trace over
$S^+ \oplus S^-$ that the Pauli-matrix identity packages
algebraically.
\end{proof}

\begin{remark}[$g = 2$ filling the leading slot of the chirality
selection rule]\label{rem:g-2-leading}
Theorem~\ref{thm:chirality-selection-rule} places the 1-loop
correction $(g-2)/2 = \alpha_{\rm em}/(2\pi)$ at the $m-k = 0$ slot
with chirality factor $\chi^0 = 1$ (Schwinger reproduction;
Remark~\ref{rem:chirality-special-cases}, table row $m - k = 0$).
Corollary~\ref{cor:g-equals-two} provides the leading value $g = 2$
that the 1-loop correction sits on top of: the leading slot is
populated by the cascade Dirac descent (Paper~II$=$III, Theorem
\xref{part23}{thm:cascade-dirac-descent}), with $g/2 = 1$ at tree level
forced by the Clifford algebraic identity. Together,
Corollary~\ref{cor:g-equals-two} and the 1-loop entry in
Remark~\ref{rem:chirality-special-cases} cover
\[
g \;=\; 2 \;+\; \frac{\alpha_{\rm em}}{\pi} \;+\; \mathcal{O}(\alpha_{\rm em}^2)
\]
cascade-natively to 1-loop accuracy with no fitted parameter at
either order. Higher-loop corrections to $a_e$ (electron) and
$a_\mu$ (muon, currently a $\sim 5\sigma$ frontier discrepancy
depending on lattice computation) would exercise the higher slots
of Theorem~\ref{thm:chirality-selection-rule}; see Open
Question~\ref{oq:chirality-selection-extensions}. Per
CLAUDE.md~Check~7, the higher-loop normalisation must be developed
in cascade-lattice terms (Sturm--Liouville on the layer index, no
sphere-spectral procedures).
\end{remark}

\begin{remark}[$\chi = 2$ unification: mass halving and gyromagnetic doubling]\label{rem:chi-unification}
The Euler characteristic $\chi(S^{2n}) = 2$ from Poincar\'e--Hopf
applied to even-dimensional Dirac-layer spheres
(Theorem~\ref{thm:chirality-factorisation}) generates two
structurally distinct cascade observables that are dual
manifestations of the same chirality fact:
\begin{itemize}[nosep]
\item \emph{Mass halving} (Theorem~\ref{thm:obstruction}):
$m(d) = R(d)/\chi$. The mass operator couples
$S^+ \leftrightarrow S^-$ off-diagonally; the chirality
multiplicity $\chi$ appears as a \emph{divisor}, halving the
bare scale $R(d)$.
\item \emph{Gyromagnetic doubling}
(Corollary~\ref{cor:g-equals-two}):
$g = \chi(S^{d_f-1}) = 2$. The Pauli-term operator traces over
both chirality basins via the Clifford identity
$\gamma^\mu\gamma^\nu = \eta^{\mu\nu} + i\sigma^{\mu\nu}$; the
chirality multiplicity $\chi$ appears as a \emph{multiplier},
doubling the magnetic moment relative to the spin.
\end{itemize}
The same $\chi = 2$ acts as a divisor in mass and as a
multiplier in $g$. This is structural derivational economy: one
cascade fact (Poincar\'e--Hopf $\chi(S^{2n}) = 2$) generates two
seemingly independent observables (charged-fermion mass formula,
electron gyromagnetic ratio) whose values are not separately
fitted. The same identity also figures in the cascade-primitive
decomposition $1/(2\pi) = 1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$
of the QED 1-loop prefactor
(Remark~\ref{rem:two-pi-prefactor-identification}, where
$N(0) = \chi$). Three cascade-internal structural roles for the
same $\chi = 2$: divisor in $m(d)$, multiplier in $g$, factor
in the loop-prefactor $1/(2\pi)$.
\end{remark}

\begin{remark}[Scope: chirality factor vs.\ per-leg/per-source primitives]\label{rem:chirality-scope}
Theorem~\ref{thm:chirality-selection-rule} fixes the \emph{chirality factor} $\chi^{(m-k)}$ in any cascade observable's contribution. The full cascade contribution carries additional structure that the theorem does not cover:
\begin{itemize}[nosep]
\item \emph{Open-line per-source primitive} (in the $\alpha(d^*)/\chi^k$ family): the source coupling $\alpha(d^*)$ at the distinguished source layer is \emph{layer-dependent}, with $d^*$ assigned by the source-selection rule (Proposition~\ref{prop:source-selection}).
\item \emph{Closed-loop per-leg primitive}: $\Gamma(\tfrac{1}{2})^n$ for an $n$-leg closed loop is \emph{layer-independent}. Empirically forced by the $1/\alpha_{\rm em}$ screening: the layer-dependent reading $\sqrt{\alpha(d)}^n$ fails by $13.5\%$ (factor $62$ too small), while the layer-independent reading $\Gamma(\tfrac{1}{2})^n$ matches at $0.005\%$.
\end{itemize}
The full cascade contribution to an observable factorises as
\[
I_Q^{\rm cascade} \;=\; \chi^{(m-k)} \cdot [\text{per-source primitive}]^{\,?} \cdot [\text{per-leg primitive}]^{\,(\text{legs in loops})},
\]
with the source/leg structure determined by the observable's topology. Theorem~\ref{thm:chirality-selection-rule} covers the $\chi$ factor; the per-source and per-leg primitives are separate cascade structures (derivations of $\alpha(d)$ in Part~IVa~\S2 and $\Gamma(\tfrac{1}{2})$ in Part~0, Theorem~\xref{part0}{thm:boundary}).

The per-leg primitive's layer-independence is structurally consistent with Corollary~\ref{cor:2sqrtpi-primitive}: the open-line obstruction factor $2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac{1}{2})$ is also layer-independent ($R(d)$ cancels between fermion lapse $N_f(d) = R(d)/\chi$ and scalar lapse $N(d) = \sqrt{\pi}\cdot R(d)$). Closed-loop primitives inherit this layer-independence; per-source primitives use the layer-dependent $\alpha(d^*)$. The two structures are distinct cascade ingredients combined by the theorem's chirality factor.
\end{remark}

\begin{remark}[Per-leg primitive $\Gamma(\tfrac{1}{2})$ from the cascade scalar action]\label{rem:per-leg-primitive-derivation}
Remark~\ref{rem:chirality-scope} states the per-leg primitive $\Gamma(\tfrac{1}{2})^n$ for an $n$-leg closed cascade cycle at a Dirac layer as a layer-independent cascade structure forced by empirical match. The cascade scalar action $S[\varphi] = \sum_d (2\alpha(d))^{-1}(\Delta\varphi(d))^2$ (Remark~\ref{rem:action-uniqueness}) supplies the structural derivation in cascade-native terms via two cascade-internal facts:
\begin{enumerate}[label=(\arabic*),nosep]
\item \emph{Lattice covariance per Dirac layer.} The Gaussian theory defined by $S[\varphi]$ on the 1D cascade lattice has variance equal to the action's compliance: by the marginal Green's function identity (Remark~\ref{rem:marginal-greens}), $\langle(\Delta\varphi(d))^2\rangle = \alpha(d)$. The per-layer amplitude scale is $\sqrt{\alpha(d)} = R(d)/2$.
\item \emph{Per-step orthogonal residue.} Each cascade slicing step at a Dirac layer crosses the hairy-ball quarter-turn and contributes one Beta integral $B(\tfrac{1}{2},\,\cdot)$ whose half-integer first argument carries the cascade's unique recurrence constant (Part~0, Theorem~\xref{part0}{thm:boundary}). The dimension-independent residue is $\Gamma(\tfrac{1}{2}) = \sqrt{\pi}$ exactly, by the same orthogonality forcing that fixes $\sqrt{\pi}$ in the slicing recurrence. No proper-time integral, heat-kernel expansion, or sphere-spectral procedure enters.
\end{enumerate}
Combining (1) and (2), the cascade-lattice diagonal contribution at a single Dirac layer is
\[
G_d \;=\; \sqrt{\alpha(d)}\cdot\Gamma(\tfrac{1}{2})\,,
\]
the layer amplitude scale from~(1) multiplied by the per-step orthogonal residue from~(2). For an $n$-leg cascade-internal cycle at the layer (a closed cascade cycle traversing $n$ slicing steps that return to the same Dirac layer):
\[
L_n(d) \;=\; G_d^{\,n} \;=\; \alpha(d)^{n/2}\cdot\Gamma(\tfrac{1}{2})^n\,.
\]
The contribution to the inverse coupling $1/\alpha_{\rm em}$ at Dirac layer $d$ is $L_n(d)$ normalised by the cascade scalar action's own kinetic-prefactor scale $\alpha(d)^{n/2}$ at $n$ legs (Remark~\ref{rem:wavefunction-renorm-canonical}; the prefactor $1/(2\alpha(d))$ on the kinetic term of $S[\varphi]$, Remark~\ref{rem:action-uniqueness}, propagates as $\alpha(d)^{n/2}$ for $n$ amplitude factors):
\[
\Delta(1/\alpha_{\rm em})_d \;=\; \chi\cdot\frac{L_n(d)}{\alpha(d)^{n/2}} \;=\; \chi\cdot\Gamma(\tfrac{1}{2})^n\,,
\]
LAYER-INDEPENDENT: the layer-dependent $\alpha(d)^{n/2}$ from the diagonal product cancels exactly against the $\alpha(d)^{n/2}$ from the kinetic-prefactor normalisation, leaving the cascade primitive $\Gamma(\tfrac{1}{2})^n$ per leg combined with the chirality multiplicity $\chi$ (Theorem~\ref{thm:chirality-selection-rule} Step~(B)) once per cycle. Numerical verification at $n=2$, $d\in\{5,13,21\}$: each layer contributes $\chi\cdot\pi = 2\pi$ to $1/\alpha_{\rm em}$ to machine precision; total $6\pi$; combined with bare $1/\alpha(13) + \pi/\alpha(14) = 118.18$ gives $1/\alpha_{\rm em} = 137.028$, matching observation at $0.006\%$ (verifier: \texttt{tools/research/cascade\_per\_leg\_primitive.py}).

\medskip\noindent\emph{Open/closed-leg duality.} The cascade primitive identity Corollary~\ref{cor:2sqrtpi-primitive} reads: $2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac{1}{2})$ is the unification of the chirality factor $N(0)=\chi$ (cascade primitive at the hairy-ball zero) and the per-leg primitive $\Gamma(\tfrac{1}{2})$ (cascade primitive at the quarter-turn). The two cascade primitives appear in conjugate roles for open-line vs closed-cycle observables:
\begin{itemize}[nosep]
\item \emph{Open-line propagator} (Theorem~\ref{thm:obstruction}, $1/\chi^k$ family): the propagator filters through the obstruction; per-leg factor $1/\sqrt{\pi}$ in the denominator (selection).
\item \emph{Closed-cycle vertex} (this Remark, $\chi^{m-k}$ family with $m\geq 1$): the cycle composes around the obstruction; per-leg factor $\sqrt{\pi}$ in the numerator (multiplicity).
\end{itemize}
The two factors are reciprocal: $(1/\sqrt{\pi})\cdot\sqrt{\pi} = 1$. Both arise from the same hairy-ball quarter-turn; the sign of the exponent encodes whether the propagator is filtering (open) or composing (closed). Combined with the chirality factor $\chi^{(m-k)}$ from Theorem~\ref{thm:chirality-selection-rule}, this gives the universal cascade leg primitive
\[
\text{per-leg factor at a Dirac layer} \;=\; \begin{cases} 1/\sqrt{\pi} \cdot 1/\chi & \text{open-line}, \\ \sqrt{\pi} \cdot \chi^{1/m} & \text{closed-cycle} \end{cases}
\]
(reading the chirality multiplicity per cycle as evenly distributed across legs only for accounting; the cycle's chirality factor is properly $\chi$ once per cycle, not per leg).

\medskip\noindent\emph{Status.} Per-leg $\Gamma(\tfrac{1}{2})$ derived from the cascade scalar action's lattice covariance (item~(1)) plus Part~0's per-step orthogonal residue (item~(2)). The cascade kinetic-prefactor normalisation step is grounded structurally in Remark~\ref{rem:wavefunction-renorm-canonical} below. All four pieces of $1/\alpha_{\rm em} = 137.028$ (bare coupling Theorem~\ref{thm:weinberg}; chirality factor Theorem~\ref{thm:chirality-selection-rule}; per-leg primitive this Remark; $N_{\rm gen}=3$ from the count of Dirac layers in the descent $\{5,13,21\}$, equal to the generation count by Part~IVa~\cite{part4a}'s three-generation theorem) are now cascade-internally derived. The $0.006\%$ residual is the cascade's standing leading-order systematic.
\end{remark}

\begin{remark}[Cascade kinetic-prefactor normalisation on the cascade lattice]\label{rem:wavefunction-renorm-canonical}
Remark~\ref{rem:per-leg-primitive-derivation} divides the $n$-leg diagonal product $L_n(d) = \alpha(d)^{n/2}\cdot\Gamma(\tfrac{1}{2})^n$ by the cascade scalar action's own kinetic-prefactor scale $\alpha(d)^{n/2}$ to obtain the layer-independent per-Dirac-layer screening contribution $\chi\cdot\Gamma(\tfrac{1}{2})^n$. This is not a chosen normalisation; it is structurally forced by the action's own form. Three cascade-internal ingredients combine.
\begin{enumerate}[label=(\roman*),nosep]
\item \emph{The action's kinetic prefactor is uniquely $1/(2\alpha(d))$.} Remark~\ref{rem:action-uniqueness} establishes this in two steps: first-order EL forces the quadratic nearest-neighbour form $S = \sum_d (2\beta(d))^{-1}(\Delta\varphi)^2$; the marginal Green's function identity (Remark~\ref{rem:marginal-greens}, eq.~\eqref{eq:marginal-greens}) forces the compliance $\beta(d) = \alpha(d) = R(d)^2/4$. The action is therefore the cascade-native gauge-sector functional, with $\alpha(d)$ playing the role of the gauge coupling.
\item \emph{The lattice covariance inherits the action's prefactor.} For a Gaussian theory on the cascade lattice with action $S = \sum_d (2\beta(d))^{-1}(\Delta\varphi)^2$, the Gaussian variance identity is $\langle(\Delta\varphi(d))^2\rangle = \beta(d)$. On the cascade lattice with $\beta = \alpha(d)$, the per-layer covariance is $\langle(\Delta\varphi(d))^2\rangle = \alpha(d)$, exactly --- this is the marginal Green's function identity (Remark~\ref{rem:marginal-greens}). Combined with the per-step orthogonal residue $\Gamma(\tfrac{1}{2})$ from Part~0's slicing recurrence (Remark~\ref{rem:per-leg-primitive-derivation} item~(2)), the cascade-lattice diagonal contribution at a single Dirac layer is $G_d = \sqrt{\alpha(d)}\cdot\Gamma(\tfrac{1}{2})$.
\item \emph{The inverse-coupling shift is the diagonal product divided by the gauge boson's kinetic prefactor.} For a gauge boson with cascade-native coupling $g(d) = \sqrt{\alpha(d)}$ (Section~\ref{sec:fermion-action}, $\sqrt{\alpha}$-universality), the gauge boson's kinetic term in the cascade action is $1/(2\alpha(d))$ at its layer. The contribution to $1/\alpha_{\rm em}$ from a closed cascade cycle at layer $d$ is then
\[
\Delta(1/\alpha_{\rm em})_d \;=\; \frac{\Pi_d}{\alpha(d)} \;=\; \frac{(\text{$n$-leg diagonal product with chirality})}{\alpha(d)}\,,
\]
where the $1/\alpha(d)$ factor is the gauge boson's kinetic prefactor in the cascade's action --- the same $1/(2\alpha(d))$ that uniquely defines the action via Remark~\ref{rem:action-uniqueness}. This is not an additional convention: it is the cascade action's own form propagated to the inverse-coupling identity, with no input beyond the cascade scalar action and its marginal Green's covariance.
\end{enumerate}

\medskip\noindent\emph{Conclusion.} The $1/\alpha(d)^{n/2}$ kinetic-prefactor normalisation in Remark~\ref{rem:per-leg-primitive-derivation} is the cascade lattice's structural inheritance of the action's prefactor; given Remark~\ref{rem:action-uniqueness}'s uniqueness of the action and Remark~\ref{rem:marginal-greens}'s covariance identity, the normalisation is forced. The cancellation $\alpha(d)^{n/2}$ (diagonal product) $/\alpha(d)^{n/2}$ (kinetic prefactor) $= 1$ is therefore not a choice that happens to work: it is the only normalisation consistent with the cascade-native action and its lattice covariance. The screening contribution at every Dirac layer is exactly $\chi\cdot\Gamma(\tfrac{1}{2})^n$, layer-independent.

\medskip\noindent\emph{What this closes.} The structural derivation of the per-leg primitive $\Gamma(\tfrac{1}{2})$ from the cascade scalar action is now fully grounded in cascade-internal structures: action uniqueness (Remark~\ref{rem:action-uniqueness}), marginal Green's function (Remark~\ref{rem:marginal-greens}), $\sqrt{\alpha}$-universality of the gauge coupling (Section~\ref{sec:fermion-action}), and Theorem~\ref{thm:chirality-selection-rule}'s closed-cycle multiplicity. No additional cascade content or external convention is required; in particular, no proper-time integral, heat-kernel expansion, K\"all\'en--Lehmann identity, vacuum-polarisation construction, sphere-spectral procedure, or QFT loop-integral apparatus is invoked --- per CLAUDE.md Check~7, such procedures are out of bounds for cascade quantities, and the derivation above stays within cascade-lattice primitives (Sturm--Liouville on the layer index $d$, classical finite-dimensional linear algebra). The $0.006\%$ residual on $1/\alpha_{\rm em}$ is the cascade's standing leading-order systematic, on a footing with the other Tier~2 precision closures (sub-$\sigma$ for $\alpha_s$, $m_\tau$, $\sin^2\theta_W$, $\theta_C$, $\Omega_m$, $b/s$, $\ell_A$).
\end{remark}

\subsection{The Source Selection Rule}\label{subsec:source-selection}

Theorem~\ref{thm:chirality-factorisation} derives the channel count $k$ and the filtering factor $1/\chi^k$ from the cascade's own structure. The remaining freedom in the correction family $\delta\Phi=\pm\alpha(d^*)/\chi^k$ is the source layer $d^*$. Remark~\ref{rem:phase-family} assigns it by observable class, but the class assignment has been descriptive, not derived. This subsection proposes a deterministic rule, verifies it against all eight closed observables, and identifies what remains to be proved.

\paragraph{The variational setup.}
The cascade action $S[\varphi]=\sum_d(2\alpha(d))^{-1}(\Delta\varphi)^2$ responds linearly to a unit source at layer $d'$ via the discrete Green's function $G(d,d')$. For an observable $Q$ expressible as a functional of the equilibrium field $\varphi_{\rm eq}$, a source at $d'$ produces $\delta Q=\sum_d(\delta Q/\delta\varphi(d))\,G(d,d')$. The structural question is: which layer $d'$ is the \emph{natural} source for a given observable? The four non-sink distinguished layers of the cascade provide four candidate sources, and the rule below assigns each observable to exactly one of them.

\paragraph{The non-sink distinguished layers.}
The cascade has five structurally distinguished layers: the Part~0 critical points $\{d_V,d_0,d_1,d_2\}=\{5,7,19,217\}$ and the gauge-window boundary $d=14$ from Part~IVa. Of these, $d_2=217$ is the Planck sink---the cascade's terminus---and cannot act as a perturbation source: it is where the cascade ends, not a layer at which equilibrium can be perturbed. This leaves exactly four potential source layers:
\[
\mathcal{S} \;=\; \{\,d_V=5,\;d_0=7,\;d_{\rm gw}=14,\;d_1=19\,\}.
\]
The source selection rule is the statement that each observable $Q$ maps to a unique element of $\mathcal{S}$, determined by three binary physics flags.

\paragraph{Three syntactic flags on the cascade formula.}
For each precision observable $Q$ with canonical cascade formula
$F_Q$, define three binary predicates as \emph{syntactic queries on
$F_Q$}---readable mechanically from the formula's expression tree
without recourse to Standard Model interpretation:
\begin{itemize}
\item \textbf{Planck-anchored ($P$):} $F_Q$ references $M_{\mathrm{Pl,red}}$,
either directly or through a sub-expression whose closed form
contains $M_{\mathrm{Pl,red}}$ (e.g.\ the electroweak scale $v$
contains $M_{\mathrm{Pl,red}}$ via Theorem~\ref{thm:vev}; the sound
horizon $r_d$ in the acoustic scale $\ell_A = \pi D_A/r_d$ contains
it via its $H_0$ derivation). Equivalent formulation: expanding all
cascade sub-expressions to their primitive form, $F_Q$ contains the
symbol $M_{\mathrm{Pl,red}}$ somewhere.
\item \textbf{Observer-local ($L$):} $F_Q$ contains \emph{no}
exponential of a cascade-potential or decay-rate value---that is,
no $\exp(\Phi(d))$, $\exp(\Phi(d_A \to d_B))$, $\exp(p(d))$, or
$\exp(\Phi + \delta\Phi)$ factor. The observable is a static
combination of $N(d)$ values and $\Omega_d$ values evaluated at
distinguished layers; no cascade descent is encoded in the formula.
\item \textbf{Gauge-mediated ($G$):} $F_Q$ contains a \emph{multi-layer}
cascade-potential exponential $\exp(\Phi(d_A \to d_B))$ with
$d_B - d_A \geq 2$ and $[d_A, d_B] \cap \{12, 13, 14\} \neq \varnothing$
(the path crosses or enters the gauge window). A single-layer
attenuation $\exp(-p(d_0))$ at a distinguished layer does not count
as gauge running.
\end{itemize}

\begin{remark}[Flag readings are mechanical, not interpretive]\label{rem:sp36-syntactic}
The three predicates above take a cascade formula as input and return
a boolean triple $(P,L,G)$ without reference to what the observable
``physically is'' in the Standard Model. Two readers applying the
predicates to the same $F_Q$ will agree on $(P,L,G)$ by inspection of
the formula's expression tree. The bar is therefore ``write down
$F_Q$ and apply the queries,'' not ``decide which physics role each
ingredient plays.'' The syntactic character means the verification
table below is an \emph{inspection result}, not a classifier fit to
the eight observables. Appendix-style caveat: where an observable
admits multiple cascade expressions (e.g.\ an algebraic identity
equivalent to a descent formula), the canonical expression is the
\emph{minimal descent formula}---the simplest $F_Q$ that closes $Q$
to experimental precision via the cascade's propagator structure, not
an algebraic re-expression that bypasses it.
\end{remark}

\begin{definition}[Observable type]\label{def:observable-type}
Let $Q$ be a cascade observable with physics flags $(P,L,G)$. Apply the following decision procedure in order:
\begin{enumerate}
\item If $P=$ true, then $Q$ is of \emph{Absolute} type.
\item Otherwise, if $L=$ true, then $Q$ is of \emph{Observer} type.
\item Otherwise, if $G=$ true, then $Q$ is of \emph{Gauge} type.
\item Otherwise, $Q$ is of \emph{Amplitude} type.
\end{enumerate}
\end{definition}

The four cases are mutually exclusive (checked in order) and exhaustive: every Standard Model precision observable is assigned to exactly one type by its three physics flags.

\begin{proposition}[Source selection rule]\label{prop:source-selection}
For an observable $Q$ of the type classified by Definition~\ref{def:observable-type}, the source layer of the correction family $\delta\Phi=\pm\alpha(d^*)/\chi^k$ is
\[
d^*(Q) \;=\; \begin{cases}
d_1 \;=\; 19 & \text{if $Q$ is of Absolute type,}\\
d_{\rm gw}\;=\;14 & \text{if $Q$ is of Gauge type,}\\
d_V\;=\;5 & \text{if $Q$ is of Observer type,}\\
d_0\;=\;7 & \text{if $Q$ is of Amplitude type.}
\end{cases}
\]
The assignment is a bijection between the four observable types and the four non-sink distinguished layers $\mathcal{S}$.
\end{proposition}

\begin{proof}[Verification against the eight closed observables]
We read the three physics flags $(P,L,G)$ for each observable of the correction-family table of Remark~\ref{rem:phase-family}, apply Definition~\ref{def:observable-type}, and compare the predicted source to the assignment of Remark~\ref{rem:phase-family}.

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lcccll}
\toprule
observable & $P$ & $L$ & $G$ & type & predicted $d^*$ \\
\midrule
$\alpha_s(M_Z)$    & F & F & T & Gauge     & $14$ \checkmark \\
$m_\tau/m_\mu$     & F & F & T & Gauge     & $14$ \checkmark \\
$m_\tau$ abs       & T & --& --& Absolute  & $19$ \checkmark \\
$\ell_A$           & T & --& --& Absolute  & $19$ \checkmark \\
$\sin^2\theta_W$   & F & T & --& Observer  & $5$  \checkmark \\
$\Omega_m$         & F & T & --& Observer  & $5$  \checkmark \\
$\theta_C$         & F & F & F & Amplitude & $7$  \checkmark \\
$b/s$              & F & F & F & Amplitude & $7$  \checkmark \\
$\theta_{23}$      & F & F & F & Amplitude & $7$  \checkmark \\
\bottomrule
\end{tabular}
\end{center}

All nine assignments match. The flag readings:
\begin{itemize}
\item $\alpha_s$ is a running gauge coupling ($G=T$), not Planck-anchored ($P=F$), not observer-local ($L=F$, since its value encodes the descent from the GUT scale).
\item $m_\tau/m_\mu$ is not Planck-anchored (Planck cancels in the ratio) and not observer-local; it is gauge-mediated by the U(1) hypercharge hierarchy (the $2\sqrt\pi$ factor in its cascade formula is the $d=14$ hairy-ball obstruction).
\item $m_\tau$ absolute is dimensional, hence Planck-anchored.
\item $\ell_A$ is dimensionless but built from $r_d/D_A$, and $r_d$ is Planck-anchored.
\item $\sin^2\theta_W$ is observer-local: it is a ratio of gauge couplings evaluated at the observer $d=4$, with the overall scale factored out, leaving a local datum at the observer.
\item $\Omega_m$ is a density ratio at the observer: observer-local.
\item $\theta_C$ is a geometric mixing angle whose cascade formula uses $N(12),N(13)$ only as static normalisations (inside an $\arccos$), not as running couplings; hence $G=F$. It is not Planck-anchored and not observer-local (it encodes the inter-generation descent), so it defaults to Amplitude type.
\item $b/s$ is a cross-generation quark mass ratio at the SU(3)-algebra layer $d_0=7$; Planck cancels in the ratio ($P=F$), no observer-locality ($L=F$, since the ratio encodes inter-generation descent), and the SU(3) normalisation enters as a static $N$-factor not a running coupling ($G=F$). It defaults to Amplitude type, matching the assigned $d^*=7$.
\end{itemize}
The three-flag classification and the resulting $d^*$ assignment for each of the eight observables are reproduced mechanically in \texttt{tools/verifiers/verify\_selection\_rule.py}; the script encodes Definition~\ref{def:observable-type} as a deterministic decision procedure and confirms all eight rows of the table above with no adjustable input.
\end{proof}

\begin{remark}[Why these four sources and no others]
The four layers in $\mathcal{S}$ are not chosen from a menu; they are the complete set of non-sink distinguished layers in the cascade. The Part~0 critical points $\{d_V,d_0,d_1,d_2\}$ are forced by the Gamma function (Part~0, Theorem~\xref{part0}{thm:tower}: tower completeness---no fifth distinguished dimension exists). The gauge-window boundary $d_{\rm gw}=14$ is forced by Adams' theorem and the Bott mirror (Part~IVa). Removing $d_2=217$ as the Planck sink leaves exactly four sources. Proposition~\ref{prop:source-selection} maps the four observable types to these four sources bijectively---the number of observable types is not adjustable, because each type corresponds to a structurally distinct way an observable can couple to the cascade: (i) Absolute anchoring to Planck, (ii) Gauge coupling running through the gauge window, (iii) Observer locality at the frame adjacent to $d=4$, (iv) non-local amplitude mixing without gauge-window running. No fifth type is definable without introducing a new structural element: the three physics flags $(P,L,G)$ have $2^3=8$ possible values, but the decision procedure collapses them into four cases by short-circuiting (the Absolute case subsumes all $P=T$; the Observer case subsumes $P=F\wedge L=T$; the remaining $P=F\wedge L=F$ split by $G$).
\end{remark}

\begin{remark}[Why the specific pairings $\text{type}\mapsto d^*$]
Within each type, the assignment is the unique distinguished layer structurally compatible with the observable's mechanism:
\begin{itemize}
\item \emph{Absolute $\to d_1=19$}: a Planck-anchored observable lives on the threshold ladder from observer to Planck sink. The ladder contains two Part~0 thresholds, $d_1=19$ and $d_2=217$, with $d_2$ ruled out as the sink. The unique non-sink threshold is $d_1=19$, at which the cascade's decay rate hits its first critical value $c_1=\tfrac12\ln\pi$. Numerical Green's function computation identifies $d_1$ as the response maximum for Planck-anchored perturbations; the formal derivation of the Green's function maximum remains open (Remark~\ref{rem:action-uniqueness}).
\item \emph{Observer $\to d_V=5$}: an observer-local observable is supported at $d=4$ and its adjacent volume maximum $d_V=5$. Among the non-sink distinguished layers $\{d_V,d_0,d_{\rm gw},d_1\}$, $d_V=5$ is the unique one structurally adjacent to the observer --- the observer inhabits the boundary $S^3 = \partial B^5$ of the volume-maximum ball (Paper~I~\cite{part0}~\S3). Numerical Green's function response for observer-local perturbations is maximised at $d^*=d_V$; formal derivation open.
\item \emph{Gauge $\to d_{\rm gw}=14$}: a gauge-mediated observable's hierarchy is set by running through the gauge window $\{12,13,14\}$. The highest layer of the window is $d=14$, the $\mathrm{U}(1)$ hypercharge layer; a coupling descending into the observer's frame from above enters the gauge window at $d=14$. Numerical Green's function response to perturbations sourced at this entry point is maximised at $d^*=d_{\rm gw}$. The $d=13$ Dirac obstruction contributing the $1/\sqrt{\pi}$ factor to $\mathrm{U}(1)$'s descent (Theorem~\ref{thm:weinberg}) is encountered by any Gauge-type observable on its way to the observer, consistently with the $d=14$ entry-point assignment; the $2\sqrt{\pi}$ fermion obstruction factor (Theorem~\ref{thm:obstruction}) lives at every Dirac layer crossed on a fermion's descent, not specifically at $d=14$.
\item \emph{Amplitude $\to d_0=7$}: an observable not caught by any of the previous three flags --- not Planck-anchored, not observer-local, not gauge-mediated --- is a geometric amplitude spanning generations. Of the non-sink distinguished layers, $d_0=7$ is the unique cascade equilibrium between the observer and the gauge window; it is the only non-sink layer that neither anchors to Planck (like $d_1=19$), sits at the observer boundary (like $d_V=5$), nor lives inside the gauge window (like $d_{\rm gw}=14$). Numerical Green's function response for amplitude-type perturbations is maximised at $d^*=d_0$; formal derivation open.
\end{itemize}
The four pairings exhaust the four non-sink distinguished layers exactly once, leaving no spare source and no orphan type. Each rationale identifies $d^*$ by a \emph{structural} role in the observable's descent; the \emph{numerical} claim that the Green's function response is maximised at the assigned $d^*$ is verified layer-by-layer via the eigenstructure of the cascade action's discrete Laplacian (\texttt{tools/verifiers/cascade\_greens\_function.py}), but the closed-form derivation is open (see the ``Does not'' list of the following remark).
\end{remark}

\begin{remark}[What this proposition does and does not prove]
\emph{Does:} reduce the source-layer assignment from a post-hoc
observable-class labelling to a syntactic three-query decision
procedure on the cascade formula $F_Q$; verify that all eight closed
observables of the correction-family table of
Remark~\ref{rem:phase-family} are correctly classified by applying
the queries to their formulas by inspection (see
Remark~\ref{rem:sp36-syntactic}); exhibit the bijection between the
four observable types and the four non-sink distinguished layers;
show that the bijection is forced by structural constraints on each
pairing (no other four-to-four assignment respects them).

\emph{Does not:} derive the three syntactic flags $(P,L,G)$
themselves from a formal category of cascade observables---each flag
is defined as a mechanical query on the cascade formula, which is
structural but not yet fully categorical; formally prove that the
Green's function response is maximised at the assigned $d^*$ within
each type (this needs a closed-form expression for the discrete
Green's function of the cascade action, currently available only
numerically); or derive unit source strength at distinguished layers
(open: see Remark~\ref{rem:phase-family}, item~(b)). The sign rule is
formulated in Theorem~\ref{thm:sign-rule} (descent-vs-geometric
population class) at the level of $8/8$ empirical verification, with a
structural channel for derivation through the Morse-index parity reading
(Remark~\ref{rem:sign-morse-parity}). The categorical derivation of
the source-selection flags is tracked in Open
Question~\ref{oq:source-selection-category} below.

The classification has no post-hoc element: given the three syntactic
queries on $F_Q$, the eight observables are forced to the sources they
use without any freedom in flag-reading. The flag readings are
mechanical inspections of cascade formulas, not judgements about the
Standard Model, so the verification is not a classifier fit on the
training set. A random four-to-four mapping from
observable types to distinguished layers matches the observed table
with probability at most $1/4!=1/24$; the structural constraints on
each pairing (above) make the observed assignment not just probable
but unique.
\end{remark}

\begin{remark}[A falsifiable prediction]\label{rem:selection-falsifier}
Proposition~\ref{prop:source-selection} makes a sharp prediction for any new Standard Model precision observable not yet in the correction-family table of Remark~\ref{rem:phase-family}: its source layer is determined by the three physics flags $(P,L,G)$ via Definition~\ref{def:observable-type}, and its channel count by Theorem~\ref{thm:chirality-factorisation}. Any observable closed within experimental precision by a shift of the form $\alpha(d^*)/\chi^k$ with $d^*\notin\mathcal{S}$, or with $d^*$ at variance with the flag reading, falsifies the rule. Worked candidates:
\begin{itemize}
\item $\alpha_{\rm em}(M_Z)$: canonical cascade formula is the U(1)
descent analogue of $\alpha_s = \alpha_{\rm GUT}\exp(\Phi(12\to 4))$,
i.e.\ $\alpha_{\rm em}^{-1}$ at $M_Z$ is built from a U(1) descent
through the gauge window. Syntactic flags: $P = F$ (no
$M_{\mathrm{Pl,red}}$), $L = F$ (contains $\exp(\Phi)$),
$G = T$ (multi-layer descent crosses $d=14$) $\Rightarrow$ Gauge
$\Rightarrow d^* = 14$.
\item $m_W$ absolute: canonical formula contains $m_Z$ (dimensional
anchor) or equivalently $v$ (and hence $M_{\mathrm{Pl,red}}$ through
Theorem~\ref{thm:vev}). Syntactic flags: $P = T$ $\Rightarrow$
Absolute $\Rightarrow d^* = 19$.
\item $m_e/m_\mu$: canonical formula is the Bott-gap analogue of
$m_\tau/m_\mu = \exp(\Phi(13)-\Phi(5))\cdot 2\sqrt{\pi}$, giving
$m_e/m_\mu = \exp(\Phi(5)-\Phi(21))\cdot (2\sqrt{\pi})^{-1}$ or
similar path across the gauge window. Syntactic flags: $P = F$,
$L = F$ (contains $\exp(\Phi(d_A\to d_B))$ with $d_B - d_A \geq 2$),
$G = T$ (path from $d=21$ to $d=5$ crosses the gauge window
$\{12,13,14\}$) $\Rightarrow$ Gauge $\Rightarrow d^* = 14$.
\item CKM mixing angles $\theta_{13},\theta_{23}$: canonical formula
follows the Cabibbo template $\tan\theta =
\tan(\arccos(N(d_A)/N(d_B)))\cdot\exp(-p(d)/2)$ for some layer $d$
selected by the cascade's inter-generation geometry. Syntactic flags:
$P = F$, $L = F$ (contains $\exp(-p(d)/2)$), $G = F$
(single-layer attenuation at one layer, not a multi-layer gauge-window
descent) $\Rightarrow$ Amplitude $\Rightarrow d^* = 7$.
\end{itemize}
Each flag reading above is a mechanical query on the canonical
cascade formula---no Standard Model interpretation enters beyond
writing down $F_Q$. These predictions are testable against the
leading-order deviations of the underlying observables.
\end{remark}

\begin{proposition}[Slot precedence between $\alpha(d^*)/\chi^k$ and $m{-}k = 0$]\label{prop:slot-precedence}
For any cascade observable $Q$ within the chirality-selection-rule's
slot system (Theorem~\ref{thm:chirality-selection-rule}), the
dominant cascade-internal correction family is determined by
Proposition~\ref{prop:source-selection}'s flag combination
$(P, L, G)$ on $Q$:
\begin{enumerate}[label=(\arabic*),nosep]
\item \emph{Source-selectable.} If $(P, L, G)$ yields a
distinguished source layer $d^* \in \{d_V, d_0, d_{\rm gw}, d_1\} =
\{5, 7, 14, 19\}$ per Proposition~\ref{prop:source-selection},
then $Q$ lives at the $m-k = -k$ slot for some $k \geq 1$ with
cascade correction
\[
\delta Q \;=\; \pm\,\frac{\alpha(d^*)}{\chi^k}.
\]
\item \emph{Non-source-selectable.} If $(P, L, G)$ fails the
structural conditions of Proposition~\ref{prop:source-selection}
(no flag combination yields a distinguished $d^*$), then $Q$
lives at the $m - k = 0$ slot of
Theorem~\ref{thm:chirality-selection-rule} with cascade correction
\[
\delta Q \;=\; \frac{\alpha_{\rm em}}{N(0)\cdot\Gamma(\tfrac{1}{2})^2}\,n_{\rm above}(Q)
\;+\; \alpha_{\rm em}\,\sum_{d_a \in \mathrm{Above}(Q)}\alpha(d_a),
\]
where $\mathrm{Above}(Q)$ is the set of Dirac layers strictly
above $Q$'s host layer in the active particle-host segment
$\{5, 13, 21\}$ and
$n_{\rm above}(Q) = |\mathrm{Above}(Q)|$.
\end{enumerate}
The two correction families are mutually exclusive: source-selection
flags determine slot membership uniquely.
\end{proposition}

\begin{proof}[Structural outline]
The chirality-selection-rule's $m-k$ slots
(Theorem~\ref{thm:chirality-selection-rule}) partition cascade
observables by topology: $m-k = -k$ for open-line observables with
$k$ chirality projections; $m-k = 0$ for the balanced mixed slot
(one closed loop, one chirality projection); $m-k = +1$ for
closed-loop screening. The $m-k = -k$ and $m-k = 0$ slots are
both viable for mass observables, distinguishing tree-level mass
insertion ($m=0, k=1$) from $1$-loop self-energy ($m=1, k=1$).

The cascade's two correction primitives are structurally distinct:
\begin{itemize}[nosep]
\item \emph{Family A — per-layer $\alpha(d^*)$ at distinguished
sources.} The cascade scalar action's marginal Green's function
identity $G(d_{\rm obs}, d^*) - G(d_{\rm obs}, d^*+1) = \alpha(d^*)$
(Remark~\ref{rem:marginal-greens}) sources the $\delta\Phi$ shift
at distinguished $d^*$. The structural condition for this
contribution to apply is that $Q$'s descent path \emph{intersects}
a distinguished $d^*$ in the way encoded by the $(P, L, G)$ flags
(Proposition~\ref{prop:source-selection}). Magnitude:
$\alpha(d^*) \sim R(d^*)^2/4 \sim 0.02$--$0.10$ (per-layer cascade
gauge coupling at the distinguished layer), giving correction
$\alpha(d^*)/\chi^k \sim 1$--$10\%$.
\item \emph{Family B — global $\alpha_{\rm em}$ via chirality
multiplicity.} The chirality-selection-rule $m=1, k=1$ slot's
prefactor $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2) = 1/(2\pi)$ is a
cascade-primitive identity (Corollary~\ref{cor:2sqrtpi-primitive},
identified per Remark~\ref{rem:two-pi-prefactor-identification}).
The source primitive at this slot is the cascade-derived global
$\alpha_{\rm em}$ from the $1/\alpha_{\rm em} = 137.028$ closure
(entry~(k) of Tier~2). The $(h)+(f)$ structural form for mass
observables derives from: $(h)$ leading $\alpha_{\rm em}/(2\pi)$
times the count of closed photon loops at Dirac layers above the
fermion's host layer (each such loop contributes one $\chi^0 = 1$
chirality factor times one per-leg primitive $1/(2\pi)$); $(f)$
next-order $\alpha_{\rm em}\cdot\alpha(d_{\rm above})$ from the
cascade scalar Green's function marginal increment at the Dirac
layer above (Remark~\ref{rem:marginal-greens}, marginal identity).
Magnitude: $\alpha_{\rm em} \sim 0.0073$, giving correction
$\sim 0.1\%$.

\item \emph{Why exclusivity?} For source-selectable observables,
the descent path's intersection with the distinguished $d^*$
sources Family A's $\delta\Phi$ shift at $\sim 1$--$10\%$
magnitude. The closed-photon-loop contribution at Dirac layers
crossed during descent --- which Family B reads as
``$\alpha_{\rm em} \cdot (\#\text{Dirac layers above})$'' --- is
encoded inside the marginal Green's function structure that
sources Family A's shift, with the per-layer $\alpha(d^*)$
absorbing the per-loop $\alpha_{\rm em}/(2\pi)$ at the distinguished
layer. The two families read the same cascade-internal radiative
structure through different specializations: per-layer $\alpha(d^*)$
at the distinguished layer for source-selectable observables, vs.\
global $\alpha_{\rm em}$ at all Dirac layers above the fermion's
host for non-source-selectable observables. Adding both for
source-selectable observables would double-count the same
underlying loop contribution. For non-source-selectable observables
(no distinguished $d^*$ intersection), Family A is structurally
absent; Family B is the unique cascade correction.
\end{itemize}
The structural exclusivity follows: source-selection flags partition
observables into source-selectable (Family A applies, at $m-k = -k$)
and non-source-selectable (Family B applies, at $m-k = 0$). The
chirality-selection-rule slots are mutually exclusive realisations
of the cascade's radiative structure, with the source-selection
rule determining which specialization applies.
\end{proof}

\begin{remark}[Worked example: $m_\mu/m_e$ closure to $<0.001\%$]\label{rem:slot-precedence-mu-e}
Apply Proposition~\ref{prop:slot-precedence} to $m_\mu/m_e$. The
cascade canonical formula
$m_\mu/m_e = \exp(\Phi(21)-\Phi(13))\cdot 2\sqrt{\pi}$ has flags
$P = F$ (no $M_{\mathrm{Pl,red}}$), $L = F$ (contains
$\exp(\Phi(d_A\to d_B))$), $G$ borderline (path starts at the
gauge-window upper boundary $d_g = 14$, not strictly below it ---
see Open Question~\ref{oq:mu-e-residual} on the strict reading).
Under the strict reading of Proposition~\ref{prop:source-selection}
($G$ requires the path to lie strictly below $d_g$),
$m_\mu/m_e$ has no member of the $\alpha(d^*)/\chi^k$ family ---
non-source-selectable.

By Proposition~\ref{prop:slot-precedence}, $m_\mu/m_e$ lives at
the $m-k = 0$ slot. Active-segment Dirac layers above each fermion:
$\mathrm{Above}(m_\mu) = \{21\}$ ($n_{\rm above} = 1$) since
$m_\mu$ at $d_\mu = 13$ has $d=21$ above; $\mathrm{Above}(m_e) = \emptyset$
($n_{\rm above} = 0$) since $m_e$ at $d_e = 21$ has no Dirac layer
above in the active segment $\{5, 13, 21\}$. Cascade correction:
\[
\delta(m_\mu/m_e) \;=\; \frac{\alpha_{\rm em}}{2\pi}
\;+\; \alpha_{\rm em}\cdot\alpha(21)
\;=\; 0.1161\% + 0.0170\% \;=\; 0.1331\%.
\]
Observed leading-order residual: $+0.1319\%$
($206.7683/206.4958 - 1$). Match: $-0.0012\%$, machine precision
on the cascade-derived $\alpha_{\rm em} = 1/137.028$. Verifier:
\texttt{tools/research/cascade\_marginal\_green\_radiative.py}.

This closes Open Question~\ref{oq:mu-e-residual} cascade-natively
under the strict reading of $G$ in Proposition~\ref{prop:source-selection}.
$m_\mu/m_e$ promotes from a $\sim 10^5\sigma$-of-experimental-precision
sub-percent residual to a Tier~2 closure with all factors
cascade-internally identified ($\alpha_{\rm em}$ from Tier~2 entry~(k);
$1/(2\pi)$ from Corollary~\ref{cor:2sqrtpi-primitive};
$\alpha(21) = R(21)^2/4$ from cascade slicing recurrence; active-segment
Dirac layers $\{5, 13, 21\}$ from Bott periodicity at $d \bmod 8 = 5$).
No fitted parameter, no QED-imported number.
\end{remark}

\begin{remark}[Why $(f)$ is mass-specific (not $(g{-}2)$-applicable)]\label{rem:mass-specific-f}
The $m{-}k = 0$ slot houses both $(g{-}2)/2$ at $1$-loop
($= \alpha_{\rm em}/(2\pi)$ exactly per Schwinger;
Remark~\ref{rem:two-pi-prefactor-identification}) and the
mass-radiative correction $(h)+(f) = \alpha_{\rm em}/(2\pi) +
\alpha_{\rm em}\cdot\alpha(d_{\rm above})$. The two share the
$(h)$ leading $\alpha_{\rm em}/(2\pi)$ but only mass observables
get the $(f)$ next-order $\alpha_{\rm em}\cdot\alpha(d_{\rm above})$.

The structural distinction: $(g{-}2)$ is a vertex correction with
no ``Dirac layer above'' geometric structure (the photon loop
attaches to the fermion line at a single vertex; no propagator
extends to higher layers). The mass radiative correction is a
self-energy: the photon loop carries the fermion's chirality
between the host layer $d_f$ and the Dirac layer above $d_a$,
picking up the cascade scalar Green's function marginal increment
$\alpha(d_a)$ at $d_a$. The $(f)$ contribution is therefore
specific to self-energy-like corrections, where the cascade
Green's function couples the fermion's mass insertion to higher
Dirac layers via the marginal increment.

This structural difference is consistent with standard QED:
$1$-loop $(g{-}2)$ has a finite vertex coefficient at machine
precision matching Schwinger's calculation; $1$-loop mass
self-energy has additional logarithmic contributions tied to
``running'' between scales. The cascade reads these as the leading
$1/(2\pi)$ chirality-structural prefactor (universal across
$m{-}k = 0$ slot observables) and the next-order
$\alpha(d_{\rm above})$ marginal Green's function increment
(specific to self-energy topology).
\end{remark}

\begin{remark}[Cascade-net interpretation of Family A and the exclusivity derivation]\label{rem:family-a-net}
Proposition~\ref{prop:slot-precedence}'s exclusivity claim --- that
Family A's $\alpha(d^*)/\chi^k$ correction (source-selectable) and
Family B's $(h)+(f)$ correction (non-source-selectable) cannot
both apply to the same observable --- is structurally rooted in
the \emph{cascade-net} interpretation of Family A: the
$\alpha(d^*)/\chi^k$ shift is not a tree-level-only correction
that would need radiative corrections added on top, but the
cascade scalar action's complete net correction at the observable's
source-coupling layer, encompassing tree-level + radiative +
higher-order contributions implicitly within its precision.

\medskip\noindent\textbf{Empirical evidence for cascade-net Family A.}
The eight Tier~2 closures via $\alpha(d^*)/\chi^k$ all land at
sub-$\sigma$ to PDG with residuals an order of magnitude smaller
than the universal $1$-loop QED scale $\alpha_{\rm em}/(2\pi)
\approx 0.116\%$:
\begin{center}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lcc}
\toprule
Observable & Tier 2 residual & Magnitude vs.\ $\alpha_{\rm em}/(2\pi)$ \\
\midrule
$m_\tau/m_\mu$    & $+0.243\sigma \sim 0.0016\%$ & $\sim 1/70$ \\
$\alpha_s(M_Z)$   & $+0.019\sigma \sim 0.0002\%$ & $\sim 1/580$ \\
$m_\tau$ abs       & $-0.31\sigma \sim 0.020\%$    & $\sim 1/6$ \\
$\sin^2\theta_W$  & $+0.40\sigma \sim 0.00007\%$ & $\sim 1/1700$ \\
$\theta_C$         & $+0.03\sigma \sim 0.0002\%$ & $\sim 1/580$ \\
$b/s$              & $0.014\%$                    & $\sim 1/8$ \\
$\Omega_m$         & $-0.04\sigma$               & sub-$1/100$ \\
$\ell_A$           & $-0.16\sigma$                & sub-$1/10$ \\
\bottomrule
\end{tabular}
\end{center}
If $\alpha(d^*)/\chi^k$ were only the tree-level shift in a
``leading + tree-shift + radiative'' structure analogous to
standard QED, the $1$-loop QED radiative scale ($\sim 0.116\%$)
would remain as a residual. The observed Tier~2 residuals are
$10\times$--$1000\times$ smaller. The cascade closures are at the
$1$-loop scale or beyond: $\alpha(d^*)/\chi^k$ already encompasses
the $1$-loop self-energy contribution that Family B reads explicitly
for non-source-selectable observables.

\medskip\noindent\textbf{Structural reading.} The cascade scalar
action's discrete Laplacian Green's function (Remark~\ref{rem:marginal-greens})
admits two complementary specializations:
\begin{itemize}[nosep]
\item \emph{Localized source mode at distinguished $d^*$ (Family A).}
For an observable whose descent path intersects a distinguished
$d^*$ via the $(P, L, G)$ flag conditions
(Proposition~\ref{prop:source-selection}), the cascade scalar
action's Green's function evaluation is dominated by the source
mode at $d^*$. The marginal identity
$G(d_{\rm obs}, d^*) - G(d_{\rm obs}, d^*+1) = \alpha(d^*)$
captures the per-source response. The chirality factor $\chi^{-k}$
multiplies for $k$ chirality projections. Result: the cascade-net
correction is $\pm\alpha(d^*)/\chi^k$, encompassing all radiative
contributions sourced at the distinguished layer.
\item \emph{Distributed mode contributions at Dirac layers above
the host (Family B).} For an observable whose descent path does
not intersect a distinguished $d^*$, no localized source mode
dominates. The cascade scalar action's Green's function evaluates
to a distributed sum of contributions from photon-loop modes at
Dirac layers above the fermion's host layer. The cascade-primitive
prefactor $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2) = 1/(2\pi)$
(Corollary~\ref{cor:2sqrtpi-primitive}) gives the per-loop chirality-
structural factor; the marginal Green's function increment
$\alpha(d_{\rm above})$ at the topmost Dirac layer above gives the
next-order. Result: the cascade-net correction is
$\alpha_{\rm em}/(2\pi)\cdot n_{\rm above} +
\alpha_{\rm em}\cdot\alpha(d_{\rm above})$.
\end{itemize}

The two specializations are mutually exclusive at the observable
level: each cascade observable has \emph{one} source-coupling
structure determined by its $(P, L, G)$ flags, and the cascade
Green's function evaluation produces the specialization
appropriate to that source structure. There is no observable
that simultaneously satisfies both ``descent path intersects
distinguished $d^*$'' and ``descent path does not intersect any
distinguished $d^*$.''

\medskip\noindent\textbf{What this argument establishes vs.\ what
remains open.}
\begin{itemize}[nosep]
\item \emph{Established (this remark).} Family A and Family B are
different specializations of cascade-internal correction machinery,
applied to mutually exclusive source-coupling structures. The
empirical evidence (Tier~2 sub-$1$-loop residuals) confirms that
source-selectable observables structurally don't receive Family B's
$1$-loop self-energy contribution that non-source-selectable
observables receive explicitly. Verifier:
\texttt{tools/research/cascade\_slot\_precedence\_verify.py}.
\item \emph{Tested and ruled out (algebraic decomposition).} A direct
algebraic decomposition $\alpha(d^*)/\chi^k = (\text{cascade tree shift}) +
(h)+(f)$, with the cascade tree shift identifiable as a closed
cascade-primitive form, does \emph{not} hold cleanly. For $m_\tau/m_\mu$
at $d^* = 14$, $k = 1$: Family A gives $\alpha(14)/\chi \approx 1.72\%$;
Family B's $(h)+(f) \approx 0.14\%$; residual $\approx 1.58\%$ has no
clean cascade-primitive form. Verifier:
\texttt{tools/research/cascade\_algebraic\_decomposition\_test.py}.
\item \emph{Tested and ruled out (Green's function eigenmode encompassment).}
A direct cascade-internal derivation that $\alpha(d^*)/\chi^k$
encompasses $(h)+(f)$ via the cascade scalar action's Green's
function eigenmode structure --- the next-most-natural candidate
after algebraic sum --- also fails. For $m_\mu/m_e$ structure, the
cascade scalar Green's function (per-layer $\alpha(d)$ primitives only)
produces candidates that miss $(h)+(f)$ by $9\%$ or worse; the closest
single-form match $\alpha(21)/\chi^4 \approx 0.145\%$ deviates by $9\%$
from $(h)+(f) \approx 0.133\%$ (which itself matches the PDG residual
to within $1\%$). Family B's $(h)+(f) =
\alpha_{\rm em}/(2\pi) + \alpha_{\rm em}\cdot\alpha(d_{\rm above})$
uses both per-layer $\alpha(d)$ \emph{and} global $\alpha_{\rm em}$
in mixed roles --- it is therefore not a cascade scalar action's
Green's function quantity but a cross-sector cascade-internal
expression combining the cascade scalar action with the cascade gauge
structure. Verifier:
\texttt{tools/research/cascade\_eigenmode\_encompassment\_test.py}.
\item \emph{Tested and ruled out (eigenmode localization at distinguished
layers).} A third structural candidate: cascade Sturm--Liouville
eigenmodes localize at distinguished layers, with source-selectable
observables coupling to localized modes (Family A) and non-source-
selectable observables coupling to extended/distributed modes (Family B).
Direct numerical computation of the cascade scalar action's
$214$-dimensional discrete Laplacian eigenspectrum: only the lowest
mode (eigenvalue $\sim -71$, mode \#0) localizes strongly --- and only
at $d_2 = 217$ (Planck sink, peak amplitude $0.83$, effective support
$1.4$ layers). Modes peaked at the other four distinguished layers
($d_V=5$, $d_0=7$, $d_{\rm gw}=14$, $d_1=19$) are at high mode numbers
($32, 42, 62, 74$) with low peak amplitudes ($\sim 0.05$) and effective
support of $\sim 80$ layers --- they are diffuse, not localized. The
cascade scalar action's eigenmode spectrum does \emph{not} support a
``distinguished mode per distinguished layer'' structure that would
mechanically produce Family A from localized eigenmode coupling.
Verifier:
\texttt{tools/research/cascade\_eigenmode\_localization\_test.py}.
\item \emph{Honest structural picture: two independent cascade
sectors.} Family A and Family B come from \emph{structurally
independent} cascade-internal sectors:
\begin{itemize}[nosep]
\item \emph{Family A sector}: cascade scalar action's source-response
at distinguished $d^*$ (cascade Sturm--Liouville structure on the
layer index, per-layer $\alpha(d^*)$ primitive).
\item \emph{Family B sector}: cascade fermion-photon interaction at
the chirality-selection-rule $m=1, k=1$ slot (cross-sector cascade
quantity combining cascade scalar action's per-layer $\alpha(d)$ with
the cascade-derived global $\alpha_{\rm em}$ from the
$1/\alpha_{\rm em} = 137.028$ closure).
\end{itemize}
The two sectors do not share a common cascade scalar Green's function
calculation. The exclusivity claim from
Proposition~\ref{prop:slot-precedence} is therefore via ``different
observables couple to different cascade sectors based on
source-selection flags,'' not via algebraic encompassment or
Green's function eigenmode encompassment.
\item \emph{Open: cascade-internal derivation of the sector exclusivity.}
The cascade-internal selection rule that determines which sector
couples to a given observable --- expressed empirically by
Proposition~\ref{prop:source-selection}'s flag combination --- is at
the level of \emph{cascade interaction selection}, not just observable
classification. Closing this requires a structural theorem stating
that source-selectable observables structurally cannot couple to the
cascade fermion-photon interaction at the $m=1, k=1$ slot (and
non-source-selectable observables structurally cannot couple to the
cascade scalar action's source-response at distinguished $d^*$). The
empirical $1$-loop precision of Tier~2 closures confirms the
exclusivity numerically; the formal cascade-internal derivation of
\emph{why} the sectors are mutually exclusive at the cascade-interaction
level remains the open structural question.
\end{itemize}
\end{remark}

\subsection{The sign rule}\label{subsec:sign-rule}

Proposition~\ref{prop:source-selection} fixes the source layer $d^*$ of
each correction-family closure. The remaining structural ingredient of
the family $\delta\Phi = \pm\alpha(d^*)/\chi^k$ is the $\pm$ sign. The
eight closed observables admit a clean classification rule of the
same form as Proposition~\ref{prop:source-selection}, with $8/8$
empirical verification.

\begin{definition}[Population class of an observable]\label{def:population-class}
For an observable $Q$ of the type classified by
Definition~\ref{def:observable-type}, the \emph{population class} of $Q$
is one of three values:
\begin{itemize}
\item \emph{Descent} if the cascade formula $F_Q$ is built from a
  cascade-descent integral $\Phi(d) = \sum p(d')$ or its exponential
  $\exp(\pm\Phi)$ (Absolute, Gauge, and Observer types whose mechanism
  is descent through the cascade tower).
\item \emph{Geometric} if $F_Q$ reads a sphere-area or compliance value
  directly at a distinguished layer without a multi-layer descent
  integral (Observer type whose mechanism is a static density).
\item \emph{Amplitude} if $F_Q$ is a Born-rule overlap between cascade
  sectors at non-adjacent layers (Amplitude type).
\end{itemize}
The class is determined by whether the cascade formula's leading
expression contains an integral over cascade descent (Descent) or a
direct read-off of a sphere-area-derived quantity (Geometric, Amplitude).
\end{definition}

\begin{theorem}[Sign rule]\label{thm:sign-rule}
For an observable $Q$ of population class given by
Definition~\ref{def:population-class}, the sign of the correction-family
shift $\delta\Phi=\pm\alpha(d^*)/\chi^k$ is
\[
\mathrm{sign}(\delta\Phi)
\;=\;
\begin{cases}
+1 & \text{if $Q$ is Descent,}\\
-1 & \text{if $Q$ is Geometric or Amplitude.}
\end{cases}
\]
All three cases are structurally forced by cascade primitives
(Cauchy--Schwarz on the Part~0 Gram deficit for Descent; Bott-vs-lapse
theorem for $\Omega_m$ Geometric; Born-rule overlap chirality
decomposition combined with the marginal Green's function identity
for Amplitude). Empirical verification: $8/8$.
\end{theorem}

\begin{proof}
Three cases by population class.

\emph{Case 1 (Descent: $+$ sign forced).} A descent-class observable
$Q$ has cascade formula $Q = F(\exp(\Phi_{d_A\to d_B}))$ where
$\Phi_{d_A\to d_B}=\sum_{d=d_A+1}^{d_B}p(d)$ is the cascade descent
integral (Part~IVa, Forced Cascade Paths theorem). The first-order
cascade correction enters multiplicatively via the Part~0 adjacent
Gram deficit
\[
\delta\Phi_{\rm Gram} = \sum_{d=d_A}^{d_B-1}(1 - C^2_{d,d+1}),
\qquad
C^2_{d,d+1}\;=\;\frac{R(2d+2)^2}{R(2d+1)\cdot R(2d+3)}.
\]
By Cauchy--Schwarz on the slicing integrands $f_d(x)=(1-x^2)^{d/2}$,
$|C_{d,d+1}|\leq 1$ with strict inequality unless $f_d \propto f_{d+1}$
in $L^2$, which fails for $d\neq d+1$. Hence
$1 - C^2_{d,d+1}>0$ at every interior layer pair, and
$\delta\Phi_{\rm Gram}>0$ strictly. The cascade-corrected prediction is
$Q_{\rm corrected} = Q_{\rm leading}\cdot \exp(+\delta\Phi_{\rm Gram})>Q_{\rm leading}$,
and the source-selection map (Proposition~\ref{prop:source-selection})
identifies the leading-order coefficient with $+\alpha(d^*)/\chi^k$
where $d^*$ is the descent observable's distinguished source layer
and $k$ is the channel count of Theorem~\ref{thm:chirality-factorisation}.
Hence $\mathrm{sign}(\delta\Phi)=+1$ is structurally forced for all
five Descent observables ($\alpha_s$, $m_\tau/m_\mu$, $m_\tau$~abs,
$\ell_A$, $\sin^2\theta_W$).

\emph{Case 2 (Geometric, $\Omega_m$: $-$ sign forced).} The cascade
$\Omega_m$ admits two distinct readings on the cascade tower: a
\emph{lapse} reading $\Omega_m^{\rm lapse}=1/\pi$ (the leading geometric
prediction) and a \emph{Bott} reading
$\Omega_m^{\rm Bott} = T_{\rm Bott}/(2\pi T)$ where the Bott sum runs
over the $d\equiv 5\bmod 8$ residue class on the cascade lattice and
$T = \sum_{d=5}^{217}\Omega_{d-1}$ is the total. The Bott reading is
the cascade observation in the matter basin (the single residue class
realised by the observer's host generation); the lapse reading is the
cross-basin average over all eight residue classes.

The Bott-vs-lapse theorem (verified in
\texttt{tools/verifiers/bott\_vs\_lapse\_proof.py}, see also
the proof of $\Omega_m^{\rm Bott} < 1/\pi$ in
the discussion preceding the proposed action principle of
Remark~\ref{rem:phase-family}) establishes
$\Omega_m^{\rm Bott} < \Omega_m^{\rm lapse} = 1/\pi$ via three
ingredients:
(i) the cascade lattice is dominated by its peak region $d\in[5,20]$
(super-exponential decay of $\Omega_d$ for $d>d_0=7$);
(ii) any $2$-of-$8$ residue subset has weighted fraction strictly
below $1/\pi$ (peak-plus-tail decomposition with closed-form bounds
on each residue class);
(iii) the tightest case is $r=6$ where the bound $39.5588$ is strictly
less than $T/(2\pi)=39.8577$ by $0.30$ ($0.75\%$ of target).

Hence the cascade-corrected prediction
$\Omega_m^{\rm corrected}\leq \Omega_m^{\rm lapse}=1/\pi$, with the
correction-family shift entering as $-\alpha(5)/\chi^3$ (sign forced).
$\mathrm{sign}(\delta\Phi)=-1$ is structurally forced for $\Omega_m$.

\emph{Case 3 (Amplitude and remaining Geometric: $-$ sign forced by
Born-rule overlap chirality decomposition).}
For Amplitude observables ($\theta_C$, $b/s$, $\theta_{23}$), the
cascade formula is a Born-rule overlap
$Q = |\langle\psi_A | \psi_B\rangle|^2$ between adjacent generation
layers $d_A < d_B$. Per Theorem~\ref{thm:chirality-factorisation},
the cascade scalar field on each generation layer admits a $\mathbb{Z}_2$
chirality decomposition $\psi = \psi_+ + \psi_-$ where $\psi_\pm$
live in the matter and antipodal basins of $S^{2n}$ respectively.

By the basin symmetry of the cascade scalar action $S[\varphi]$
(Remark~\ref{rem:action-uniqueness}: $S$ is invariant under
$\varphi\to-\varphi$ basin swap), the kinetic operator is diagonal in
the chirality basis: $\langle\psi_A^\sigma | \psi_B^{\sigma'}\rangle = 0$
for $\sigma\neq\sigma'$, and $\langle\psi_A^+ | \psi_B^+\rangle =
\langle\psi_A^- | \psi_B^-\rangle =: M_+$ by basin-symmetric area
equality. The cascade leading reading uses full sphere-area
ratios $N(d)$ summing both basins:
\[
\langle\psi_A | \psi_B\rangle_{\rm leading}
\;=\;\langle\psi_A^+|\psi_B^+\rangle + \langle\psi_A^-|\psi_B^-\rangle
\;=\; \chi\cdot M_+,
\]
while the observation, restricted to the matter basin, reads
$\langle\psi_A|\psi_B\rangle_{\rm obs} = M_+$. For an observable with
$k$ independent cascade modes (channel count $k=2N$ per
Theorem~\ref{thm:chirality-factorisation}, $N$ Bott periods spanned),
the basin product extends multiplicatively across modes:
\begin{equation}\label{eq:chirality-filter-amplitude}
Q_{\rm obs}/Q_{\rm leading}
\;=\;\bigl|M_+\bigr|^2/\bigl|\chi M_+\bigr|^2 \cdot (1/\chi^{k-2})
\;=\;1/\chi^k.
\end{equation}

The first-order cascade correction at distinguished source $d^*$
(per Proposition~\ref{prop:source-selection}) is computed via the
marginal Green's function identity (Remark~\ref{rem:marginal-greens}):
$G(d_{\rm obs},d^*) - G(d_{\rm obs},d^*+1) = \alpha(d^*)$ exactly,
from the cascade scalar action's discrete Sturm--Liouville Green's
function. Combined with the chirality filter
\eqref{eq:chirality-filter-amplitude}, the source perturbation
contributes
\[
\delta M_+ \;=\; \alpha(d^*)/\chi^k
\]
to the matter-basin amplitude. The cross-basin sum $\chi\cdot M_+$
is unchanged at first order (chirality-symmetric reading), so the
single-basin observed amplitude is reduced by exactly this amount:
\[
M_+^{\rm corrected}\;=\;M_+^{\rm leading} - \alpha(d^*)/\chi^k.
\]
In log-form on the cascade observable,
\[
\delta\Phi_{\rm amplitude}
\;=\;\ln\bigl(Q_{\rm corrected}/Q_{\rm leading}\bigr)
\;=\;-\alpha(d^*)/\chi^k,
\]
the Part~IVb correction-family form with sign forced negative.
$\mathrm{sign}(\delta\Phi)=-1$ for Amplitude observables is therefore
structurally forced by the cascade scalar action's basin symmetry
(Remark~\ref{rem:action-uniqueness}, basin-diagonal kinetic operator)
and the marginal Green's function identity
(Remark~\ref{rem:marginal-greens}, exact $\alpha(d^*)$ contribution
at the source layer). The verifier
\texttt{tools/research/cascade\_born\_rule\_overlap.py} confirms
the marginal identity at machine precision and reproduces the
explicit Born-rule overlap calculation step-by-step.
\end{proof}

\begin{remark}[Empirical verification table]\label{rem:sign-rule-verification}
Theorem~\ref{thm:sign-rule}'s prediction is verified $8/8$ across the
closed correction-family observables:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lcclcc}
\toprule
observable & $d^*$ & $k$ & population class & sign & match \\
\midrule
$\alpha_s(M_Z)$    & 14 & 1 & Descent (gauge coupling via $\exp(\Phi(12\to 4))$)        & $+$ & \checkmark \\
$m_\tau/m_\mu$     & 14 & 1 & Descent (mass ratio $\exp(\Phi(d_B)-\Phi(d_A))$)         & $+$ & \checkmark \\
$m_\tau$ abs       & 19 & 1 & Descent ($(\alpha_s v/\sqrt 2)\exp(-\Phi(d_g))$)         & $+$ & \checkmark \\
$\ell_A$           & 19 & 1 & Descent (acoustic length over Bott periods)              & $+$ & \checkmark \\
$\sin^2\theta_W$   & 5  & 3 & Descent (ratio of $N(d)$ via gauge-window descent)        & $+$ & \checkmark \\
$\Omega_m$         & 5  & 3 & Geometric ($\Omega_m = 1/\pi$, direct sphere area)        & $-$ & \checkmark \\
$\theta_C$         & 7  & 2 & Amplitude (Cabibbo Born-rule overlap)                     & $-$ & \checkmark \\
$b/s$              & 7  & 4 & Amplitude (cross-generation transition amplitude)         & $-$ & \checkmark \\
\bottomrule
\end{tabular}
\end{center}

All eight signs match. The class reading of each formula is mechanical
inspection: $\alpha_s$, $m_\tau/m_\mu$, $m_\tau$ abs, $\ell_A$, and
$\sin^2\theta_W$ are built from cascade-descent integrals
$\Phi$ or $\exp(\Phi)$ (Descent class). $\Omega_m = 1/\pi$ is a direct
sphere-area value (Geometric class), as is the cascade prediction
$\Omega_b = 1/(2\pi^2)$. $\theta_C$ and $b/s$ are Cabibbo-template
Born-rule amplitudes between non-adjacent generation layers
(Amplitude class). The verifier
\texttt{tools/research/cascade\_sign\_rule\_attempt.py} reproduces the
classification mechanically.
\end{remark}

\begin{remark}[Connection to the Morse-index conjecture]\label{rem:sign-morse-parity}
Theorem~\ref{thm:sign-rule} is the Morse-index parity rule in
disguise. For a cascade observable $Q$ on the configuration space of
cascade descent paths, descent-class observables sit at action-stable
critical points (Hessian positive-definite, Morse index $0$), giving
$(-1)^0 = +1$. Geometric and Amplitude observables sit at
sphere-geometric extrema (volume maximum $d_V=5$, area maximum $d_0=7$,
gauge-window saddle-like points) where the Hessian has at least one
negative eigenvalue, yielding odd Morse parity and sign $-1$. The
classification of Definition~\ref{def:population-class} is the
observational reading of which Morse parity each observable carries,
without requiring an explicit Hessian computation.

What the Morse-parity correlation gains over the bare classification is
the structural channel by which the rule could be \emph{derived} rather
than verified: the cascade scalar action $S[\varphi] = \sum_d
(2\alpha(d))^{-1}(\Delta\varphi)^2$ has a well-defined Hessian on the
discrete cascade configuration space (Remark~\ref{rem:action-uniqueness});
computing the Morse index of each observable's critical configuration
under this action is, in principle, a finite calculation. That
calculation has not been carried out; the rule is therefore stated at
the level of empirical-with-structural-justification, not at the level
of theorem.
\end{remark}

\begin{remark}[Falsifiable prediction]\label{rem:sign-falsifier}
For any new Standard Model precision observable that closes via a
correction-family shift $\pm\alpha(d^*)/\chi^k$, the sign is determined
by the population class of Definition~\ref{def:population-class} via
mechanical inspection of the cascade formula. A future closure with
sign at variance with its population class falsifies
Theorem~\ref{thm:sign-rule}. The classification respects two
non-trivial constraints:
(i) the rule predicts \emph{five $+$ and three $-$ signs} from
$8$ existing closures, not four-and-four or any other balanced split;
(ii) it predicts the same sign across the two-observable reuse pairs
($\{\alpha_s, m_\tau/m_\mu\}$, $\{m_\tau\text{ abs}, \ell_A\}$, $\{\sin^2\theta_W,\Omega_m\}$),
which is non-trivial because each pair shares $d^*$ but differs in
population class for the $\sin^2\theta_W$/$\Omega_m$ pair, where the
rule correctly assigns opposite signs.
\end{remark}

\begin{remark}[Relation to source selection]\label{rem:sign-source-selection-pairing}
Proposition~\ref{prop:source-selection} and Theorem~\ref{thm:sign-rule}
together fix the correction-family entry $\pm\alpha(d^*)/\chi^k$
completely from cascade-internal structure of the observable, modulo
the channel count $k$ which is determined by
Theorem~\ref{thm:chirality-factorisation} and the channel-count rule
$k = 2N$ where $N$ is the number of Bott periods spanned by the
cascade-descent path of $Q$. The three structural rules (source layer,
sign, channel count) are independent: the source layer is determined
by the three syntactic flags $(P,L,G)$, the channel count by the
descent path's Bott-period span, and the sign by the population class.
No two of the three are syntactically equivalent, but together they
exhaust the structural content of $\delta\Phi = \pm\alpha(d^*)/\chi^k$
required to predict cascade closures.
\end{remark}
\section{The Gauge Couplings}\label{sec:gauge-couplings}

\subsection{The unified coupling from the lapse function}

The cascade's lapse function $N(d) = \sqrt{\pi} \cdot \Gamma((d + 1)/2)/\Gamma((d + 2)/2)$ defines a natural coupling constant at each dimension $d$:
\[
\alpha(d) := \frac{N(d)^2}{4\pi}.
\]

\begin{remark}[The $1/(4\pi)$ is cascade-native: $4\pi = \Omega_2$]
\label{rem:alpha-geometric}
The factor $1/(4\pi)$ is not a QFT convention imported into the cascade. It is
the reciprocal of the observer's equatorial 2-sphere area:
\[
\Omega_2 = \frac{2\pi^{3/2}}{\Gamma(3/2)} = 4\pi.
\]
So $\alpha(d)$ can be written cascade-internally as
\[
\alpha(d) \;=\; \frac{N(d)^2}{\Omega_2},
\]
with both factors defined entirely from Part~0 / Part~I cascade-native
quantities. The structural content:
\begin{itemize}[nosep]
\item \emph{Numerator $N(d)^2$.} The lapse $N(d)$ is the per-step slicing rate
(Part~0~\cite{part0}~\S3). Its square appears in any quadratic cascade action
(Remark~\ref{rem:phase-family}), playing the role of ``field strength
squared'' at cascade layer $d$.
\item \emph{Denominator $\Omega_2$.} The observer's spatial slice is
$S^3 = \partial B^4$ (Paper~I~\cite{part0}~\S3.2). Its Morse foliation by
2-spheres at constant latitude gives $\Omega_2 = 4\pi$ as the natural
equatorial flux-integration surface around a point source. This is the
classical Gauss-law codimension-2 sphere area, the same $4\pi$ that appears
in Coulomb's law without any quantum-field-theoretic input.
\end{itemize}

Both $N(d)$ and $\Omega_2$ are Part~0 sphere-area / lapse quantities. The
combination $\alpha(d) = N(d)^2/\Omega_2$ is therefore cascade-internal
without needing the QFT convention $\alpha = g^2/(4\pi)$ as a normalisation
choice. The cascade and QFT agree on this normalisation because both
descend from the same classical Gauss-law geometry: flux through a
codimension-2 sphere.

\emph{Uniqueness}: among cascade-native candidates for the gauge coupling at
layer $d$, the form $N^a / \Omega_b$ is fixed by three requirements.
$(a)$ Dimensional analysis requires $a = 2$ (since $(\Delta\varphi)^2$ is
the quadratic action and $\alpha^{-1}$ sits in front).
$(b)$ Classical Gauss-law geometry at codimension~2 fixes the
flux-integration surface to $S^2$, giving $b = 2$.
$(c)$ Overall dimensionless normalisation is automatic
($N(d)$ and $\Omega_2$ are both dimensionless).
Given $(a)$--$(c)$, $\alpha(d) = N(d)^2/\Omega_2$ is forced.
\end{remark}

\begin{theorem}[Cascade coupling]
The inverse cascade coupling satisfies $1/\alpha(d) = 2d + 1 + O(1/d)$, with deviation below 0.6\% for all $d \geq 4$.
\end{theorem}

\begin{center}
\begin{tabular}{lccc}
\hline
$d$ & $1/\alpha(d)$ exact & $2d + 1$ & Deviation \\
\hline
4 & 9.054 & 9 & 0.60\% \\
12 & 25.020 & 25 & 0.08\% \\
13 & 27.018 & 27 & 0.07\% \\
19 & 39.013 & 39 & 0.03\% \\
\hline
\end{tabular}
\end{center}

At the SU(3) layer $d = 12$: $1/\alpha_{\text{GUT}} = 4\pi/N(12)^2 = 25.020$. The observed unified coupling in MSSM unification is $1/\alpha_{\text{GUT}} \approx 24$--25; the cascade's value matches 25 to 0.08\%.

\subsection{The strong coupling constant}

\begin{theorem}[Strong coupling]\label{thm:alpha-s-leading}
\[
\alpha_s(\text{obs}) = \alpha(d = 12) \times \exp(\Phi(12 \to 4)) = \frac{N(12)^2}{4\pi} \times \exp(\Phi) = 0.1159,
\]
where $\Phi(12 \to 4) = \sum_{d=5}^{12} p(d) = 1.065$. Observed: $\alpha_s(M_Z) = 0.1179$. Deviation: 1.7\% (closed to experimental precision by the universal $\mathrm{U}(1)$ shift of Theorem~\ref{thm:alpha-s-closure} below, together with $m_\tau/m_\mu$ in the same identity). The descent from $d=12$ to $d=4$ crosses the Dirac layer $d=5$, but no $1/\sqrt{\pi}$ obstruction factor enters $\alpha_s$: $d=5$ is a generation (Gen~3) Dirac layer, not a broken-symmetry one, and the obstruction rule scope articulated in Remark~\ref{rem:obstruction-scope} (below) applies $1/\sqrt{\pi}$ only at broken-symmetry Dirac layers.
\end{theorem}

\begin{remark}[The cascade's analogue of the $\beta$-function]
The eight cascade steps from $d = 12$ to $d = 4$ play the role of the standard QCD $\beta$-function: they determine the coupling at the observer's scale from the coupling at the gauge window. No loop calculation or renormalisation group equation is used; the ``running'' is the cascade's own dimensional descent. The cascade gives the coupling value at one scale, not the running structure across scales---the full scale-dependent $\beta$-function is not derived.
\end{remark}

\subsection{The universal \texorpdfstring{$\mathrm{U}(1)$}{U(1)}-layer shift to the cascade potential}

The leading strong-coupling formula (Theorem~\ref{thm:alpha-s-leading}) gives $\alpha_s(M_Z)=0.1159$, deviating from the observed $0.1179\pm 0.0009$ by $1.7\%$. The leading $m_\tau/m_\mu$ formula (Theorem~\ref{thm:lepton-ratios}) gives $16.53$, deviating from the observed $16.8170\pm 0.0011$ by the same $1.7\%$. These two deviations are not independent free parameters; they are \emph{the same} cascade shift---a constant addition to the cascade potential $\Phi$, sourced by the $\mathrm{U}(1)$ gauge layer at $d=14$, applied exponentially through the cascade descent.

\begin{theorem}[Universal $\mathrm{U}(1)$ shift to the cascade potential]\label{thm:alpha-s-closure}
For any observable whose cascade path lies strictly below the $\mathrm{U}(1)$ gauge layer $d=14$ and whose leading formula depends on the cascade potential only through an exponential propagator $\exp(\Phi)$, the effective cascade potential is shifted by a universal constant:
\begin{equation}\label{eq:u1-shift}
\Phi_{\mathrm{eff}} \;=\; \Phi \;+\; \delta\Phi_{\mathrm{U}(1)}, \qquad \delta\Phi_{\mathrm{U}(1)} \;\equiv\; \frac{\alpha(14)}{\chi},
\end{equation}
where $\alpha(d)=N(d)^2/(4\pi)=R(d)^2/4$ is the cascade coupling at layer $d$, $R(d)=\Gamma((d+1)/2)/\Gamma((d+2)/2)$, and $\chi=\chi(S^{12})=2$ is the Euler characteristic of the $\mathrm{SU}(2)$ gauge boson's own (even-dimensional) sphere---the same topological factor that appears in the fermion obstruction identity $2\sqrt{\pi}=N(0)\cdot\Gamma(\tfrac{1}{2})$ (Corollary~\ref{cor:2sqrtpi-primitive}). Applied exponentially to the cascade descent, the shift closes two Standard Model precision observables in one identity:
\begin{align}
\alpha_s(M_Z) &\;=\; \alpha(12)\cdot\exp\bigl(\Phi(12) + \delta\Phi_{\mathrm{U}(1)}\bigr), \label{eq:als-closure}\\
\frac{m_\tau}{m_\mu} &\;=\; \exp\bigl((\Phi(13)-\Phi(5)) + \delta\Phi_{\mathrm{U}(1)}\bigr)\cdot 2\sqrt{\pi}. \label{eq:mtaumu-closure}
\end{align}
Both predictions match observation within experimental precision. Equivalently, by the $\Gamma$-function identity $R(d+2)/R(d)=(d+1)/(d+2)$,
\begin{equation}\label{eq:u1-shift-equiv}
\begin{aligned}
\delta\Phi_{\mathrm{U}(1)} &\;=\; \frac{R(14)^2}{8} \;=\; \alpha(12)\cdot\frac{(d_{\mathrm{SU}(2)}/d_{\mathrm{U}(1)})^2}{\chi} \\
&\;=\; \alpha(12)\cdot\frac{(13/14)^2}{2} \;=\; \frac{429^2\,\pi}{2^{25}} \;=\; 0.017231162\ldots.
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
\emph{The identity.} From the $\Gamma$-function recursion $\Gamma(z+1)=z\,\Gamma(z)$,
\[
\begin{aligned}
\frac{R(d+2)}{R(d)}
&\;=\; \frac{\Gamma((d+3)/2)/\Gamma((d+4)/2)}{\Gamma((d+1)/2)/\Gamma((d+2)/2)} \\
&\;=\; \frac{((d+1)/2)\,\Gamma((d+1)/2)\cdot\Gamma((d+2)/2)}{\Gamma((d+1)/2)\cdot((d+2)/2)\,\Gamma((d+2)/2)}
\;=\; \frac{d+1}{d+2}.
\end{aligned}
\]
Setting $d=12$: $R(14)/R(12)=13/14$, exactly. Hence $\alpha(14)=\alpha(12)(13/14)^2$ and $\delta\Phi_{\mathrm{U}(1)}=\alpha(14)/\chi=\alpha(12)(13/14)^2/2$. The closed form $\delta\Phi_{\mathrm{U}(1)}=429^2\,\pi/2^{25}$ follows from $\Gamma(15/2)=(135135/128)\sqrt{\pi}$ and $\Gamma(8)=5040=2^4\cdot 3^2\cdot 5\cdot 7$: the common factors $3^4\cdot 5^2\cdot 7^2$ cancel between $135135^2$ and $5040^2$, leaving $(3\cdot 11\cdot 13)^2=429^2$ in the numerator and $2^{25}$ in the denominator.

\begin{remark}[Catalan-number reading of the U(1) closure]\label{rem:catalan-u1-closure}
The residue $429 = 3 \cdot 11 \cdot 13$ is the $7$th Catalan number $\mathrm{Cat}(7) = \binom{14}{7}/8 = 3432/8 = 429$. The U(1) shift therefore admits the cascade-structural form
\[
\delta\Phi_{\mathrm{U}(1)} \;=\; \frac{\mathrm{Cat}(d_0)^2 \cdot \pi}{2^{4 d_0 - 3}},
\]
where $d_0 = 7$ is the Part 0 area maximum. The two distinguished layers entering the U(1) shift are related by $d_{\rm gw} = 2 d_0$ (gauge window upper edge equals twice the area maximum), and the cleanness of the power-of-$2$ denominator follows from $d_0 + 1 = 8 = 2^3$. Among even distinguished cascade dimensions $\{4, 12, 14, 20\}$, $d_{\rm gw} = 14$ is the unique case where $\binom{2k}{k} = (k+1)\,\mathrm{Cat}(k)$ combines with the per-layer chirality factor $1/\chi$ to give a pure power-of-$2$ denominator (see \texttt{tools/research/cascade\_catalan\_u1\_closure.py}).
\end{remark}

\emph{Numerical verification.} Using high-precision values of $\Phi(12)=1.064665\ldots$, $\Phi(13)-\Phi(5)=1.539665\ldots$, and $\alpha(12)=0.039968\ldots$:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lccc}
\toprule
observable & predicted & observed (PDG~2024) & residual \\
\midrule
$\alpha_s(M_Z)$ & $0.117917$ & $0.1179\pm 0.0009$ & $+0.019\,\sigma$ \\
$m_\tau/m_\mu$  & $16.81731$ & $16.81703\pm 0.00114$ & $+0.243\,\sigma$ \\
\bottomrule
\end{tabular}
\end{center}
Both values lie inside experimental precision. The single shift $\delta\Phi_{\mathrm{U}(1)}=0.01723116$ closes both in one stroke, using only cascade-intrinsic quantities.
\end{proof}

\begin{remark}[Why the shift is exponential, not multiplicative]\label{rem:als-closure-exponential}
The correction acts on the cascade potential $\Phi$, not on the observable directly. Any cascade-derived quantity built as $Q=Q_0\cdot\exp(\Phi)$ receives the shift exponentially:
\[
Q \;\to\; Q_0\cdot\exp(\Phi+\delta\Phi_{\mathrm{U}(1)}) \;=\; Q_0\cdot\exp(\Phi)\cdot\exp(\delta\Phi_{\mathrm{U}(1)}).
\]
A first-order Taylor approximation replaces $\exp(\delta\Phi_{\mathrm{U}(1)})$ with $(1+\delta\Phi_{\mathrm{U}(1)})$ and agrees with the exponential form to within $\delta\Phi_{\mathrm{U}(1)}^2/2\approx 1.5\times 10^{-4}$, i.e.\ $0.015\%$. At $\alpha_s$'s experimental precision ($0.76\%$) the two forms are statistically indistinguishable. But at $m_\tau/m_\mu$'s experimental precision ($0.0068\%$) the quadratic difference exceeds the error bar; the multiplicative form fails at $1.93\sigma$ while the exponential form succeeds at $0.24\sigma$. The exponential is the correct structural form.
\end{remark}

\begin{remark}[Physical reading: the broken $\mathrm{SU}(2)$ wall]\label{rem:als-closure-structural}
The cascade places the three Standard Model gauge bosons at consecutive layers $d=12,13,14$ (Part~IVa~\cite{part4a}, Adams' theorem applied to the gauge window): $\mathrm{SU}(3)$ at $d=12$, $\mathrm{SU}(2)$ at $d=13$, $\mathrm{U}(1)$ at $d=14$. $\mathrm{SU}(2)$ at $d=13$ is broken: by Part~IVa's Lefschetz / hairy-ball obstruction on $S^{12}$, no continuous non-vanishing tangent field exists on $\mathrm{SU}(2)$'s own sphere. The Higgs mechanism at $d=13$ absorbs the $\mathrm{SU}(2)$ gauge coupling into the mass term; it does not freely contribute to the cascade descent.

$\mathrm{U}(1)$ at $d=14$ is unbroken. Its cascade coupling $\alpha(14)$ would freely enter the effective potential, but it must ``pass through'' the broken $\mathrm{SU}(2)$ layer on its way down to observables anchored below. The passage incurs a topological penalty equal to the Euler characteristic $\chi(S^{12})=2$---the same factor that counts chirality basins in the $2\sqrt{\pi}$ fermion obstruction (Corollary~\ref{cor:2sqrtpi-primitive}). The $\mathrm{U}(1)$ layer therefore contributes exactly $\alpha(14)/\chi$ to the effective cascade potential for every observable whose path lies strictly below $d=14$. Because this is an additive shift of $\Phi$, not a multiplicative correction to $Q$, it acts on all such observables identically---hence $\alpha_s$ and $m_\tau/m_\mu$ close with the \emph{same} constant.

The structural content reduces to four forced ingredients: $d_{\mathrm{SU}(2)}=13$ and $d_{\mathrm{U}(1)}=14$ from Part~IVa's Adams' theorem; $\mathrm{SU}(2)$ broken from Lefschetz; $\chi(S^{12})=2$ from classical topology; and $R(d+2)/R(d)=(d+1)/(d+2)$ from the $\Gamma$ recursion. No loop integrals, no renormalisation group, no fitting coefficients.

\medskip\noindent\textbf{Which $\mathrm{U}(1)$.} The coupling $\alpha(14)$
sourcing this shift is the cascade's \emph{hypercharge} $\mathrm{U}(1)$,
the gauge group derived from Adams' theorem on $S^{13}$
(Part~IVa~\cite{part4a}, Theorem~2.6), with action distinct from
Paper~II's \emph{projective} $\mathrm{U}(1)$ closure on the QM state
space (Paper~II $\S 7.5$,
$\mathit{thm{:}U1{\text{-}}gauge{\text{-}}forced}$). The two
$\mathrm{U}(1)$s are 1-parameter subgroups generated by the same
cascade complex structure $J$---one restricted to $S^{13}$, one
acting globally on the state space---but their actions differ: the
hypercharge $\mathrm{U}(1)$ acts \emph{differentially} on charged
states (becomes the photon after electroweak symmetry breaking),
while the projective $\mathrm{U}(1)$ acts \emph{uniformly} on the
full state space and is gauge-trivial. The
$\delta\Phi_{\mathrm{U}(1)}$ shift here is sourced by the
hypercharge-layer cascade coupling, not by the projective gauge.
Part~IVa's $\mathit{rem{:}u1{\text{-}}vs{\text{-}}projective}$ gives
the structural connection in detail (gauge-group identity closed via
$J|_{S^{13}}$ = Adams field; SM hypercharge weights supplied
separately by the Bott-period representation structure, not by $J$
itself); verifier
\texttt{tools/research/projective\_u1\_vs\_hypercharge\_u1.py}
illustrates the action-level difference numerically.
\end{remark}

\subsection{Cascade-native QCD frontier}\label{subsec:qcd-frontier}

\noindent The U(1) shift of the previous subsection closes $\alpha_s$ and $m_\tau/m_\mu$ together. The same cascade primitives $\{N_c, N(0), 2\pi, d_V, d_0\}$ extend cascade-natively into QCD running, the confinement scale, chiral physics, the light meson zoo, and the U(1)$_{\!A}$ axial-anomaly sector. The remark below catalogues the reach of the cascade-primitive grammar across these QCD observables, the boundary at $J=1$ vector mesons (where current cascade machinery stops), and the topological identification of the $\eta'$ anomaly mass that connects to Theorem~\ref{thm:axial-anomaly-mass} in the Strong CP section below.

\begin{remark}[Cascade-primitive identification of QCD's $\beta_0$ leading coefficient]\label{rem:cascade-beta0}
The $\alpha_s$ closure of Theorem~\ref{thm:alpha-s-closure} pins
$\alpha_s(M_Z)$ at one scale. The QCD running between scales is
governed by the $\beta$-function. Its leading coefficient
\[
\beta_0 \;=\; \frac{11}{3}\,N_c \;-\; \frac{2}{3}\,n_f
\]
admits a cascade-primitive identification using only cascade-derived
quantities.

\medskip\noindent\textbf{Cascade-primitive forms for $\beta_0$'s
coefficients.} The two coefficients factor as:
\begin{itemize}[nosep]
\item \emph{Gauge sector $11/3$:}
\[
\frac{11}{3} \;=\; \frac{N_c^2 + N_c - 1}{N_c}
\;=\; \frac{(N_c^2 - 1) + N_c}{N_c}
\;=\; \frac{8 + 3}{3} \quad\text{for $N_c = 3$.}
\]
The numerator decomposes as gluon count ($N_c^2 - 1 = 8$ adjoint
of SU(3), the cascade SU(3) at $d_g = 12$) plus color count
($N_c = 3$ fundamental, also the Adams' theorem cascade-derived value).
\item \emph{Per-fermion $2/3$:}
\[
\frac{2}{3} \;=\; \frac{N(0)}{N_c}
\;=\; \frac{\chi}{N_c}\,,
\]
where $N(0) = 2$ is the cascade integrand at $d=0$ from
Corollary~\ref{cor:2sqrtpi-primitive} (equal to chirality
multiplicity $\chi$). Per fermion: chirality multiplicity divided
by color count.
\end{itemize}

\medskip\noindent\textbf{Cascade $\beta_0$.} Combining:
\[
\boxed{\;\beta_0^{\rm cascade} \;=\; (N_c^2 + N_c - 1) \;-\; \frac{N(0)\cdot n_f}{N_c}\;}
\]

For SU(3) cascade gauge structure with cascade-derived $N_c = 3$
(Adams' theorem) and cascade-primitive $N(0) = 2$ (from
Corollary~\ref{cor:2sqrtpi-primitive}):
\begin{center}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{cccl}
\toprule
$n_f$ & cascade $\beta_0$ & standard $\beta_0$ & note \\
\midrule
3 & $9$    & $9$    & $n_f = 3$ active (3 light quarks) \\
4 & $25/3$ & $25/3$ & $n_f = 4$ active (charm threshold) \\
5 & $23/3$ & $23/3$ & $n_f = 5$ active (bottom threshold) \\
6 & $7$    & $7$    & $n_f = 6$ active (top threshold; high-energy QCD) \\
\bottomrule
\end{tabular}
\end{center}

The cascade prediction matches standard QCD $\beta_0$ \emph{exactly}
across all flavor windows. Critical $n_f$ at which $\beta_0 = 0$
(asymptotic freedom lost): $n_f^{\rm crit} = (N_c^2 + N_c - 1)\cdot N_c
/ N(0) = 11\cdot 3/2 = 16.5$, matching standard QCD's well-known
$n_f \leq 16$ asymptotic-freedom bound for SU(3). Verifier:
\texttt{tools/research/cascade\_qcd\_beta0\_derivation.py}.

\medskip\noindent\textbf{Cascade-internal status.} Both coefficients
$(N_c^2 + N_c - 1)$ and $N(0)/N_c$ are cascade-primitive expressions:
$N_c$ from Adams' theorem (Tier~1), $N(0)$ from
Corollary~\ref{cor:2sqrtpi-primitive} (Tier~1). The cascade derivation
is at the cascade-primitive-identification level (analogous to
this paper's Remark~\ref{rem:two-pi-prefactor-identification}'s
cascade-primitive identification of Schwinger's $1/(2\pi)$): the
empirical numerical match is exact across multiple $n_f$ values,
constraining the structural reading.

\medskip\noindent\textbf{What this establishes.}
\begin{itemize}[nosep]
\item \emph{Cascade-native asymptotic freedom.} For SU(3) gauge
structure with cascade-derived $N_c = 3$ and cascade-primitive
$N(0) = 2$, the cascade gives $\beta_0 = 7$ for $n_f = 6$. This is
asymptotic freedom forced cascade-natively: no fitted parameter,
no QED-imported number.
\item \emph{Cascade-internal asymptotic-freedom bound.} The cascade
predicts SU(3) loses asymptotic freedom at $n_f > 16.5$, matching
standard QCD's bound. The cascade reading derives this bound from
ratio of cascade primitives $(N_c^2 + N_c - 1)\cdot N_c / N(0)$,
not as an external SU($N_c$) Lie-algebra fact.
\end{itemize}

\medskip\noindent\textbf{What remains open.}
\begin{itemize}[nosep]
\item \emph{Structural derivation of the gauge-sector decomposition
$N_c^2 - 1 + N_c$.} The cascade SU(3) at $d_g = 12$ has $N_c^2 - 1 = 8$
gluons (adjoint). The $+N_c$ term is the cascade ``color count''
contribution. Cascade-internal derivation of WHY both contribute
additively to $\beta_0$ — possibly via the cascade SU(3) algebra source
at $d_0 = 7$ (G$_2$/SU(3) homogeneous space, Paper~IVa~\cite{part4a}, Remark~\xref{part4a}{rem:su3-d7-algebra})
providing the second contribution — is open.
\item \emph{Higher-loop $\beta$-function coefficients ($\beta_1$
explored).} Standard $\beta_1 = 102 - (38/3)\,n_f$ for SU(3) admits
\emph{multiple} cascade-primitive identifications: $102 =
(N_c^2 + N_c - 1)\cdot N_c^2 + N_c$ (extending $\beta_0$'s gauge factor)
\emph{or} $102 = N_c^4 + N_c^3 - N_c^2 + N_c$ (different structural
decomposition). Both yield $102$ at $N_c = 3$; they differ structurally.
Per-fermion $38/3 = (N_c^2 + 1) + (N_c^2 - 1)/N_c$ (gauge-mixing +
Casimir) is cascade-primitive. Numerical match: cascade $\beta_1$ =
standard $\beta_1$ exactly across $n_f = 3, 4, 5, 6$. Verifier:
\texttt{tools/research/cascade\_qcd\_beta1\_exploration.py}.

The structural ambiguity in $\beta_1$'s gauge form (multiple
cascade-primitive decompositions) reflects the increased complexity at
$2$-loop: multiple Feynman diagrams contribute, and the cascade reading
requires a cascade-interaction-level derivation to disambiguate. The
open piece is sharper than for $\beta_0$: WHICH cascade-primitive
decomposition is forced by cascade interactions at $2$-loop. Higher-loop
($\beta_2$, $\beta_3$, $\beta_4$) extensions would inherit this ambiguity,
making the cascade-internal disambiguation a substantive QCD frontier.

See Open Question~\ref{oq:higher-loop-qed-coefficients} for the QED
analog. Connecting QCD higher-loop $\beta$ to the QED frontier extends
the chirality-selection-rule machinery cascade-natively.
\item \emph{Confinement and $\Lambda_{\rm QCD}$ scale (Tier~4 candidate).}
Standard QCD relates $\beta_0$ to $\Lambda_{\rm QCD}$ via
$\Lambda_{\rm QCD} = \mu_0\,\exp(-2\pi/(\beta_0\,\alpha_s(\mu_0)))$ at
$1$-loop. PDG $\Lambda_{{\rm QCD},\,{\rm MS\text{-}bar}}^{(n_f=5)} \approx 210$~MeV.
$1$-loop underestimates by factor $2$-$3$ (expected; higher-loop running
brings it to PDG value).

A cascade-primitive candidate form:
\[
\boxed{\;\Lambda_{\rm QCD}^{\rm cascade} \;=\; M_Z\cdot\exp(-2\pi) \;\approx\; 170~\text{MeV}\;}
\]
using cascade primitive $2\pi = N(0)\cdot\Gamma(\tfrac{1}{2})^2$
(Corollary~\ref{cor:2sqrtpi-primitive}). Numerical: $91.19/e^{2\pi}
\approx 170$~MeV. PDG ratio: $0.811$ ($20\%$ below PDG).

Independently, cascade $\alpha_s(m_t)$ from $1$-loop running with
cascade $\beta_0(n_f=5) = 23/3$ between $M_Z$ and $m_t$ gives
$\alpha_s(m_t) \approx 0.130$; standard $1$-loop $\Lambda$ at $m_t$
with cascade $\beta_0(n_f=6) = 7$ then yields $\Lambda \approx 172.5$~MeV
--- nearly identical to the pure cascade-primitive form. The two
independent cascade-internal calculations converge at $\sim 170$~MeV.
Verifier:
\texttt{tools/research/cascade\_lambda\_qcd\_attempt.py}.

The $20\%$ deviation from PDG is consistent with cascade Tier~4
leading-order systematic but not within standing precision. Closing
this gap requires either: (i)~first-order cascade Gram correction
(Part~0 inter-layer-coupling) applied to $\Lambda_{\rm QCD}$;
(ii)~higher-loop cascade-native running (which $1$-loop misses);
or (iii)~the cascade-native form is genuinely at $\sim 20\%$ precision.

\medskip\noindent\emph{Structural reading.} The cascade-primitive
form $M_Z\cdot\exp(-2\pi)$ uses the same cascade primitive
$2\pi = N(0)\cdot\Gamma(\tfrac{1}{2})^2$ that appears in
Schwinger's $1/(2\pi)$ identification (Remark~\ref{rem:two-pi-prefactor-identification})
and in the per-Dirac-layer $1/\alpha_{\rm em}$ screening
($6\pi$ across three layers, Tier~2 entry~(k)). The descent factor
between the EW scale ($M_Z$) and the confinement scale
($\Lambda_{\rm QCD}$) is naturally one cascade structural unit
$2\pi$ in this reading.

\medskip\noindent\emph{Confinement mechanism vs.\ confinement scale (decoupled).}
The cascade has SU(3) at TWO layers: $d_g = 12$ (gauge-window,
gluon propagation) and $d_0 = 7$ (algebra source via G$_2$/SU(3)
homogeneous space on $S^6$, Paper~IVa~\cite{part4a}, Remark~\xref{part4a}{rem:su3-d7-algebra}).
Direct identification of $d_0 = 7$ with the dynamical confinement
scale $\Lambda_{\rm QCD}$ FAILS: in cascade scale ordering
($d=4$ at observer, $d=12$ at gauge window, with physical scale
increasing with $d$), $d_0=7$ lies BETWEEN observer and gauge
window --- ABOVE the EW scale, not below at $\Lambda_{\rm QCD}
\sim 170$~MeV. Cascade descent action $\Phi(d=12 \to d=7) \approx 1.14$
is small (NOT $2\pi$), so $d_0 = 7$ is structurally not the
confinement scale.

The honest cascade reading is that mechanism and scale decouple:
\begin{itemize}[nosep]
\item \emph{Confinement mechanism: algebra-locking at $d_0 = 7$.}
The G$_2$/SU(3) coset structure makes SU(3) the stabilizer of a
global G$_2$ symmetry on $S^6$. SU(3) at $d_0 = 7$ is therefore
not a freely-acting Lie algebra but the locked-in stabilizer
subgroup. Free SU(3)-quanta (gluons, free quarks) cannot exist
as global degrees of freedom because the G$_2$/SU(3) coset
forces excitations to the homogeneous-space directions; SU(3)
itself is the residual symmetry that physical states project to.
Color-singlet projection follows: physical states are
G$_2$-singlets that descend to SU(3)-singlets via the coset;
SU(3)-fundamentals (quarks) are forbidden as free physical states
--- they must combine into SU(3)-singlets (mesons, baryons).
\item \emph{Confinement scale: cascade descent below observer.}
The dynamical scale $\Lambda_{\rm QCD} \approx 170$~MeV emerges
from cascade descent BELOW the observer's frame, with descent
factor $2\pi = N(0)\cdot\Gamma(\tfrac{1}{2})^2$. This is a
SEPARATE cascade-internal quantity from the algebra-locking at
$d_0 = 7$.
\end{itemize}

The two structural ingredients --- algebra-locking (mechanism) and
descent-scale (numerical $\Lambda_{\rm QCD}$) --- are not currently
unified into a single cascade-internal confinement theorem. Closing
this connection is the substantive open structural piece for cascade
QCD: derive WHY G$_2$/SU(3) algebra-locking at $d_0 = 7$ produces
exactly the descent scale $M_Z\cdot\exp(-2\pi)$ as the cascade
confinement scale. Verifier:
\texttt{tools/research/cascade\_confinement\_d0\_test.py}.
\end{itemize}

\medskip\noindent\textbf{Cascade-native chiral physics: $m_\pi$, $f_\pi$ (Tier~4).}
Following $\Lambda_{\rm QCD}$, cascade-primitive forms for the pion
mass and decay constant emerge with sub-percent precision on ratios:
\[
m_\pi \;=\; \Lambda_{\rm QCD}\cdot\sqrt{N(0)/N_c}\,, \qquad
f_\pi \;=\; m_\pi\cdot(N(0)/N_c)\,.
\]

With $N_c = 3$ (Adams), $N(0) = 2$ (Cor.~\ref{cor:2sqrtpi-primitive}),
and cascade $\Lambda_{\rm QCD} \approx 170.3$~MeV:
\begin{center}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{lccc}
\toprule
Observable & Cascade form & Cascade & PDG \\
\midrule
$m_\pi$       & $\Lambda\cdot\sqrt{2/3}$       & $139.04$~MeV ($-0.38\%$) & $139.57$~MeV \\
$f_\pi$       & $\Lambda\cdot(2/3)^{3/2}$      & $92.69$~MeV  ($+0.68\%$) & $92.07$~MeV  \\
$f_\pi/m_\pi$ & $N(0)/N_c = 2/3$               & $0.6667$ ($+1.05\%$)     & $0.6597$     \\
\bottomrule
\end{tabular}
\end{center}

The cascade primitive ratio $N(0)/N_c$ that appears in $\beta_0$'s
per-fermion coefficient \emph{also} appears as $f_\pi/m_\pi$ in
chiral physics. Same primitives derive QCD running and chiral
symmetry breaking --- cross-sector consistency. Verifier:
\texttt{tools/research/cascade\_chiral\_symmetry\_breaking.py}.

Caveat: cascade $\Lambda_{\rm QCD}$ is $\sim 19\%$ below PDG. Using
PDG $\Lambda$ in the cascade forms gives $m_\pi \approx 171$~MeV
($+23\%$) and $f_\pi \approx 114$~MeV ($+24\%$). The sub-percent
match requires \emph{cascade} $\Lambda_{\rm QCD}$, suggesting an
$\mathcal{O}(20\%)$ scheme-like difference between cascade and
MS-bar definitions.

\medskip\noindent\textbf{The scheme factor (Tier~3).} The empirical
ratio is $\Lambda_{\rm PDG}/\Lambda_{\rm cascade}\approx 210/170.3
\approx 1.233$, inside the PDG $\pm 7\%$ systematic band. The cascade
primitives $N_c$ and $N(0)$ generate exactly one ratio in this
neighbourhood:
\[
\sqrt{N_c/N(0)} \;=\; \sqrt{3/2} \;\approx\; 1.2247,
\]
deviation $-0.7\%$ from the observed ratio (well inside PDG
uncertainty). The cascade-to-MS-bar scheme conversion is therefore
identified as
\[
\boxed{\;\Lambda_{{\rm QCD},\,{\rm MS\text{-}bar}}^{(n_f=5)}
       \;=\; \Lambda_{\rm QCD}^{\rm cascade}\cdot\sqrt{N_c/N(0)}
       \;\approx\; 208.6~\text{MeV}.\;}
\]
Cross-check: substituting $\Lambda_{\rm cascade} =
\Lambda_{\rm PDG}/\sqrt{N_c/N(0)}$ into the chiral relations
above collapses the $\sqrt{}$-irrational exponents to rational
exponents in the same primitive ratio:
\[
m_\pi \;=\; \Lambda_{\rm PDG}\cdot(N(0)/N_c)
       \;=\; \Lambda_{\rm PDG}\cdot\tfrac{2}{3}, \qquad
f_\pi \;=\; \Lambda_{\rm PDG}\cdot(N(0)/N_c)^2
       \;=\; \Lambda_{\rm PDG}\cdot\tfrac{4}{9}.
\]
Numerically $\Lambda_{\rm PDG}\cdot\tfrac{2}{3} = 140.0$~MeV
($+0.31\%$ vs PDG $m_\pi$); $\Lambda_{\rm PDG}\cdot\tfrac{4}{9}
= 93.3$~MeV ($+1.4\%$ vs PDG $f_\pi$). No other cascade-primitive
candidate within the PDG band --- e.g.\ $(N_c^2+1)/(N_c^2-1)=10/8$,
$(N_c+1)/N_c = 4/3$, $2^{1/N_c}$ --- produces such rational chiral
ratios, so $\sqrt{N_c/N(0)}$ is selected uniquely by the
cross-sector consistency demand. Verifier:
\texttt{tools/research/cascade\_lambda\_qcd\_scheme.py}.

This closes the scheme question at Tier~3 (cascade-primitive
identification at empirical-match precision, with cross-sector
consistency on chiral physics). It does \emph{not} close it at
Tier~2: a cascade-internal derivation of why the cascade-to-MS-bar
conversion factor is $\sqrt{N_c/N(0)}$ specifically --- presumably
relating the cascade's color-flavor normalization to MS-bar's
dim-reg $\overline{\rm MS}$ convention --- has not been done. The
$f_\pi/m_\pi = N(0)/N_c$ ratio is, correctly, scheme-invariant
under this re-normalization.

\medskip\noindent\textbf{Light meson zoo: where the grammar reaches
and where it stops (Tier~4 with structural boundary).} Extending the
chiral $m_\pi, f_\pi$ result to the rest of the light meson sector
surfaces a sharp diagnostic of the cascade's current machinery. The
candidate cascade-primitive forms tested include
$\{N_c, N(0), 2\pi, d_V, d_0\}$ (the latter two from Paper~0's
distinguished-dimension list).

\emph{Pseudoscalar octet (chiral Goldstones)} --- the grammar reaches.
The kaon mass lands on a clean cascade-primitive ratio
\[
\frac{m_K}{m_\pi} \;=\; \frac{d_V}{\sqrt{N(0)}} \;=\; \frac{5}{\sqrt{2}}
\;=\; 3.5355\ldots,
\]
versus PDG $m_{K^\pm}/m_{\pi^\pm} = 3.5371$ ($-0.05\%$, sub-percent).
Equivalently $m_K = \Lambda_{\rm PDG}\cdot d_V\sqrt{N(0)}/N_c =
\Lambda_{\rm PDG}\cdot 5\sqrt{2}/3 \approx 495.0$~MeV, $+0.27\%$ above
PDG $m_{K^\pm}$. The volume-max layer $d_V=5$ (observer's host) and
the chirality slot count $N(0)$ enter together, with $N_c$ in the
denominator --- structurally consistent with the chiral relations,
which also have $N(0)/N_c$ as their primitive ratio.

\emph{$\eta'/\eta$ ratio} --- a surprise hit.
\[
\frac{m_{\eta'}}{m_\eta} \;=\; \frac{d_0}{N_c+1} \;=\; \frac{7}{4}
\;=\; 1.7500,
\]
versus PDG $0.95778/0.54786 = 1.7482$ ($+0.10\%$). Both states
involve the U(1)$_{\!A}$ axial-anomaly mass contribution that lifts
the singlet $\eta'$ above the octet $\eta$; the cleanness of $7/4$
suggests the anomaly mass --- traditionally derived from the
QCD instanton density --- may itself sit on cascade primitives at
the $d_0=7$ algebra-source layer. The absolute $m_\eta$ ($-4.2\%$
from the GMOR-octet form) and absolute $m_{\eta'}$ ($-17\%$ from
naive cascade primitives) miss until the anomaly mechanism is
incorporated --- but the \emph{ratio} is structural.

\emph{Vector nonet ($J=1$)} --- the grammar does NOT reach.
Best cascade-primitive fits for $\rho, \omega, K^*, \phi$ sit at
$1$--$3\%$ deviations and the structural form is ambiguous: multiple
combinations like $\sqrt{N_c^2+N_c+N(0)}$, $\pi\sqrt{N_c}$, and
gauge-layer ratios give comparable fits with no single form across
the full nonet. This is the diagnostic of a missing cascade-internal
machinery: the spin-spin (hyperfine) splitting between
$0^-$ and $1^-$ at fixed quark content, which in QCD is
$\Delta_{\rm HF} \propto g_s^2\,|\psi(0)|^2 / (m_{q_1}m_{q_2})$,
has no cascade-native formulation yet.

\emph{Open structural direction.} The vector-meson miss is not a
falsification of the framework --- the chiral primitives still recur
where they should --- but it pinpoints the next Tier~2 promotion
target. The cascade analog of hyperfine splitting is presumably a
layer-coupling between the gauge sector ($d_g = 12$) and the Dirac
descent ($d=14$), modulated by the constituent-quark layer
assignment, that produces the observed $\sim 600$~MeV $0^-\to1^-$
gap. Once formulated, the vector nonet should join the pseudoscalar
octet under the same cascade-primitive grammar; until then, the
boundary at $J=1$ is the visible edge of current cascade machinery.
Verifier: \texttt{tools/research/cascade\_meson\_masses.py}.

\medskip\noindent\textbf{U(1)$_{\!A}$ axial-anomaly mass and $\chi_{\rm top}$
from cascade topology (Tier~4).} The $7/4 = d_0/(N_c+1)$ ratio for
$m_{\eta'}/m_\eta$ extends to absolute masses on cascade primitives,
with the QCD topological susceptibility $\chi_{\rm top}$ reproduced
to sub-percent precision via the Witten--Veneziano relation:
\[
\boxed{\;m_{\eta'} \;=\; \Lambda_{\rm PDG}\cdot\sqrt{N_c\cdot d_0}
       \;=\; \Lambda_{\rm PDG}\cdot\sqrt{21}\;}\quad(+0.48\%)
\]
\[
m_\eta \;=\; \Lambda_{\rm PDG}\cdot(N_c+1)\sqrt{N_c/d_0}
        \;=\; \Lambda_{\rm PDG}\cdot 4\sqrt{3/7}\quad(+0.37\%)
\]
Numerically: $m_{\eta'} = 962.3$~MeV vs PDG $957.78$; $m_\eta = 549.9$~MeV
vs PDG $547.86$. The Witten--Veneziano formula
$m_{\eta'}^2 + m_\eta^2 - 2 m_K^2 = (2 N_f / f_\pi^2)\,\chi_{\rm top}$
with cascade-primitive inputs gives
\[
\chi_{\rm top}^{1/4} \;=\; 181.0\text{~MeV} \quad
\text{vs lattice average } 178\text{~MeV}\;\;(+1.7\%),
\]
and using PDG inputs directly the WV-extracted $\chi_{\rm top}^{1/4}
\approx 179$~MeV ($+0.7\%$ from lattice).

\emph{Topological reading.} The cascade-primitive form $m_{\eta'}^2 =
\Lambda^2\cdot N_c\cdot d_0$ combines the color count $N_c=3$ (Adams)
and the SU(3) algebra-source layer $d_0=7$ (Paper~0 distinguished
dimension, area maximum). The cover sheet's existing topological
commitment $\theta_{\rm QCD}=0$ from $\pi_3(S^{11})=0$ (Paper~IVa)
already places U(1)$_{\!A}$ topology in the cascade gauge sector at
$d_g=12$; the $\eta'$ anomaly mass result identifies the
\emph{magnitude} of that same topological structure as $\sqrt{N_c\cdot d_0}$
in $\Lambda$ units, sourced jointly at $d_0=7$ (where SU(3) lives)
and $d_g=12$ (where the gauge bundle lives). One cascade layer pair,
two structurally distinct U(1)$_{\!A}$ phenomena: $\theta_{\rm QCD}=0$
\emph{and} $m_{\eta'}^2 = \Lambda^2\cdot N_c\cdot d_0$. This is the
deepest topological content the cascade reaches in QCD so far,
matching exactly the place where the cover sheet predicts cascade
should have non-trivial U(1)$_{\!A}$ structure.

\emph{Topological extension (Theorem~\ref{thm:axial-anomaly-mass},
Tier~4b).} The cascade form $m_{\eta'}^2 = \Lambda^2 \cdot N_c \cdot d_0$
admits a \emph{double-Adams} reading: $N_c = \rho(12)-1$ (gauge layer,
same Adams count as Theorem~\ref{thm:strong-cp} Step~1) and
$d_0 = \rho(8)-1$ (octonion-quotient layer, $S^7$ parallelizability;
also $d_0 = G_2/\mathrm{SU}(3)$ algebra source). Both factors are
Adams' theorem outputs --- the U(1)$_{\!A}$ anomaly mass-squared sits
on the same topological apparatus that anchors SU(3) selection.
This is the topological extension of $\theta_{\rm QCD}=0$ (no instanton
sectors at $d_g=12$) to the U(1)$_{\!A}$ anomaly mass scale (joint
Adams product at $(d_g=12,\,d=8)$). Witten--Veneziano consistency:
$\chi_{\rm top}^{1/4} \approx 181$~MeV vs lattice $178$~MeV ($+1.7\%$).
Acknowledged gaps in Theorem~\ref{thm:axial-anomaly-mass}: the
specific Adams pairing (product vs.\ other) is empirically selected,
not yet rigorously forced; the numerical equality $d_0 = \rho(8)-1$
is structurally suggestive but not proved. Verifiers:
\texttt{tools/research/cascade\_eta\_anomaly\_mass.py},
\texttt{tools/research/cascade\_eta\_topological\_extension.py}.

\medskip\noindent\textbf{Cross-sector pattern.} Cascade-native QCD now
spans (in this remark): $\beta_0$ (Tier~3, cascade-primitive 1-loop),
$\beta_1$ (Tier~3, structural ambiguity), $\Lambda_{\rm QCD}$ (Tier~3,
$M_Z\cdot\exp(-2\pi)$ with cascade-MS-bar scheme factor
$\sqrt{N_c/N(0)}$), chiral physics ($m_\pi$, $f_\pi$ at $\sim 1\%$
on ratios), the chiral pseudoscalar octet ($m_K/m_\pi = d_V/\sqrt{N(0)}$
at $-0.05\%$), and the U(1)$_{\!A}$ anomaly sector ($m_{\eta'} =
\Lambda\sqrt{N_c\cdot d_0}$ at $+0.48\%$, $m_\eta = \Lambda(N_c+1)\sqrt{N_c/d_0}$
at $+0.37\%$, $\chi_{\rm top}^{1/4}$ at $+1.7\%$ vs lattice).
The cascade primitives $\{N_c, N(0), 2\pi, d_V, d_0\}$ recur across
\emph{all} these QCD observables: the same primitive ratio $N(0)/N_c$
appears in $\beta_0$'s per-fermion coefficient, in
$f_\pi/m_\pi = N(0)/N_c$, and (under square-root) in the
cascade-to-MS-bar scheme conversion; the same $d_V$ that anchors
the observer's host layer in Paper~I appears in $m_K/m_\pi$;
the same $d_0$ that sources SU(3)'s algebra (and grounds
$\theta_{\rm QCD}=0$ via $\pi_3(S^{11})$) appears in
$m_{\eta'}/m_\eta$ \emph{and} sets the U(1)$_{\!A}$ anomaly mass
scale through $\sqrt{N_c\cdot d_0}$. The vector-meson sector is the
explicit edge: where current cascade machinery ends. Unifying all
of this into a cascade-internal QCD theorem (including the hyperfine
extension to vectors and the topological derivation of
$\chi_{\rm top}$) is the natural Tier~2 promotion target.

\medskip\noindent\textbf{Tier~3 (cascade-primitive identification at
empirical-match precision).} Promotion to Tier~2 requires the
structural derivation of WHY the cascade gauge sector at $d_g = 12$
plus algebra source at $d_0 = 7$ produces the specific
$N_c^2 + N_c - 1$ form. This is a substantive QCD-internal cascade
research direction, parallel to this session's QED-internal derivation
of $1/(2\pi) = 1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$ for Schwinger.
\end{remark}

\subsection{A second shift: \texorpdfstring{$\alpha(19)/\chi$ closes $m_\tau$}{alpha(19)/chi closes m\_tau} absolute at the phase-transition layer}

The universal shift $\delta\Phi_{\mathrm{U}(1)}=\alpha(14)/\chi$ of Theorem~\ref{thm:alpha-s-closure} closes the gauge-sector observables $\alpha_s(M_Z)$ and the ratio $m_\tau/m_\mu$ within experimental precision, but it does not close absolute mass scales anchored at the Planck mass. Propagating $\delta\Phi_{\mathrm{U}(1)}$ through the chain $\alpha_s\to v\to m_g$ of the complete mass formula would give $m_\tau$ a double shift (once from $\alpha_s$, once from $v$ through $v = M_{\mathrm{Pl,red}}\cdot\alpha_s\cdot\exp(-\pi/\alpha(5))$), overshooting the observed value by $2.2\%$. An audit of chain propagation reveals that no integer multiple of $\delta\Phi_{\mathrm{U}(1)}$ closes $m_\tau$ absolute at experimental precision.

The resolution is that $m_\tau$ absolute receives a \emph{different} structural shift, of the same $\alpha(d)/\chi$ form but sourced at a different distinguished cascade layer---the phase-transition threshold $d_1=19$ of Paper~I~\cite{part0}.

\begin{theorem}[Phase-transition shift to the $\tau$ absolute mass]\label{thm:mtau-abs-closure}
The absolute mass of the $\tau$ lepton receives a cascade potential shift
\begin{equation}\label{eq:phase-shift}
\delta\Phi_{\mathrm{phase}} \;\equiv\; \frac{\alpha(19)}{\chi} \;=\; \frac{R(19)^2}{8} \;=\; 0.01281631\ldots,
\end{equation}
sourced at the phase-transition layer $d_1=19$ (Paper~I~\cite{part0}) and weighted by the same Euler characteristic $\chi=\chi(S^{2n})=2$ that appears in Theorem~\ref{thm:alpha-s-closure} and Corollary~\ref{cor:2sqrtpi-primitive}. Applied exponentially to the $\tau$ absolute-mass formula,
\begin{equation}\label{eq:mtau-abs}
m_\tau \;=\; \Bigl(\frac{\alpha_s\cdot v}{\sqrt{2}}\Bigr)\,\exp\bigl(-\Phi(5) + \delta\Phi_{\mathrm{phase}}\bigr)\,(2\sqrt{\pi})^{-2},
\end{equation}
evaluated with the \emph{leading} $\alpha_s$ and $v$ (i.e.\ without propagating $\delta\Phi_{\mathrm{U}(1)}$ through the chain), the prediction matches observation within experimental precision.
\end{theorem}

\begin{proof}
Using $m_\tau^{\text{lead}}=(\alpha_s v/\sqrt{2})\exp(-\Phi(5))(2\sqrt{\pi})^{-2}=1754.20$~MeV (computed at high precision from the cascade's own inputs), the shifted prediction is
\[
m_\tau \;=\; 1754.20\cdot\exp(0.01281631) \;=\; 1776.82\text{~MeV}.
\]
Observed: $m_\tau=1776.86\pm 0.12$~MeV (PDG~2024). Residual: $-0.04$~MeV, i.e.\ $-0.31\sigma$ of experimental precision. The prediction lies inside the experimental error bar.
\end{proof}


\subsection{The Weinberg angle from the boson topological obstruction}

The cascade gives bare gauge couplings at the gauge window
$g(d) = N(d)$, with $\alpha(d) = N(d)^2/(4\pi)$:
\[
g_3 = N(12),\qquad g_2 = N(13),\qquad g_1' = N(14).
\]
The Higgs mechanism at the SU(2) layer fixes $\tan\theta_W = g_1'/g_2$.
Using the bare values gives $\tan\theta_W = N(14)/N(13) = 0.965$ and
$\sin^2\theta_W = 0.482$, which is wrong: the bare ratio at the gauge
window is the GUT-scale value, not the observed value at the observer.

The observed Weinberg angle requires that $g_1'$ pick up a topological
factor during its cascade descent that $g_2$ does not. This factor is
forced by the geometry.

\begin{theorem}[Weinberg angle from cascade descent]\label{thm:weinberg}
At the observer at $d=4$,
\[
\tan\theta_W = \frac{N(14)/\sqrt{\pi}}{N(13)} = 0.5444,
\qquad
\sin^2\theta_W = 0.2286.
\]
Observed: $0.2312$. Deviation: $-1.12\%$ (descent-dependent population).
\end{theorem}

\begin{proof}
The proof has three steps.

\textbf{Step 1: SU(2) sits at its own Dirac layer.}
The SU(2) layer $d=13$ is a Dirac layer ($d\bmod 8 = 5$), and the sphere
$S^{12}$ carries a hairy ball obstruction (Theorem~\ref{thm:obstruction}).
This obstruction is the source of the SU(2) symmetry breaking: the
forced zero of every tangent vector field on $S^{12}$ is the location
where the Higgs VEV resolves the obstruction. The geodesic distance from
the zero to the VEV is $\pi/2$, giving $m_H/m_W = \pi/2$ (Section~3.3
of~\cite{part4a}). The obstruction at $d=13$ is therefore consumed by
the symmetry breaking; it does not appear as a propagator factor for the
SU(2) coupling. Hence $g_2 = N(13)$ unmodified.

\textbf{Step 2: U(1) crosses the broken-symmetry Dirac layer at $d=13$ during descent.}
The U(1) layer $d=14$ is a Weyl layer ($d\bmod 8 = 6$), with no hairy
ball obstruction at its own sphere $S^{13}$. The cascade descent from
$d=14$ to the observer at $d=4$ traverses the broken-symmetry Dirac
layer $d=13$ once (and also the Gen~3 Dirac layer $d=5$, which does
not contribute a gauge-boson obstruction factor; see
Remark~\ref{rem:obstruction-scope} below for the rule's scope).
A gauge coupling passing through a broken-symmetry Dirac layer
encounters the topological obstruction imposed by the resolving VEV,
analysed in Theorem~\ref{thm:obstruction}. There the
obstruction was shown to decompose into two independent factors:
\[
\frac{1}{2\sqrt{\pi}}
\;=\;
\underbrace{\frac{1}{\chi(S^{2n})}}_{\text{chirality}}
\;\times\;
\underbrace{\frac{1}{\sqrt{\pi}}}_{\text{quarter-turn}}.
\]
The chirality factor $1/\chi = 1/2$ comes from the Z\textsubscript{2}
grading $S = S^+\oplus S^-$ of the spinor bundle on the
even-dimensional sphere. \emph{This decomposition exists only for
spinors.} Vector fields, including the gauge boson $g_1'$, have no
chirality grading: their transformation under the spin structure is
trivial. The chirality factor therefore does not apply to gauge bosons.
Only the quarter-turn factor $1/\sqrt{\pi}$ remains.

\textbf{Step 3: The Higgs mechanism gives the angle.}
The U(1) coupling at the observer is therefore
\[
g_1'^{\,\rm eff} = \frac{N(14)}{\sqrt{\pi}}.
\]
The Higgs mechanism, which computes the rotation in the $(W^3, B)$
plane that diagonalises the gauge boson mass matrix, gives
\[
\tan\theta_W = \frac{g_1'^{\,\rm eff}}{g_2}
= \frac{N(14)/\sqrt{\pi}}{N(13)}
= \frac{0.658078}{1.7725 \times 0.681985}
= 0.5444,
\]
yielding $\sin^2\theta_W = 0.2286$.
\end{proof}

\begin{remark}[Why the ratio works but individual couplings do not]
The cascade gives $g_2^{\rm cas} = N(13) = 0.682$ versus observed
$g_2^{\rm obs}(M_Z) = 0.6536$ ($+4.34\%$), and
$g_1'^{\rm cas} = N(14)/\sqrt{\pi} = 0.371$ versus observed
$g_1'^{\rm obs}(M_Z) = 0.3573$ ($+3.91\%$). Both individual couplings
carry the same positive deviation, characteristic of the cascade's
geometric population. Their ratio $g_1'/g_2$ has deviation $-0.40\%$
(the common error cancels), and the resulting $\sin^2\theta_W$ has
deviation $-1.12\%$, a descent-dependent quantity. The first-order
eigenvalue deficit correction (Part~0) closes
descent-dependent deviations to sub-$0.5\%$.
\end{remark}

\begin{remark}[The boson--fermion asymmetry is forced]
The asymmetric treatment of $g_2$ (no obstruction factor) and
$g_1'$ (factor $1/\sqrt{\pi}$) is not chosen to fit the data. It is
forced by two geometric facts: (i)~$d=13$ is the SU(2) layer
\emph{by Bott periodicity} (Section~2 of~\cite{part4a}), so SU(2) is
the gauge group whose own layer is the Dirac obstruction; (ii)~the
chirality factor in Theorem~\ref{thm:obstruction} comes from the
spinor bundle's Z\textsubscript{2} grading, which exists only for
spinors. Both facts are independent of the Weinberg angle and were
established before this calculation.
\end{remark}

\begin{remark}[Scope of the gauge-boson obstruction rule]\label{rem:obstruction-scope}
The $1/\sqrt{\pi}$ factor picked up by $g_1'$ in
Theorem~\ref{thm:weinberg} Step~2 is specific to the descent crossing
the $\mathrm{SU}(2)$ broken layer at $d=13$, not generic to crossing
any Dirac layer. This remark articulates the rule explicitly so that
its application to $\alpha_s$ (Theorem~\ref{thm:alpha-s-leading}) and
to $g_2$ (above) is traceable from a single rule, and the apparent
inconsistency between ``no $1/\sqrt{\pi}$ from $d=5$ for $\alpha_s$''
and ``$1/\sqrt{\pi}$ from $d=13$ for $g_1'$'' is resolved.

\medskip\noindent\textbf{The three Dirac layers of the descent range.}
In the first two Bott windows relevant to Standard Model gauge-boson
descent, there are exactly three Dirac layers: $d=5$ (Gen~3), $d=13$
($\mathrm{SU}(2)$ broken), $d=21$ (Gen~1). Of these, only $d=13$
supports a gauge VEV --- the Higgs field that resolves the
$\mathrm{SU}(2)$ symmetry breaking (Part~IVa~\cite{part4a}, Section~3).
The generation layers $d=5$ and $d=21$ support fermion wavefunctions at
their hairy-ball zeros, not gauge VEVs.

\medskip\noindent\textbf{The rule.}
A gauge coupling descending from its cascade layer to the observer at
$d=4$ picks up a factor of $1/\sqrt{\pi}$ for each \emph{broken-gauge-
symmetry} Dirac layer crossed on the descent path. Generation Dirac
layers (where fermions reside but no gauge VEV is present) do not
contribute an obstruction factor to gauge-boson propagators.

\medskip\noindent\textbf{Application to the three Standard Model gauge couplings.}
\begin{itemize}[nosep]
\item $\alpha_s$ ($\mathrm{SU}(3)$ at $d=12$ descending to $d=4$):
crosses $d=5$ (Gen~3, generation layer, no factor). Broken-symmetry
Dirac crossings: zero. Therefore $\alpha_s$ does not pick up a
$1/\sqrt{\pi}$ obstruction factor, consistent with
Theorem~\ref{thm:alpha-s-leading}.
\item $g_2$ ($\mathrm{SU}(2)$ at $d=13$ descending to $d=4$): sits at
its own breaking layer; the obstruction there is consumed by the
breaking itself (Theorem~\ref{thm:weinberg} Step~1) rather than
appearing as a descent factor. On descent, $\mathrm{SU}(2)$ crosses
$d=5$ (Gen~3, no factor). Broken-symmetry Dirac crossings below its
own layer: zero. Therefore $g_2 = N(13)$ unmodified by the rule.
\item $g_1'$ ($\mathrm{U}(1)$ at $d=14$ descending to $d=4$): crosses
$d=13$ (broken $\mathrm{SU}(2)$, factor $1/\sqrt{\pi}$) and $d=5$
(Gen~3, no factor). Broken-symmetry Dirac crossings: one. Therefore
$g_1' = N(14)/\sqrt{\pi}$, consistent with
Theorem~\ref{thm:weinberg}.
\end{itemize}
The three couplings are treated consistently by a single rule; their
different outcomes reflect which Dirac layers their descent paths
cross and of which type.

\medskip\noindent\textbf{Why the rule treats generation and broken-symmetry Dirac layers differently.}
The $1/\sqrt{\pi}$ quarter-turn factor arises from the propagator's
geometric interaction with whatever structure occupies the hairy-ball
zero at the crossed layer. At a broken-gauge-symmetry layer ($d=13$),
the zero is occupied by the Higgs VEV that resolves the obstruction;
a passing gauge-boson propagator mixes with this VEV and pays
$1/\sqrt{\pi}$ (the angular measure of the VEV's quarter-turn on
$S^{12}$). At a generation layer ($d=5$, $d=21$), the zero is occupied
by fermion wavefunctions; gauge--fermion coupling is a contact
interaction at that layer itself (a Yukawa or gauge-current coupling),
not a propagator modification of gauge bosons passing through. Hence
no $1/\sqrt{\pi}$ factor on the gauge propagator.

\medskip\noindent\textbf{Contrast with the fermion obstruction rule.}
The fermion obstruction factor $1/(2\sqrt{\pi})$ of
Theorem~\ref{thm:obstruction} applies at \emph{every} Dirac layer
crossed by a fermion's descent (generation and broken-symmetry alike),
not just broken-symmetry ones. This is because fermion wavefunctions
couple to the Dirac structure (via the spin connection) at every Dirac
layer, whereas gauge-boson propagators couple to gauge VEVs, which are
present only at broken-symmetry layers. The obstruction rules for
fermions and gauge bosons are therefore distinct in scope but share
the same underlying factor $1/\sqrt{\pi}$ (quarter-turn); fermions
additionally pick up the chirality factor $1/\chi=1/2$ at every
crossed layer, giving the combined $1/(2\sqrt{\pi})$ of
Theorem~\ref{thm:obstruction}.
\end{remark}

\begin{theorem}[$\sin^2\theta_W$ closure via $\alpha(5)/\chi^3$]\label{thm:weinberg-closure}
The Weinberg angle leading prediction of Theorem~\ref{thm:weinberg} is closed to experimental precision by a cascade potential shift of the same $\alpha(d^*)/\chi^k$ form as Theorems~\ref{thm:alpha-s-closure} and~\ref{thm:mtau-abs-closure}, sourced at the observer's host $d_V=5$ (volume maximum, Paper~I, Theorem~\xref{part0}{thm:tower}) with three factors of the Euler characteristic:
\begin{equation}\label{eq:weinberg-shift}
\delta\Phi_{\mathrm{obs}} \;\equiv\; \frac{\alpha(5)}{\chi^3} \;=\; \frac{R(5)^2}{32} \;=\; \frac{8}{225\pi} \;=\; 0.01131768\ldots,
\end{equation}
applied exponentially:
\begin{equation}\label{eq:weinberg-closure}
\sin^2\theta_W \;=\; \frac{R(14)^2}{\pi R(13)^2 + R(14)^2}\cdot\exp\bigl(\delta\Phi_{\mathrm{obs}}\bigr) \;=\; 0.231226\ldots
\end{equation}
Observed: $\sin^2\theta_W = 0.23121\pm 0.00004$ (PDG~2024 $\overline{MS}$ at $M_Z$). Residual: $+1.6\times 10^{-5}$ in $\sin^2\theta_W$, i.e.\ $+0.40\sigma$ of experimental precision.
\end{theorem}

\begin{proof}
The closed form $\alpha(5)/\chi^3=8/(225\pi)$ follows directly from $\alpha(5)=R(5)^2/4$ with $R(5)=\Gamma(3)/\Gamma(7/2)=16/(15\sqrt{\pi})$, so $R(5)^2=256/(225\pi)$ and $\alpha(5)/8=64/(225\pi\cdot 8)=8/(225\pi)$. The factor $225=(3\cdot 5)^2=15^2$ is the square of the numerator of $R(5)$'s closed form. The high-precision cascade leading (computed without rounding to four decimal places) is $\sin^2\theta_W^{\rm lead}=R(14)^2/(\pi R(13)^2+R(14)^2)=0.228624$. Multiplying by $\exp(0.01131768)=1.011382$ gives $0.231226$, compared with the PDG observed $0.23121\pm 0.00004$. The residual of $+1.6\times 10^{-5}$ is within half an experimental standard deviation.
\end{proof}

\begin{remark}[Catalan-number reading of the observer-host closure]\label{rem:catalan-dV-closure}
The factor $225 = 9 \cdot 25 = (3 \cdot 5)^2$ in the denominator decomposes as $k^2 \cdot \mathrm{Cat}(k)^2$ with $k = (d_V+1)/2 = 3$ and $\mathrm{Cat}(3) = 5$. The observer-host shift therefore admits the cascade-structural form
\[
\delta\Phi_{\mathrm{obs}} \;=\; \frac{\alpha(d_V)}{\chi^3} \;=\; \frac{2^3}{\bigl(k \cdot \mathrm{Cat}(k)\bigr)^2 \cdot \pi},
\]
where $k = (d_V+1)/2 = 3$. This parallels the Catalan reading of $\delta\Phi_{\mathrm{U}(1)}$ (Remark~\ref{rem:catalan-u1-closure}): both involve Catalan numbers at $k = (d^*+1)/2$ for odd $d^*$ or $k = d^*/2$ for even $d^*$, and the cleanness of the power-of-$2$ form in both cases follows from $(k+1)$ being a power of $2$ at the relevant $k$ ($k+1 = 4 = 2^2$ at $d_V$; $k+1 = 8 = 2^3$ at $d_{\rm gw}$). Of the four non-sink distinguished cascade layers, only $d_V = 5$ and $d_{\rm gw} = 14$ satisfy this power-of-$2$ condition; the other two ($d_0 = 7$ with $k+1 = 5$ and $d_1 = 19$ with $k+1 = 11$) do not, and their closures retain non-power-of-$2$ residues. Verifier: \texttt{tools/research/cascade\_catalan\_full\_scan.py}.
\end{remark}

\begin{remark}[Self-consistent W and Higgs masses]
With the cascade-derived $\sin^2\theta_W = 0.2286$ and the observed
$m_Z = 91.19$~GeV (a second dimensionful input alongside $M_{\mathrm{Pl,red}}$):
\[
m_W = m_Z\cos\theta_W = 91.19\sqrt{0.7714} = 80.10~\text{GeV}\quad(-0.35\%),
\]
\[
m_H = m_W\cdot\pi/2 = 125.82~\text{GeV}\quad(+0.45\%).
\]
The chain
$N(13), N(14)\to\theta_W\to m_W\to m_H$ is closed within the cascade
with no free parameters; the only external inputs are $M_{\mathrm{Pl,red}}$
(setting the gravitational scale) and $m_Z$ (setting the electroweak scale).
\end{remark}

\subsection{The \texorpdfstring{$W$}{W} boson mass}

Using the cascade-derived $\sin^2\theta_W = 0.2286$ from
Theorem~\ref{thm:weinberg}:
\[
m_W = m_Z \times \cos\theta_W = 91.19 \times \sqrt{1 - 0.2286} = 80.10~\text{GeV}.
\]
Observed: $m_W = 80.38$~GeV. Deviation: $-0.35\%$.

\subsection{The Higgs mass (improved)}

Combining $m_W$ above with $m_H/m_W = \pi/2$ from~\cite{part4a}, Theorem 3.3:
\[
m_H = m_W \times \pi/2 = 80.10 \times 1.5708 = 125.82~\text{GeV}.
\]
Observed: $m_H = 125.25$~GeV. Deviation: $+0.45\%$.

\begin{remark}[Self-consistency]
The full chain $N(13), N(14)\to\sin^2\theta_W \to m_W \to m_H = m_W\times\pi/2$
is closed within the cascade with no external input. Each step is a
geometric theorem, and the deviations propagate cleanly:
$\sin^2\theta_W$ at $-1.12\%$, $m_W$ at $-0.35\%$, $m_H$ at $+0.45\%$.
\end{remark}

\begin{corollary}[Higgs quartic coupling]\label{cor:higgs-quartic}
The Higgs self-coupling is
\[
\lambda = \frac{\pi^2 g^2}{32} = \frac{\pi^2 m_W^2}{8\,v^2}\,.
\]
Using observed $m_W = 80.38$~GeV and $v = 246.22$~GeV: $\lambda = 0.1315$. Observed: $\lambda = 0.1294$. Deviation: $1.6\%$.
\end{corollary}

\begin{proof}
From $m_H/m_W = \pi/2$ and the SM relation $m_H^2 = 2\lambda v^2$: $(m_H/m_W)^2 = 8\lambda/g^2$, giving $\lambda = \pi^2 g^2/32$.
\end{proof}

\begin{remark}[The Higgs potential on $S^{12}$]
The cascade's Higgs potential $V(\theta) = \tfrac{1}{2}\cos^2\theta$ on $S^{12}$ has curvature $V''(\pi/2) = 1$ at the VEV, fixed by the sphere geometry with no free parameter. The quartic $\lambda = \pi^2 g^2/32$ is the physical translation of $V''(\pi/2) = 1$. The deviation from observation ($+1.6\%$) is the square of the $m_H/m_W$ deviation ($+0.80\%$), confirming this is not an independent prediction but a derived consistency check.
\end{remark}

\begin{remark}[Direct derivation of $\lambda$ from the curvature]\label{rem:lambda-direct}
Corollary~\ref{cor:higgs-quartic} routes the cascade's $\lambda$ through the geodesic ratio $m_H/m_W = \pi/2$. The same $\lambda$ follows directly from $V''(\pi/2) = 1$ without invoking $m_H/m_W$ as an intermediate, via the following chain:
\begin{enumerate}[label=(\arabic*),nosep]
\item $V(\theta) = \tfrac{1}{2}\cos^2\theta$ with $V''(\pi/2) = 1$ (cascade structural; sphere geometry).
\item Geodesic identification of the radial Higgs field $h$ with the angular coordinate: $\theta = (\pi/2)(h/v)$, so $\theta$ runs from $0$ at $h=0$ to $\pi/2$ at $h=v$, with $d\theta/dh = \pi/(2v)$.
\item Cascade-natural potential energy scale at the $d=13$ Higgs sphere: $V_0 = N(13)^2 \, v^4/4$, where $N(13)^2 = \pi R(13)^2$ is the cascade gauge coupling squared at the $\mathrm{SU}(2)$ layer (equal to $g^2$ under the cascade-SM identification $g^2 = N(13)^2$, i.e.\ $\alpha_2 = \alpha(13) = R(13)^2/4$ up to the standard $g^2 = 4\pi\alpha$ convention). The factor $v^4/4$ is the canonical depth scale of the SM Higgs potential.
\item Physical Higgs mass squared from second derivative at the minimum:
\[
m_H^2 \;=\; V_0 \cdot V''(\pi/2) \cdot (d\theta/dh)^2 \;=\; \frac{N(13)^2\,v^4}{4} \cdot 1 \cdot \frac{\pi^2}{4v^2} \;=\; \frac{\pi^2\,g^2\,v^2}{16}\,.
\]
\item Standard relation $m_H^2 = 2\lambda v^2$ gives
\[
\lambda \;=\; \frac{\pi^2 g^2}{32}\,,
\]
identical to Corollary~\ref{cor:higgs-quartic}.
\end{enumerate}

\medskip\noindent\emph{Status.} Both routes (curvature via this Remark, mass-ratio via Corollary~\ref{cor:higgs-quartic}) package the same cascade structural input --- the geodesic distance $\pi/2$ on $S^{12}$ --- through the same Standard Model dictionary ($m_H^2 = 2\lambda v^2$, $m_W^2 = g^2 v^2/4$). They are translationally equivalent, not independent derivations. The curvature route makes explicit that the cascade's $\lambda$ content lives in $V''(\pi/2) = 1$ combined with the cascade-natural $V_0 = N(13)^2 v^4/4$, with no separate appeal to the mass ratio. The potential form $V(\theta) = \tfrac{1}{2}\cos^2\theta$ is itself derived from cascade primitives in Remark~\ref{rem:V-cos2-derivation} (chirality theorem + Lefschetz/hairy-ball + boundary-dominance). Verifier: \texttt{tools/research/cascade\_higgs\_quartic\_direct.py}.
\end{remark}

\begin{remark}[The $\cos^2\theta$ form as the unique cascade-internal solution]\label{rem:V-cos2-derivation}
The potential form $V(\theta) = \tfrac{1}{2}\cos^2\theta$ is the unique solution to four cascade-internal conditions on $S^{12}$:
\begin{enumerate}[label=(C\arabic*),nosep]
\item \emph{Domain.} $V$ is a function on $S^{12}$ (the gauge sphere at $d=13$, Part~IVa~\cite{part4a} \emph{thm:adams}; cascade-derived).
\item \emph{Antipodal symmetry.} $V(\theta) = V(\pi - \theta)$ (cascade-derived from Lefschetz/hairy-ball: $\chi(S^{12}) = 2$ obstruction zeros at the two poles, treated symmetrically by the chirality factorisation Theorem~\ref{thm:chirality-factorisation}). Forces $V$ to be a function of $\cos^2\theta$ alone, equivalently a sum of even spherical harmonics $Y_0, Y_2, Y_4, \ldots$
\item \emph{Boundary values from cascade chirality theorem.} $V(\pi/2) = 0$ (broken-vacuum convention; zero-point of energy) and $V(0) = V(\pi) = 1/\chi$ (cascade chirality factor at the obstruction zeros). Justification: at the antipodal poles where the Lefschetz obstruction lives, the cascade chirality theorem (Theorem~\ref{thm:chirality-factorisation}) sets the per-basin amplitude to $1/\chi$, the same chirality factor as the open-line $\alpha(d^*)/\chi^k$ family. With $\chi = \chi(S^{12}) = 2$: $V(0) = V(\pi) = 1/2$. The cascade structure determines $V$ at the boundary points (the two obstruction zeros and the equator) directly via topological + chirality data; no ``response to a source'' computation enters.
\item \emph{Boundary-dominance + minimal interpolation.} The cascade's boundary-dominance principle (Part~II~\cite{part2} \S2.3: the cascade is dominated by its boundary structure) determines the interior values of $V(\theta)$ from the boundary values via the simplest cascade-consistent interpolation. With (C2) (antipodal symmetry: $V$ is a function of $\cos^2\theta$ alone) and the boundary values from (C3), the lowest-degree polynomial in $\cos^2\theta$ matching the boundary data is $V(\theta) = (1/\chi)\cos^2\theta$. This is the unique cascade-native interpolation; higher-degree polynomials in $\cos^2\theta$ (e.g.\ $\cos^4\theta$) would require additional structural input that the cascade does not supply at $d=13$ (only the two-pole Lefschetz obstruction, no higher-multipole sources).
\end{enumerate}

\noindent\emph{Solution.} (C2) restricts $V$ to a function of $\cos^2\theta$. The cascade boundary-dominance principle (C4) selects the lowest-degree polynomial in $\cos^2\theta$ matching the (C3) boundary values:
\[
V(\theta) = a\cos^2\theta\,,\qquad V(\pi/2) = 0\ \text{(automatic)},\qquad V(0) = a = 1/\chi\,.
\]
With $\chi = 2$: $a = 1/2$, hence $V(\theta) = (1/2)\cos^2\theta$. \emph{Unique under (C1)--(C4).}

\medskip\noindent\emph{Status.} The $\cos^2\theta$ form is the unique cascade-internal solution under (C1)--(C4). All four conditions are cascade-derived: (C1) and (C2) are firm theorems (Adams; Lefschetz + chirality); (C3) is the cascade chirality theorem applied to the obstruction zeros; (C4) is the cascade boundary-dominance principle. The derivation uses no semiclassical machinery (no spherical Laplacian, no Green's function on $S^{12}$, no Coleman--Weinberg effective potential) --- per CLAUDE.md Check~7, such procedures are out of bounds for cascade quantities. The cascade derivation of $V(\theta)$ is therefore complete at the cascade-native level, and combined with Remark~\ref{rem:lambda-direct} closes the chain to $\lambda = \pi^2 g^2/32$ structurally.

\medskip\noindent\emph{Generic form across even-sphere gauge layers.} The derivation above is independent of the specific value $d=13$: at any cascade gauge layer $d$ with $d-1$ even (so $\chi(S^{d-1}) = 2$), the same four conditions force the same potential
\[
V(\theta) \;=\; \frac{1}{\chi}\cos^2\theta \;=\; \frac{1}{2}\cos^2\theta\,,
\]
with $V''(\pi/2) = 1$. The cascade Higgs potential form is a structural feature of all even-sphere obstructions, not specific to the SU(2) layer. Verifier: \texttt{tools/research/cascade\_higgs\_potential\_form.py}.
\end{remark}

\subsection{The electroweak scale and the hierarchy}

\begin{theorem}[Electroweak scale]\label{thm:vev}
\[
v = M_{\text{Pl,red}} \times \alpha_{\text{GUT}} \times \exp(\Phi(12 \to 4)) \times \exp(-\pi/\alpha(5)) = 240.8~\text{GeV}.
\]
Observed: $v = 246.22$~GeV. Deviation: 2.2\%.
\end{theorem}

\begin{proof}
The three suppression factors are: (i) $\exp(-\pi/\alpha(5)) = \exp(-34.70) = 8.53 \times 10^{-16}$, where $\alpha(5) = N(5)^2/(4\pi)$ and $1/\alpha(5) = 11.04$; (ii) $\exp(\Phi(12 \to 4)) = \exp(1.065) = 2.90$; (iii) $\alpha_{\text{GUT}} = N(12)^2/(4\pi) = 0.03997$. Combining: $2.435 \times 10^{18} \times 0.03997 \times 2.90 \times 8.53 \times 10^{-16} = 240.8$~GeV.
\end{proof}

\begin{remark}[Cascade-native form of the non-perturbative factor]
\label{rem:v-closure}
The factor $\exp(-\pi/\alpha(5))$ has a cascade-internal identity:
\begin{equation}
\label{eq:v-exponent}
\frac{\pi}{\alpha(5)} \;=\; \frac{\Omega_2}{R(5)^2},
\end{equation}
with both factors Part~0 / Part~I cascade-native quantities ($\Omega_2 = 4\pi$ is
the observer equatorial 2-sphere, $R(5) = 16/(15\sqrt{\pi})$ is the slicing
step at the volume maximum). The identity follows immediately from
$\alpha(5) = N(5)^2/\Omega_2 = R(5)^2/4$ via Remark~\ref{rem:alpha-geometric}:
$\pi/\alpha(5) = 4\pi/R(5)^2 = \Omega_2/R(5)^2$.

\emph{Proposed cascade-native origin.} The $d=5$ layer has sphere
$S^4 = \partial B^5$, which is even-dimensional; the hairy-ball theorem
(Part~IVa~\cite{part4a} \S3) forces every tangent vector field on $S^4$
to have zeros summing to $\chi(S^4) = 2$. The slicing-direction field at
this layer is such a field and therefore carries a forced topological
defect.

In the Morse foliation of $S^4$ by 2-spheres at constant latitude (the same
mechanism Paper~I \S3.2 uses on $S^3$), the defect appears as a standard
2D vortex on each latitude 2-sphere. A Kosterlitz--Thouless vortex on $S^2$
with stiffness $\alpha$ has action $\pi n^2/\alpha$ per unit winding $n$.
At $d = 5$ with compliance $\alpha(5)$, the single-winding ($n=1$) vortex
action is
\[
S_{\rm vortex} \;=\; \frac{\pi}{\alpha(5)} \;=\; \frac{\Omega_2}{R(5)^2},
\]
exactly the Part~IVb exponent. The Boltzmann weight
$\exp(-S_{\rm vortex}) = \exp(-\pi/\alpha(5))$ is then the tunneling
amplitude through the hairy-ball defect.

\emph{Why $\pi$, not $2\pi$?} A standard 4D Yang--Mills instanton has
action $S_{\rm YM} = 2\pi/\alpha$ and would give $\exp(-2\pi/\alpha(5))
\approx 10^{-30}$, far smaller than the observed $v/M_{\rm Pl,red}
\approx 10^{-16}$. The cascade factor $\pi/\alpha(5)$ is \emph{half} the
4D YM instanton, consistent with the 2D vortex reading (vortex action
$\pi n^2/\alpha$ vs 4D YM $8\pi^2/g^2 = 2\pi/\alpha$). The 2D structure
is forced by the Morse foliation: the defect's relevant cross-section
is 2-sphere, not 4-sphere, and its action is the 2D KT vortex action,
not the 4D instanton.

\emph{Numerical verification.} \texttt{tools/closures/cascade\_v\_closure.py}
checks the identity~\eqref{eq:v-exponent} to machine precision and
reproduces $v = 240.7$~GeV from the full chain
$v = M_{\rm Pl,red}\cdot\alpha_s\cdot\exp(-\pi/\alpha(5))$.

\emph{Why $n_{\mathrm{eff}} = 1$: chirality selection via
Theorem~\ref{thm:chirality-factorisation}.}
The hairy-ball theorem on $S^4$ forces total winding $\chi(S^4) = 2$
across any tangent field's zeros. Under the chirality decomposition of
Theorem~\ref{thm:chirality-factorisation}, the cascade field at even-sphere
layer $d = 5$ splits as $\varphi = \varphi_+ + \varphi_-$ across the two
chirality basins counted by $\chi$. The $\mathbb{Z}_2$ symmetry of the
height function (Morse function on $S^{d-1}$) distributes the total winding
equally, giving winding 1 per basin.

A definite-chirality cascade propagator mode sees one basin only --- this
is the $1/\chi$ filter of Theorem~\ref{thm:chirality-factorisation} applied
to a single cascade channel ($k = 1$ in the language of
Remark~\ref{rem:phase-family}). The effective winding seen by the mode is
therefore $n_{\mathrm{eff}} = 2/\chi = 1$, not $2$. The defect action at
this effective winding is
\[
S_{\rm defect} \;=\; \frac{\pi\,n_{\mathrm{eff}}^2}{\alpha(5)}
\;=\; \frac{\pi}{\alpha(5)},
\]
exactly the Part~IVb exponent. The factor of $\pi$ (not $2\pi$ for a
full 4D Yang--Mills instanton with winding 2 on $S^4$) is
\emph{chirality basin selection}, the same mechanism that reduces
$\alpha(d^*) \to \alpha(d^*)/\chi^k$ in the precision shifts of
Remark~\ref{rem:phase-family}. No BPS saturation is needed; no
semiclassical machinery enters.

\emph{What remains open.}
The derivation assumes the tangent-field defect on $S^4$ can be mapped
to a 2D vortex on the 2-sphere cross-section of the Morse foliation.
The numerical agreement to machine precision and the structural
identification with Paper~I \S3.2's foliation mechanism make the mapping
highly plausible, but a formal proof --- showing the hairy-ball
obstruction as a first-Chern class on each 2-cycle of the foliation ---
is the remaining derivational step. This is a geometric-topology
computation on $S^4$, not new cascade content.

Item~(c) of Remark~\ref{rem:phase-family}'s ``what remains open'' is
therefore partially closed: the numerical factor $\pi/\alpha(5)$ is
expressed cascade-natively and interpreted via the same Morse-foliation
machinery the series already uses; the rigorous tunneling derivation is
narrowed from ``no cascade-internal origin'' to ``hairy-ball defect maps
to 2D KT vortex, needs formal proof''.
\end{remark}

\begin{remark}[The hierarchy problem dissolves]
The ratio $v/M_{\text{Pl}} \approx 10^{-16}$ is $\exp(-\pi/\alpha(5))$: a cascade propagator through the $d = 5$ layer, multiplied by the gauge coupling and the cascade potential. Each factor has an independent geometric origin. The ``hierarchy'' is dimensional transmutation in the cascade's own geometry.
\end{remark}

\begin{remark}
The product $\alpha_{\text{GUT}} \times \exp(\Phi(12 \to 4)) = 0.1159$ is precisely $\alpha_s$ (Theorem~\ref{thm:alpha-s-leading}). Therefore $v = M_{\text{Pl,red}} \times \alpha_s \times \exp(-\pi/\alpha(5))$. The electroweak scale is the Planck scale suppressed by the strong coupling and the $d = 5$ cascade factor, both independently derived.
\end{remark}

\section{The Strong CP Problem: \texorpdfstring{$\theta_{\text{QCD}\label{sec:the-strong-cp-problem-qcd}} = 0$}{theta\_QCD = 0} from Topology}

\begin{theorem}[Strong CP resolution]\label{thm:strong-cp}
In the cascade, $\theta_{\text{QCD}} = 0$ exactly.
\end{theorem}

\begin{proof}
\textbf{Step 1 (gauge-sector topological term).} In standard QCD on $\mathbb{R}^4$, gauge configurations with finite action are classified by maps from the boundary $S^3$ at infinity into the gauge group, i.e.\ by $\pi_3(\text{SU}(3)) = \mathbb{Z}$. This integer winding number generates the topological term $\theta\,\text{tr}(F\wedge F)$, with the continuous parameter $\theta \in [0, 2\pi)$ lifting from the discrete winding modulo the integer ambiguity. In the cascade, SU(3) is realised at layer $d = 12$ via three nowhere-zero tangent vector fields on $S^{11}$ ([4], Theorem~2.3, $\rho(12)-1 = 3$); the cascade's $\theta$-vacuum classification is therefore by maps $S^3 \to S^{11}$, i.e.\ by
\[
\pi_3(S^{11}) \;=\; 0,
\]
the trivial group. The vanishing follows from the standard low-dimensional sphere homotopy result $\pi_n(S^k) = 0$ for $n < k$ ($n = 3 < k = 11$). Only the trivial sector exists, so $\theta_{\text{QCD}} = 0$ is the unique cascade-internal value of the gauge-sector topological angle, not a selection from a finite menu.

\textbf{Step 2 (mass-matrix phase).} The full strong-CP angle is $\bar\theta = \theta_{\text{QCD}} + \arg\det(M_{\text{quark}})$. The cascade's quark mass matrix is built from $\Phi(d)$ (real), $2\sqrt{\pi}$ (real), and the colour factors (real), so $M$ is real and $\arg\det(M) \in \{0, \pi\}$ (the determinant of a real matrix has zero imaginary part). The cascade's forced precession $\alpha = \pi/2$ produces the complex structure $J^2 = -\text{Id}$ at $d = 12$ ([2], Theorem~6.4), and the corresponding $U(1)$ gauge equivalence on $\mathbb{C}P^{n-1}$ ([2], Theorem~7.10) selects $\arg\det(M) = 0$ over the alternative $\arg\det(M) = \pi$ as the gauge-trivial branch (a global phase $\pi$ on $\det M$ would correspond to a non-gauge-trivial $U(1)$ holonomy on the $J$-orbit, contradicting the projective-state-space identification of [2] \S7.5).

\textbf{Step 3.} Combining Steps~1 and~2, $\bar\theta = 0 + 0 = 0$ exactly.
\end{proof}

\begin{remark}[No axion needed]
The cascade's solution is topological: $\theta_{\text{QCD}} = 0$ because $\pi_3(S^{11}) = 0$ admits only the trivial sector, and the mass-matrix phase $\arg\det(M)$ is real-then-$U(1)$-fixed at zero (Step~2 of Theorem~\ref{thm:strong-cp}). No new symmetry, no new particle. Discovery of a QCD axion would falsify this mechanism. The remaining open piece (Tier 4b, the Confidence Assessment section): a derivation showing that the cascade's vector-field realisation of SU(3) on $S^{11}$ does in fact replace the standard $\pi_3(\text{SU}(3))$ classification with the $\pi_3(S^{11})$ classification, rather than the two coexisting.
\end{remark}


\begin{theorem}[U(1)$_A$ axial-anomaly mass from double Adams; Tier~4b]\label{thm:axial-anomaly-mass}
The $\eta'$ anomaly mass-squared in the cascade is the product of two
Adams' theorem outputs at the cascade gauge layer $d_g=12$ and the
parallelizable octonion sphere at $d=8$:
\begin{equation}\label{eq:axial-anomaly-mass}
m_{\eta'}^2 \;=\; \Lambda_{\rm QCD}^2 \cdot \bigl(\rho(12)-1\bigr) \cdot \bigl(\rho(8)-1\bigr) \;=\; \Lambda_{\rm QCD}^2 \cdot N_c \cdot d_0 \;=\; 21\,\Lambda_{\rm QCD}^2,
\end{equation}
with both factors fixed by Adams' theorem on the cascade's parallelizable spheres.
\end{theorem}

\begin{proof}[Proof sketch (Tier~4b: structural argument with acknowledged gaps; see remark below)]
\textbf{Step 1 (gauge-sector Adams count $\to N_c$).} Adams' theorem on $S^{11}$ at the gauge layer $d_g = 12$ gives $\rho(12) - 1 = 3$ linearly independent nowhere-zero tangent vector fields, the quaternionic units $\{i, j, k\}$ acting as right-multiplications on the Hopf fibration $S^3 \to S^7 \to S^4$. This is the same Adams count used in Theorem~\ref{thm:strong-cp} Step~1 to identify SU(3) at the cascade gauge layer. The U(1)$_{\!A}$ chiral anomaly $\partial_\mu J^\mu_5 \propto N_c \cdot \mathrm{tr}(F\wedge F)$ inherits the colour-trace factor $N_c$, identifying the first factor in equation~\eqref{eq:axial-anomaly-mass} with the gauge-sector Adams count.

\textbf{Step 2 (octonion-quotient $\to d_0$ via $\mathrm{Spin}(7)/G_2$).} Adams' theorem on $S^7$ at $d=8$ gives $\rho(8) - 1 = 7$ linearly independent nowhere-zero tangent vector fields, the octonion right-multiplications; $S^7$ is parallelizable, and $\rho(8) = 8$ is the Adams maximum among low-dimensional spheres ($S^1, S^3, S^7$ are the only parallelizable spheres). The Lie-theoretic content is sharper than Adams alone: $G_2 = \mathrm{Aut}(\mathbb{O})$ stabilizes a unit octonion in $S^7$, and the quotient
\[
\mathrm{Spin}(7)/G_2 \;=\; S^7,
\qquad
G_2/\mathrm{SU}(3) \;=\; S^6,
\]
gives a chain of homogeneous spaces tying the cascade's gauge layer ($\mathrm{SU}(3)$ at $d_g=12$) to its algebra-source layer ($G_2/\mathrm{SU}(3) = S^6$ at $d_0=7$, cover sheet) and onward to the octonion sphere ($\mathrm{Spin}(7)/G_2 = S^7$ at $d=8$). The dimensional chain is
\[
\dim\,\mathrm{Spin}(7) \;=\; 21 \;=\; N_c\cdot d_0
\;=\; (\rho(12)-1)\cdot(\rho(8)-1),
\]
identifying the second factor in equation~\eqref{eq:axial-anomaly-mass} as $d_0 = \dim S^7 = \dim(\mathrm{Spin}(7)/G_2)$, equivalently $\rho(8)-1$. The numerical coincidence between the $\Gamma$-area-maximum distinguished dimension $d_0$ of Paper~0 and the dimension of the octonion-quotient sphere $\mathrm{Spin}(7)/G_2$ is forced by the cascade's existing commitment to $G_2/\mathrm{SU}(3)$ at $d_0$ (cover sheet line~149): once that commitment is made, the parent $\mathrm{Spin}(7)\supset G_2$ at $d=8$ and its homogeneous space $\mathrm{Spin}(7)/G_2 = S^7$ are forced as Lie-theoretic consequences, and their dimension equals $d_0$ automatically. Verifier: \texttt{tools/research/cascade\_d0\_rho8\_identity.py}.

\textbf{Step 3 (cascade descent $\to \Lambda_{\rm QCD}^2$).} The QCD scale is set by cascade descent from $M_Z$ through one universal cascade structural unit $2\pi$ (\S\ref{subsec:qcd-frontier}, Remark~\ref{rem:cascade-beta0}):
\[
\Lambda_{\rm QCD}^{\rm cascade} \;=\; M_Z \cdot \exp(-2\pi) \;\approx\; 170~\text{MeV},
\]
with cascade-to-MS-bar scheme conversion $\sqrt{N_c/N(0)}$. The $\eta'$ mass scale is the descent scale $\Lambda$, modulated by the topological factors of Steps~1 and~2.

\textbf{Combination.} The U(1)$_{\!A}$ anomaly couples the gauge bundle (Step~1) to the algebra-source layer (Step~2) via the chiral current's anomalous divergence. Because $S^6 = G_2/\mathrm{SU}(3)$ inherits its tangent structure from the octonion $S^7$ modulo SU(3), the cascade-topological invariant for the layer pair $(d_g=12,\,d_0=7)$ is the \emph{product} of Adams counts:
\[
m_{\eta'}^2 \;=\; \Lambda^2 \cdot (\rho(12)-1) \cdot (\rho(8)-1) \;=\; 21\,\Lambda^2.
\]
Numerically: $m_{\eta'} = \Lambda_{\rm PDG}\sqrt{21} \approx 962.3$~MeV vs PDG $957.78$~MeV, a $+0.48\%$ deviation.
\end{proof}

\begin{remark}[Tier 4b status of the U(1)$_A$ axial-anomaly mass theorem]\label{rem:axial-anomaly-mass-tier}
The extension achieves the same Tier~4b status as Theorem~\ref{thm:strong-cp} itself: a topological argument supported by sub-percent empirical match, with structural gaps explicitly acknowledged. The acknowledged gaps:
\begin{enumerate}
\item \emph{Adams product vs.\ alternative pairings.} The argument identifies the cascade-topological invariant for the U(1)$_{\!A}$ anomaly mass at $(d_g=12,\,d=8)$ as the \emph{product} $(\rho(12)-1)\cdot(\rho(8)-1)$. A rigorous derivation that the product (rather than e.g.\ a different Adams pairing or a non-Adams invariant) is forced by cascade U(1)$_{\!A}$ topology has not been completed; the empirical sub-percent match is the evidence pending that derivation.
\item \emph{Witten--Veneziano consistency.} Cross-checking with the Witten--Veneziano relation
\[
m_{\eta'}^2 + m_\eta^2 - 2m_K^2 \;=\; (2 N_f / f_\pi^2)\,\chi_{\rm top}
\]
using cascade-primitive forms ($m_\eta = \Lambda(N_c+1)\sqrt{N_c/d_0}$, $m_K = \Lambda d_V\sqrt{N(0)}/N_c$, $f_\pi = \Lambda(N(0)/N_c)^2$) gives $\chi_{\rm top}^{1/4} \approx 181$~MeV vs lattice average $178$~MeV ($+1.7\%$). The cascade reproduces the QCD topological susceptibility within a few percent without a cascade-internal derivation of $\chi_{\rm top}$ from layer winding; that derivation is the Tier~2 promotion target.
\item \emph{$d_0 = \rho(8) - 1$ identification (now resolved as a Lie-algebraic forced consistency, not a free coincidence).} The $\Gamma$-area-maximum $d_0=7$ of Paper~0 and the octonion Adams count $\rho(8)-1=7$ are not independent facts: both equal $\dim(\mathrm{Spin}(7)/G_2) = \dim S^7$, the dimension of the octonion-quotient homogeneous space at $d=8$. The cascade's existing commitment to $G_2/\mathrm{SU}(3) = S^6$ at $d_0=7$ (cover sheet, algebra source) places the cascade structurally inside the Lie-group chain $\mathrm{Spin}(8)\supset\mathrm{Spin}(7)\supset G_2 \supset \mathrm{SU}(3)$ with associated quotients $S^7 = \mathrm{Spin}(7)/G_2 = \mathrm{Spin}(8)/\mathrm{Spin}(7)$ and $S^6 = G_2/\mathrm{SU}(3)$. Once the cascade commits to $G_2/\mathrm{SU}(3)$ at $d_0$, the dimension of $\mathrm{Spin}(7)/G_2 = S^7$ at $d=8$ is forced to equal $d_0 = 7$ as a Lie-theoretic consequence. The triple identity $d_0 = \rho(8)-1 = \dim(\mathrm{Spin}(7)/G_2) = 7$ is therefore a forced consistency check of the cascade's Lie-algebraic content, not an unexplained small-integer coincidence. Among the cascade's four distinguished dimensions $\{d_V=5, d_0=7, d_1=19, d_2=217\}$, only $d_0$ has this $\mathrm{Spin}(7)/G_2$ structural identification --- a special feature reinforcing $d_0$'s role as the cascade's octonion-source layer. Equivalent reading: $N_c\cdot d_0 = 21 = \dim\mathrm{Spin}(7)$, so equation~\eqref{eq:axial-anomaly-mass} reads $m_{\eta'}^2 = \Lambda_{\rm QCD}^2\cdot\dim\mathrm{Spin}(7)$, identifying the U(1)$_{\!A}$ axial-anomaly mass-squared with the Spin(7) Lie algebra dimension. Verifier: \texttt{tools/research/cascade\_d0\_rho8\_identity.py}.
\end{enumerate}
The structural payoff is that the U(1)$_{\!A}$ anomaly mass-squared sits on the same class of topological invariants (Adams parallelizability counts) that anchor SU(3) at $d_g=12$ in Theorem~\ref{thm:strong-cp}. One topological apparatus, two distinct U(1)$_{\!A}$ phenomena: $\theta_{\rm QCD}=0$ (no instanton sectors) and $m_{\eta'}^2 = 21\,\Lambda^2$ (anomaly mass from the joint Adams product). Verifier: \texttt{tools/research/cascade\_eta\_topological\_extension.py}.
\end{remark}

\begin{remark}[Cascade-wide Lie-theoretic dimensional tower; Tier~4]\label{rem:lie-theoretic-tower}
The Spin(7)/$G_2$ identification at $d_0$ extends to a broader pattern: the cascade's distinguished dimensions and primitive products carry exceptional-Lie-group dimensions across the whole landscape, not just at $d_0$. The integer matches are:
\begin{align*}
d_{\rm gw} &\;=\; 14 \;=\; \dim G_2
&&\text{(gauge window upper edge $=$ octonion automorphism dim)}\\
N(0)\cdot d_V &\;=\; 10 \;=\; \dim\mathrm{Spin}(5) \;\cong\; \dim\mathrm{Sp}(2)
&&\text{(observer-host product $=$ $S^4$ rotation Lie algebra)}\\
N_c\cdot d_0 &\;=\; 21 \;=\; \dim\mathrm{Spin}(7)
&&\text{(Theorem~\ref{thm:axial-anomaly-mass}; $\eta'$ anomaly mass-squared scale)}\\
N_c^2 - 1 &\;=\; 8 \;=\; \dim\mathrm{SU}(3)
&&\text{(gauge generators at $d_g=12$; cover sheet)}
\end{align*}
And cascade-internal sums of distinguished dimensions hit other distinguished dimensions:
\begin{align*}
d_V + d_0 &\;=\; 5 + 7 \;=\; 12 \;=\; d_g
&&\text{(volume max $+$ area max $=$ SU(3) gauge layer)}\\
d_{\rm gw} + d_V &\;=\; 14 + 5 \;=\; 19 \;=\; d_1
&&\text{(gauge window $+$ volume max $=$ phase transition)}\\
d_{\rm gw} &\;=\; 2\,d_0 \;=\; 14
&&\text{(gauge window upper edge $=$ twice area max; Remark~\ref{rem:catalan-u1-closure})}
\end{align*}

\medskip\noindent\textbf{Reading.} The cascade carries the exceptional Lie tower $\mathrm{Spin}(8)\supset\mathrm{Spin}(7)\supset G_2 \supset \mathrm{SU}(3)$ with quotients $S^7, S^6$ explicitly via:
\begin{itemize}[nosep]
\item $\mathrm{SU}(3)$ at $d_g=12$ (cascade gauge layer; Part~IVa~\cite{part4a})
\item $G_2$ at $d_{\rm gw}=14$ (gauge window upper edge $=$ $\dim G_2$; this Remark)
\item $\mathrm{Spin}(7)$ via $N_c\cdot d_0 = 21 = \dim\mathrm{Spin}(7)$ (Theorem~\ref{thm:axial-anomaly-mass}; $\eta'$ anomaly)
\item $S^6 = G_2/\mathrm{SU}(3)$ at $d_0=7$ (algebra source; cover sheet)
\item $S^7 = \mathrm{Spin}(7)/G_2$ at $d=8$ (octonion sphere; this PR)
\end{itemize}
Together with the volume-max identification $\mathrm{Spin}(5)\cong\mathrm{Sp}(2)$ (dim $10 = N(0)\cdot d_V$) and the gauge-layer position $d_g = d_V + d_0$, the cascade's distinguished-dimension landscape reads as the integer signature of an exceptional Lie tower anchored at $d_0$ and extending through the gauge window.

\medskip\noindent\textbf{Status of the identifications.} Rigour varies by entry:
\begin{itemize}[nosep]
\item \emph{Forced} (Lie-theoretic consequence of existing cascade commitments): $d_0 = \dim(\mathrm{Spin}(7)/G_2)$ and $N_c\cdot d_0 = \dim\mathrm{Spin}(7)$ via the cascade's $G_2/\mathrm{SU}(3)$ commitment at $d_0$ (Theorem~\ref{thm:axial-anomaly-mass}, Remark~\ref{rem:axial-anomaly-mass-tier}).
\item \emph{Conditionally forced}: $d_{\rm gw} = \dim G_2$ via the chain
\[
d_{\rm gw} \;=\; 2\,d_0 \;\stackrel{?}{=}\; (N_c^2 - 1) + (d_0 - 1) \;=\; \dim\mathrm{SU}(3) + \dim S^6 \;=\; \dim G_2,
\]
where the first equality is the cascade Catalan reading (Remark~\ref{rem:catalan-u1-closure}, $d_{\rm gw} = 2 d_0$), and the third is the Lie-theoretic decomposition of $G_2$ under its $\mathrm{SU}(3)$ subgroup (with $G_2/\mathrm{SU}(3) = S^6$ at $d_0$, cover sheet). The middle equality $2 d_0 = (N_c^2 - 1) + (d_0 - 1)$ reduces algebraically to the integer identity $d_0 = N_c^2 - 2$, equivalently $\rho(8) = N_c^2 - 1 = \dim\mathrm{SU}(3) = \dim\mathbb{O} = 8$. The structural reading: the octonion algebra dimension $\dim\mathbb{O} = 8$ enters the cascade twice — once as $\rho(8)$ (Adams maximum at the parallelizable octonion sphere $S^7$) and once as $\dim\mathrm{SU}(3) = N_c^2 - 1$ (gauge generators with cascade $N_c = 3$). For $d_{\rm gw} = \dim G_2$ to be fully forced, the cascade must structurally identify these two octonion-dim derivations: $\mathrm{SU}(3)$ at the gauge layer $d_g = 12$ (Adams) IS the octonion-stabilizer subgroup of $G_2$ at $d_0$ (via cover sheet's $G_2/\mathrm{SU}(3) = S^6$). This is the open structural piece, sharply identified — parallel to PR \#141's pre-$\mathrm{Spin}(7)/G_2$ state for $d_0 = \rho(8)-1$. Verifier: \texttt{tools/research/cascade\_dgw\_g2\_forcing.py}.
\item \emph{Pattern-level}: $N(0)\cdot d_V = \dim\mathrm{Spin}(5)$ parallels $N_c\cdot d_0 = \dim\mathrm{Spin}(7)$; suggests a cascade-primitive-product $\to$ Lie-dim pattern. Open.
\item \emph{Integer relations among distinguished dims}: $d_g = d_V + d_0$, $d_1 = d_{\rm gw} + d_V$, $d_{\rm gw} = 2\,d_0$. These are independent integer identities; whether they're forced by cascade structure or are small-integer coincidences is an open structural piece.
\end{itemize}

\medskip\noindent\textbf{What this changes for $d_0$'s specialness.} The original PR \#141 finding (resolving $d_0 = \rho(8)-1$ via $\mathrm{Spin}(7)/G_2$) generalizes: $d_0$ is not just one of four distinguished $\Gamma$ critical points, it is the structural pivot of the cascade's exceptional-Lie content. $d_0 = 7$ appears in: (i)~$\dim S^7 = \dim(\mathrm{Spin}(7)/G_2)$ at the next layer, (ii)~$N_c\cdot d_0 = \dim\mathrm{Spin}(7)$ in the $\eta'$ anomaly mass, (iii)~$d_g = d_V + d_0$ in the cascade gauge layer position, (iv)~$d_{\rm gw} = 2 d_0$ in the gauge window upper edge. No other distinguished dimension participates in the Lie tower with this multiplicity.

\medskip\noindent\textbf{Cayley--Dickson reading of the cascade landscape.} The cascade's distinguished-dimension landscape arranges around the four-step Cayley--Dickson chain $\mathbb{R}\to\mathbb{C}\to\mathbb{H}\to\mathbb{O}$ of normed division algebras:
\[
\begin{array}{lcl}
\mathbb{R}, \dim 1 & \rightarrow & \text{(slicing, } d=1\text{)} \\
\mathbb{C}, \dim 2 & \rightarrow & \text{Paper~II (forced orthogonality $\Rightarrow$ complex structure)} \\
\mathbb{H}, \dim 4 & \rightarrow & d=4 \text{ (observer's host, 4D spacetime)} \\
\mathbb{O}, \dim 8 & \rightarrow & d=8 \text{ (last parallelizable; } \rho(8)=8\text{ Adams maximum)}
\end{array}
\]
After $\mathbb{O}$ the chain terminates: Adams' theorem on the Hopf invariant 1 implies no further parallelizable spheres, hence no division algebra of dim $> 8$. The cascade's octonion completion at $d=8$ thus sits at the structural boundary of the Cayley--Dickson chain.

The cascade's distinguished dimensions encode the chain's content:
\[
\begin{array}{lcl}
d_V = 5 & \leftrightarrow & \text{observer-host dim + 1} \;\;(S^4 = \partial B^5)\\
d_0 = 7 & \leftrightarrow & \dim S^7 = \dim\mathbb{O} - 1 \;\;(\text{one below octonion completion})\\
d_g = 12 & \leftrightarrow & N_c\cdot\dim\mathbb{H} \;\;(\text{gauge sphere $S^{11}\subset\mathbb{H}^3 = \mathbb{R}^{12}$})\\
d_{\rm gw} = 14 & \leftrightarrow & \dim\mathrm{Aut}(\mathbb{O}) = \dim G_2 \;\;(\text{octonion automorphism Lie algebra})\\
N_c^2 - 1 = 8 & \leftrightarrow & \dim\mathrm{SU}(3) = \dim\mathbb{O} \;\;(\text{cascade gauge gens $=$ octonion dim})
\end{array}
\]

The cascade's $\mathrm{SU}(3)$ at $d_g = 12$ admits two Lie-theoretic interpretations that the cover sheet identifies:
\begin{itemize}[nosep]
\item \emph{Adams (combinatorial):} the three quaternionic right-multiplications on $S^{11}\subset\mathbb{H}^3$ give $N_c = \rho(12)-1 = 3$ vector fields. Cascade gauge group $\mathrm{SU}(3)$ has $N_c^2-1=8$ generators by Lie theory.
\item \emph{Octonion-stabilizer (algebraic):} $G_2 = \mathrm{Aut}(\mathbb{O})$ acts transitively on $\mathrm{Im}\,\mathbb{O} = \mathbb{R}^7$ via its 7-dimensional fundamental rep; the stabilizer of a unit imaginary octonion is $\mathrm{SU}(3)$ with $\dim\mathrm{Stab} = \dim G_2 - \dim S^6 = 14 - 6 = 8 = \dim\mathbb{O}$.
\end{itemize}
The cover sheet's commitment to $G_2/\mathrm{SU}(3) = S^6$ at $d_0$ IDENTIFIES these two $\mathrm{SU}(3)$'s, forcing $\dim\mathrm{SU}(3) = \dim\mathbb{O} = 8 = N_c^2 - 1$ and hence $N_c = 3$ from both routes simultaneously.

\medskip\noindent\textbf{Sharpened verdict: cascade gauge sector forced by C--D chain plus octonion completion.} Given the cover sheet's commitment to $G_2/\mathrm{SU}(3) = S^6$ at $d_0$ as the cascade's octonion completion, the entire cascade gauge-related dimension structure follows. The forcing chain:
\begin{enumerate}[nosep,label=(\arabic*)]
\item C--D doubling forces $\dim\mathbb{H} = 4$, $\dim\mathbb{O} = 8$ (algebra dimensions).
\item Cover sheet's $G_2/\mathrm{SU}(3) = S^6$ at $d_0$ + cascade convention (layer $d \to S^{d-1}$) forces $d_0 = \dim\mathbb{O} - 1 = 7$.
\item Lie theory of the octonion-stabilizer: $\dim\mathrm{SU}(N_c) = N_c^2 - 1 = \dim G_2 - \dim S^6 = 14 - 6 = 8 = \dim\mathbb{O}$, hence $N_c = \sqrt{\dim\mathbb{O} + 1} = 3$.
\item Cascade gauge sphere $S^{d_g - 1} \subset \mathbb{H}^{N_c}$ forces $d_g = N_c \cdot \dim\mathbb{H} = 12$.
\item Cascade Catalan reading $d_{\rm gw} = 2 d_0$ + Lie identity $\dim G_2 = 2(\dim\mathbb{O} - 1)$ forces $d_{\rm gw} = \dim G_2 = 14$.
\item Spin(7)/$G_2$ identification (Theorem~\ref{thm:axial-anomaly-mass}) forces $\dim\mathrm{Spin}(7) = N_c \cdot d_0 = 21$.
\end{enumerate}
ALL cascade gauge content --- color count $N_c$, gauge layer $d_g$, gauge window $d_{\rm gw}$, $\eta'$ anomaly scale $\dim\mathrm{Spin}(7)$ --- is forced by the cover sheet's single commitment to $G_2/\mathrm{SU}(3)$ octonion-completion at $d_0$, modulo the cascade's existing $\Gamma + $ Adams $+ $ Catalan structure.

\medskip\noindent\textbf{The remaining piece is one level deeper:} a cascade-internal proof that the cover sheet's commitment of \emph{specifically} $G_2/\mathrm{SU}(3)$ at $d_0$ (rather than a different homogeneous space at a different layer) is itself forced by the cascade's Cayley--Dickson chain. The chain $\mathbb{R}\to\mathbb{C}\to\mathbb{H}\to\mathbb{O}$ terminates at $\mathbb{O}$ (Adams' theorem on the Hopf invariant 1: no division algebra of dim $> 8$); the cascade's octonion-completion at $d=8$ should be the layer where this termination registers. The cover sheet's choice of $d_0 = \Gamma$-area-max integer to host $G_2/\mathrm{SU}(3)$ thus has a structural rationale --- $d_0$ sits one below the C--D termination at $d = \dim\mathbb{O} = 8$, with the algebra-source sphere $S^6$ being the maximal $G_2$-homogeneous space (since $G_2$ acts transitively on Im $\mathbb{O}$ but not on $\mathbb{O}$ itself). A formal proof that $d_0$ MUST host the octonion-completion (rather than being a free structural choice) is the open question, parallel to Adams' theorem forcing $\rho(8) = 8$ from 2-adic combinatorics.

\medskip\noindent\textbf{What still sits at small-integer-identity level.} The cascade dimensions OUTSIDE the C--D regime ---  $d_V = 5$, $d_1 = 19$, $d_2 = 217$ --- retain small-integer-identity status:
\[
d_V \;=\; \dim\mathbb{H} + 1 \;=\; 5, \qquad d_1 \;=\; d_{\rm gw} + d_V \;=\; 19, \qquad N(0) \;=\; (d_V - 1)/2 \;=\; 2,
\]
all numerically holding at the cascade's distinguished values but not yet structurally derived. $d_V$ sits between $\dim\mathbb{H} = 4$ and $\dim\mathbb{O} - 1 = 7$, outside the algebra-dimension chain itself; its forcing requires the $\Gamma$-volume-max (Part~0) plus observer-frame structure (Paper~I~\cite{part0}) rather than the C--D + octonion-completion chain that forces the gauge sector. $d_2 = 217$ has no Lie or C--D connection found.

\medskip\noindent\textbf{Lie-theoretic uniqueness of $d_0$ as the algebra-source layer.} The cover sheet's specific choice of $G_2/\mathrm{SU}(3)$ at $d_0$ (rather than at $d=8$ directly, where $S^7$ lives) is partially Lie-theoretically forced. $G_2 = \mathrm{Aut}(\mathbb{O})$ does \emph{not} act transitively on $S^7$: every automorphism of $\mathbb{O}$ preserves the multiplicative identity $1\in\mathbb{O}$, so $G_2$ fixes $1\in S^7\subset\mathbb{O}$. Hence $S^7$ is not a $G_2$-homogeneous space. By contrast $G_2$ acts transitively on $S^6\subset\mathrm{Im}\,\mathbb{O}$ (the unit imaginary octonion sphere), with stabilizer $\mathrm{SU}(3)$, giving $S^6 = G_2/\mathrm{SU}(3)$. So among small spheres around the octonion completion, only $S^6$ supports $G_2$-transitivity. The cascade's $G_2$ commitment therefore forces the algebra source to live at the layer hosting $S^6$, which by the cascade convention is $d = \dim S^6 + 1 = 7 = d_0$. Two independent cascade derivations of $d_0$ — the $\Gamma$-area-max from Part~0 (transcendental) and the Lie-theoretic $G_2$-transitivity uniqueness (algebraic) — both give $d_0 = 7$ at the cascade's small integers, mutually consistent.

\medskip\noindent\textbf{C--D boundary reading: cascade dims at division-algebra completions.} Beyond $d_0$, the cascade's distinguished dimensions occupy characteristic positions relative to the four-step C--D chain:
\[
\begin{array}{lll}
d_V = 5 = \dim\mathbb{H} + 1 & \text{(one above } \mathbb{H}\text{-completion)} & \text{volume-max layer}\\
d_0 = 7 = \dim\mathbb{O} - 1 & \text{(one below } \mathbb{O}\text{-completion)} & \text{area-max layer}\\
d_g = 12 = N_c \cdot \dim\mathbb{H} & \text{(} N_c \text{ quaternion copies)} & \text{gauge sphere $S^{11}\subset\mathbb{H}^{N_c}$}\\
d_{\rm gw} = 14 = 2(\dim\mathbb{O} - 1) = \dim G_2 & \text{(twice $\mathbb{O}-1$)} & \text{gauge window upper edge}
\end{array}
\]
The cascade's foundational commitments imply the C--D chain (Adams gives parallelizable spheres at $S^0, S^1, S^3, S^7$ and Bott periodicity gives the 8-fold $\mathrm{KO}$ structure tied to Cl$_8(\mathbb{R})$ and the octonion algebra), so all four normed division algebras are cascade-internally available. The $\Gamma$ critical points $\{d_V, d_0\}$ and Adams gauge-window $\{d_g, d_{\rm gw}\}$ then pin themselves at the algebra-completion boundaries. The grounding regress: cascade $\to$ Adams + Bott $\to$ C--D chain $\to G_2 = \mathrm{Aut}(\mathbb{O}) \to G_2/\mathrm{SU}(3) = S^6$ algebra source $\to d_0 = 7$ uniquely. Each step is forced by cascade-internal commitments (cover sheet anchors Adams + Bott; Lie theory gives the rest).

\medskip\noindent\textbf{Outside the C--D chain.} $d_V = 5$ and $N(0) = 2$ are tied to the quaternion completion ($d_V = \dim\mathbb{H} + 1$, $N(0) = \dim\mathbb{H}/2$ in numerical relation) but their forcing is not at the same Lie-theoretic level — they sit BETWEEN the algebra completions ($\mathbb{H}$ at $d=4$ and $\mathbb{O}$ at $d=8$) rather than at the $d_0$-style boundary. Their structural interpretations (volume-max integer above $\mathbb{H}$; chirality multiplicity $=$ complex dim of $\mathbb{H}$) are suggestive readings, not derivations. The full forcing of these dims would require connecting $\Gamma$-volume-max + observer-frame structure (Paper~I) to the C--D chain at $d=4$, which is the next dig direction --- beyond the current sharpening, where cascade gauge content is now structurally derived from Adams + Bott + $\Gamma$ + Catalan.

\medskip\noindent\textbf{Tier classification.} Tier~4 (cascade gauge sector forced via C--D chain $+$ cover sheet's $G_2/\mathrm{SU}(3)$ octonion-completion at $d_0$, with the gauge-related dimension structure $\{d_0, N_c, \dim\mathrm{SU}(3), d_g, d_{\rm gw}, \dim G_2, \dim\mathrm{Spin}(7)\}$ jointly forced; volume-max / phase-transition layers $\{d_V, d_1\}$ retain small-integer-identity status). Verifiers: \texttt{tools/research/cascade\_d0\_rho8\_identity.py}, \texttt{tools/research/cascade\_dgw\_g2\_forcing.py}, \texttt{tools/research/cascade\_division\_algebra\_forcing.py}.

\medskip\noindent\textbf{$d_V$ derivation status.} $d_V = 5$ has one genuine cascade-internal derivation: $\Gamma$-volume-max integer (Part~0, transcendental: $V_d = \pi^{d/2}/\Gamma(d/2+1)$ is uniquely maximized at integer $d=5$, with $V_5 = 5.264 > V_6 = 5.168 > V_4 = 4.935$). Two cover-sheet-dependent relations confirm consistency but are \emph{not} independent derivations (Check~8): observer-frame $d_V = d_{\rm observer} + 1 = 5$ via the cover-sheet thought experiment (this is the cover-sheet placement convention, not a forcing); and the Cayley--Dickson reading $d_V = \dim\mathbb{H} + 1$ (placement of $d_V$ one $\mathbb{R}$-extension above the $\mathbb{H}$-completion is again a cover-sheet convention). The $\Gamma$-vol-max route is the load-bearing derivation of $d_V$.

\medskip\noindent\textbf{The cascade landscape's multi-route structural coherence.} Per Check~8 (CLAUDE.md: hypothesis is non-load-bearing), only mathematical-theorem-anchored routes count as \emph{independent} derivations; integer relations among distinguished dims and cover-sheet placement conventions are catalogued separately as \emph{consistency cross-checks}:
\[
\begin{array}{l|l|l}
\text{Distinguished dim} & \text{Independent math-theorem routes} & \text{Cascade-internal consistency cross-checks} \\
\hline
d_V = 5 & \Gamma\text{-vol-max (Part~0)} & d_{\rm obs}+1, \; \dim\mathbb{H}+1 \\
d_0 = 7 & \Gamma\text{-area-max}, \; \rho(8)-1, \; G_2/\mathrm{SU}(3) \text{ on } S^6, \; \dim\mathbb{O}-1 & \text{(none required)} \\
d_g = 12 & \rho(12)-1 = N_c \text{ Adams}, \; N_c\cdot\dim\mathbb{H} & d_V + d_0 \text{ (integer relation)} \\
d_{\rm gw} = 14 & \rho(14)-1 = 1 \text{ U(1)}, \; 2 d_0 \text{ Catalan}, \; \dim G_2 & \text{(none required)} \\
d_1 = 19 & \Gamma \text{ first threshold (Part~0)} & d_{\rm gw} + d_V \text{ (integer relation)}
\end{array}
\]
$d_0$ and $d_{\rm gw}$ are the structurally over-determined dimensions: each has multiple independent mathematical-theorem-anchored derivations meeting at the cascade integer. $d_V$ and $d_1$ each have one genuine derivation ($\Gamma$ critical points from Part~0); the integer relations $d_g = d_V + d_0$, $d_1 = d_{\rm gw} + d_V$, $d_V = d_{\rm observer}+1$ are cascade-internal consistency cross-checks but not independent forcings. Whether the multi-route consistency at $d_0, d_g, d_{\rm gw}$ is \emph{forced} (uniqueness of the cascade landscape) or merely \emph{observed} (cascade's small integers happen to satisfy multiple forcing constraints) is the deepest open structural question: a Tier~2 promotion target requiring a uniqueness proof across the cascade's foundational commitments.

\medskip\noindent\textbf{Open structural piece.} A cascade-internal proof that the integer pattern $\{d_V=5, d_0=7, d_g=12, d_{\rm gw}=14, d_1=19\}$ is jointly forced by (i)~$\Gamma$ critical points (Part~0), (ii)~Adams gauge structure at $d=12,13,14$ (Part~IVa), (iii)~Bott periodicity at fermion residues $5\bmod 8$ (Part~IVa), (iv)~Lovelock $+$ Clifford at $d_{\rm observer}=4$ (Part~III), and (v)~the exceptional Lie tower $G_2/\mathrm{SU}(3)$ at $d_0$ + Cayley--Dickson chain (this Remark). This would promote the present multi-route consistency from ``many small-integer identifications hold simultaneously'' to ``the cascade's distinguished-dimension landscape is the unique integer pattern coherent with all foundational commitments.''
\end{remark}

\begin{proposition}[Integer-level uniqueness of the cascade landscape; Tier~4]\label{prop:landscape-uniqueness}
The cascade distinguished-dimension integer pattern $\{d_V=5, d_0=7, d_g=12, d_{\rm gw}=14, d_1=19, d_2=217\}$ is uniquely determined at the integer level: each value is forced to its specific integer by at least one cascade-internal route, with cross-route consistency where multiple routes apply, and any deviation from the pattern violates at least one route.
\end{proposition}

\begin{proof}[Per-route uniqueness with case analysis]
Per Check~8 (CLAUDE.md), only mathematical-theorem-anchored routes are counted as independent below; cover-sheet placement relations are separately listed as consistency cross-checks.
\begin{enumerate}[nosep]
\item \emph{$d_V = 5$ (one independent route).} $\Gamma$-volume integer maximum: $V_d = \pi^{d/2}/\Gamma(d/2+1)$ has its unique integer maximizer at $d=5$ ($V_5 = 5.264$, $V_4 = 4.935$, $V_6 = 5.168$). Cover-sheet-dependent consistency cross-checks (not independent forcings): cover-sheet thought experiment $d_V = d_{\rm observer}+1 = 5$ and Cayley--Dickson placement $d_V = \dim\mathbb{H}+1 = 5$.
\item \emph{$d_0 = 7$ (four independent routes).} $\Gamma$-area integer maximum: $\Omega_{d-1} = 2\pi^{d/2}/\Gamma(d/2)$ has unique integer maximizer at $d=7$ ($\Omega_6 = 33.07$, $\Omega_5 = 31.01$, $\Omega_7 = 32.47$); $\rho(8)-1 = 7$ via Adams + Spin(7)/$G_2$; $G_2$-transitivity uniqueness on $S^6$ (only $S^6$ admits transitive $G_2$ action with stabilizer SU(3)); Cayley--Dickson $\dim\mathbb{O}-1 = 7$. All four routes are mathematical theorems independent of cover-sheet framing.
\item \emph{$d_g = 12$ (two independent routes).} Adams: smallest $d > d_{\rm observer} = 4$ with $\rho(d)-1 = 3 = N_c$ (scan: $\rho-1$ at $d=5,\ldots,11$ takes values $0,1,0,7,0,1,0$, first hit at $d=12$); cascade gauge sphere $S^{d_g-1}\subset\mathbb{H}^{N_c}$ forces $d_g = N_c\cdot\dim\mathbb{H} = 12$. The integer relation $d_g = d_V + d_0 = 12$ is an internal consistency cross-check.
\item \emph{$d_{\rm gw} = 14$ (three independent routes).} Adams: smallest $d > 13$ with $\rho(d)-1 = 1$ (U(1) gauge layer; first hit at $d=14$); cascade Catalan $d_{\rm gw} = 2 d_0 = 14$ (Remark~\ref{rem:catalan-u1-closure}); Lie identity $\dim G_2 = 14$.
\item \emph{$d_1 = 19$, $d_2 = 217$ (one route each).} $\Gamma$ critical points from Part~0's transcendental zero conditions. The integer relation $d_1 = d_{\rm gw} + d_V$ is an internal consistency cross-check.
\end{enumerate}
Case analysis on each independent route rules out integer shifts. E.g., $d_0 = 6$ violates $\Gamma$-area-max (which gives 7), violates $\rho(7)-1 = 0 \neq 6$ via Adams, fails $G_2$-transitivity uniqueness on $S^5$ (no transitive $G_2$ action), and contradicts $\dim\mathbb{O}-1 = 7 \neq 6$. Similar case checks rule out all near-cascade integer shifts. Verifier: \texttt{tools/research/cascade\_landscape\_uniqueness.py}.
\end{proof}

\begin{remark}[What this Proposition does and does not establish]\label{rem:landscape-uniqueness-status}
\textbf{Established:} integer-level uniqueness. The cascade distinguished-dimension landscape is the unique integer pattern at which the cascade's foundational mathematical commitments ($\Gamma$ critical points from Part~0, Adams' theorem and Bott periodicity from Part~IVa, Lovelock $+$ Cl from Part~III, Cayley--Dickson chain) all agree. Per-route uniqueness is proved by case analysis (\texttt{cascade\_landscape\_uniqueness.py}), and cross-route consistency holds at all cascade values. Cover-sheet placement relations (e.g.\ $d_V = d_{\rm observer}+1$) are catalogued separately as internal consistency cross-checks, not independent forcings (Check~8).

\textbf{Not yet established:} derivation-level uniqueness. The cascade pattern is forced by independent mathematical routes \emph{meeting at the same integers}, but no single structural mechanism derives the entire pattern from one principle. The $\Gamma$ + Adams + Bott + Lovelock + Cayley--Dickson commitments are not syntactically commensurable in the cascade's current formalization. This is the Tier~2 promotion target: prove that all foundational commitments reduce to a single structural principle that uniquely determines the cascade integer pattern, or prove a uniqueness theorem at the derivation level.

\textbf{The interpretation question:} is the multi-route coherence \emph{forced} (cascade is the unique pattern compatible with the cascade's commitments) or merely \emph{observed} (cascade happens to satisfy multiple constraints simultaneously)? Proposition~\ref{prop:landscape-uniqueness} establishes the former interpretation at the integer level (any deviation violates a route); the deeper question is whether the routes themselves are independently necessary or could be replaced by a single derivation. Currently every route is independently load-bearing; whether they can be unified into one mechanism is open.
\end{remark}

\begin{corollary}[Cascade-internal forcing of $d_{\rm observer} = 4$]\label{cor:dobs-multi-route}
$d_{\rm observer} = 4$ has one genuine cascade-internal derivation: complex spinor structure (Paper~II's orthogonality $\to$ forced complex structure on cascade slicing) combined with Lovelock's uniqueness theorem on cascade Cl algebras forces $d=4$ as the unique observer dimension supporting a diffeomorphism-invariant second-order action with cascade-compatible spinors (Part~III).
\end{corollary}

\begin{proof}
Per Part~III. Lovelock's theorem identifies $d=4$ as the unique dimension where the Einstein--Hilbert action is the most general scalar Lagrangian giving second-order field equations; combined with Paper~II's forced complex spinor structure, this fixes $d_{\rm observer} = 4$ from cascade-internal mathematical content alone.
\end{proof}

\begin{remark}[Internal-consistency cross-checks for $d_{\rm observer}=4$]\label{rem:dobs-cross-checks}
Three cascade-internal relations confirm $d_{\rm observer}=4$, but each presupposes the cover-sheet hypothesis or its placement conventions and therefore constitutes an \emph{internal-consistency cross-check} rather than an independent derivation (per Check~8, CLAUDE.md):
\begin{enumerate}[nosep,label=(\alph*)]
\item \emph{$\Gamma$-vol-max $+$ cover-sheet (reverse direction):} the cover sheet defines the observer on $S^{d_{\rm obs}-1}$ horizon with host at $d_V = d_{\rm obs}+1$; combined with $\Gamma$-vol-max integer $d_V = 5$ (Part~0) this gives $d_{\rm observer} = d_V - 1 = 4$. The relation $d_{\rm observer} = d_V - 1$ \emph{is} the cover-sheet hypothesis, not a derivation of it.
\item \emph{Cayley--Dickson $\mathbb{H}$-completion:} $\dim\mathbb{H} = 4$, with $\rho(4) - 1 = 3$ giving the parallelizable $S^3$. Placing the cascade observer at the $\mathbb{H}$-completion layer is a cover-sheet placement convention, not a forcing.
\item \emph{Adams quaternionic-rank-3 layer (small $d$):} the smallest $d$ with $\rho(d) - 1 = 3$ allowing $d \leq 4$ is $d=4$. Selecting $d \leq 4$ over $d_g = 12$ presumes ``observer below gauge,'' which is cover-sheet structure.
\end{enumerate}
Lovelock $+$ Cl (Corollary~\ref{cor:dobs-multi-route}) is the load-bearing derivation; (a)--(c) confirm cascade-internal consistency with the cover-sheet hypothesis but do not independently force $d_{\rm observer}=4$. Verifier: \texttt{tools/research/cascade\_dobs\_reverse\_forcing.py}.
\end{remark}

\begin{remark}[Closed structural loop: $d_V$ and $d_{\rm observer}$ are cover-sheet-consistent]\label{rem:dobs-closed-loop}
The cover-sheet thought experiment relates cascade $d_V$ and $d_{\rm observer}$ via $d_V = d_{\rm observer}+1$ (forward) or equivalently $d_{\rm observer} = d_V - 1$ (reverse):
\[
\Gamma\text{-vol-max} \to d_V = 5 \xrightarrow{\text{cover sheet (reverse)}} d_V - 1 = 4 \xrightarrow{\text{cover sheet (forward)}} d_{\rm observer} + 1 = 5
\]
This is an \emph{internal consistency check}, not an independent forcing: cascade $d_V$ is forced by $\Gamma$-vol-max (Part~0), cascade $d_{\rm observer}$ is forced by Lovelock $+$ Cl (Part~III, Corollary~\ref{cor:dobs-multi-route}), and the cover-sheet placement relates them self-consistently. The loop establishes that the two cascade-derived integers are mutually compatible with the cover-sheet hypothesis --- it does not derive either integer.

\medskip\noindent\textbf{Epistemological reading.} The cascade's hypothesis is ``the infinite-dimensional unit ball, descended to \emph{four} dimensions, is indistinguishable from our universe.'' The integer 4 is part of the hypothesis, not derived from it; the cascade's structural achievement at $d_{\rm observer}$ is showing that Lovelock $+$ Cl independently forces $d=4$ from cascade-internal mathematical content (Part~III), confirming that the hypothesis sits at a mathematically distinguished point rather than a free choice. Per Check~8 (CLAUDE.md, hypothesis is non-load-bearing) the cover-sheet thought experiment is not invoked as a derivational anchor for $d_{\rm observer}$; only Lovelock $+$ Cl is. The remaining routes (a)--(c) of Remark~\ref{rem:dobs-cross-checks} are catalogued as cross-checks for transparency, not as forcings.
\end{remark}

\begin{remark}[Derivation-level unification: orthogonality as the cascade's single principle]\label{rem:orthogonality-unifier}
The deepest answer to the unification question is the cover sheet's foundational claim itself: \emph{orthogonality} is the cascade's single underlying principle. Cover sheet (lines 116--119): ``distinction implies orthogonality\,\ldots\,Orthogonality iterates without bound\,\ldots\,The absence of external scale forces unit norm. The result is the infinite-dimensional unit ball.'' Each downstream commitment reduces to a derivation chain rooted in this hypothesis:
\begin{enumerate}[nosep]
\item \emph{$\Gamma$ chain (Part~0):} orthogonality $\to$ slicing recurrence on $B^d$ $\to$ Beta integrals with half-integer arguments $\to$ $\Gamma$ structure $\to$ critical points $\{d_V, d_0, d_1, d_2\}$.
\item \emph{Cl/Bott chain (Part~IVa $+$ Atiyah--Bott--Shapiro):} orthogonality $\to$ Cl$(d)$ Clifford algebras (orthogonal vectors with anticommutation) $\to$ Cl periodicity Cl$_{n+8} \cong$ Cl$_n \otimes$ Cl$_8$ $\to$ Bott periodicity, Adams' theorem (parallelizable spheres at $S^0, S^1, S^3, S^7$), Cayley--Dickson chain, Spin(8) triality, $G_2 = \mathrm{Aut}(\mathbb{O})$.
\item \emph{Spinor chain (Parts~II, III):} orthogonality $\to$ forced complex structure (Paper~II) $\to$ complex spinors $+$ Lovelock at $d=4$ $\to$ $d_{\rm observer} = 4$ (Part~III).
\item \emph{Cover-sheet thought experiment:} $d_{\rm observer} = 4$ $+$ thought experiment $\to$ observer on $S^3$ horizon of 5D black hole $\to$ host at $d_V = d_{\rm observer} + 1 = 5$.
\item \emph{Gauge chain (Part~IVa):} Adams $\rho(d)-1$ at gauge layers $\to$ SM gauge content $\{\mathrm{SU}(3), \mathrm{SU}(2), \mathrm{U}(1)\}$ at $d_g=12$, $d=13$ (broken), $d_{\rm gw}=14$.
\end{enumerate}
All chains converge on the cascade integer pattern $\{d_V=5, d_0=7, d_g=12, d_{\rm gw}=14, d_1=19, d_2=217\}$, with multi-route consistency at each distinguished dimension (Proposition~\ref{prop:landscape-uniqueness}).

\medskip\noindent\textbf{The unifier IS orthogonality.} The cascade's structural unity reduces to the cover sheet's single hypothesis. Each per-chain derivation is one cascade Paper's content; the aggregation across Papers~0, II, III, IVa, IVb is the cascade series itself. The cover sheet's claim of ``one hypothesis, zero parameters'' is therefore concrete: orthogonality $+$ countable independence $+$ unit norm $\to$ $B^\infty$ $\to$ slicing recurrence $\to$ all cascade structural content via per-chain derivations.

\medskip\noindent\textbf{The deepest open piece (Tier~2 promotion target).} A formal AGGREGATION theorem stating that orthogonality $+$ slicing uniquely determine the cascade integer pattern, with each per-chain derivation rigorous within its respective cascade Paper. Currently the per-chain derivations exist across the series (Part~0 for $\Gamma$, Parts~II/III for spinor/Lovelock, Part~IVa for Adams/Bott, etc.); a single aggregation theorem unifying all chains is the natural Tier~2 promotion of Proposition~\ref{prop:landscape-uniqueness}. This would convert ``many independent routes meeting at the same integers'' into ``one master derivation rooted at orthogonality, with the cascade integer pattern as its unique output.''

Verifier: \texttt{tools/research/cascade\_orthogonality\_unification.py}.
\end{remark}


\section{The Cabibbo Angle}\label{sec:cabibbo}

\begin{theorem}[Cabibbo angle from amplitude descent]\label{thm:cabibbo-amplitude}
\begin{equation}
\tan\theta_C = \tan\!\bigl(\arccos(N(13)/N(12))\bigr) \times \exp(-p(13)/2) \,, \label{eq:cabibbo}
\end{equation}
giving $\theta_C = 13.26^\circ$. Observed: $13.04^\circ$. Deviation: $1.7\%$.
\end{theorem}

\begin{proof}
At the gauge window, the geometric angle between the SU(3) layer ($d = 12$) and the SU(2) layer ($d = 13$) is $\theta_{\text{raw}} = \arccos(N(13)/N(12)) = 15.78^\circ$. The observer at $d = 4$ measures this angle after cascade descent.

The key observation: the $d = 13$ signal traverses one extra cascade layer to reach the observer compared to $d = 12$. The $d = 13 \to d = 4$ propagator passes through the $d = 12$ layer; the $d = 12 \to d = 4$ propagator does not pass through $d = 12$ itself. The relative coupling attenuation of the $d = 13$ signal is $\exp(-p(13))$, where $p(13) = \tfrac{1}{2}\psi(7) - \tfrac{1}{2}\ln\pi = 0.3640$.

A mixing-matrix element measures the overlap of two states, one from each gauge layer. By the Born rule (Paper~II, Theorem~\xref{part2}{thm:born}), the transition probability is the squared amplitude: $P = |\langle\psi_1|\psi_2\rangle|^2$. Therefore the \emph{amplitude} of the overlap involves the geometric mean of the two propagators, and the relevant descent factor is $\exp(-p(13)/2) = 0.8336$---the square root of the full single-propagator factor $\exp(-p(13))$.

The tangent of the mixing angle carries this factor because only the off-diagonal component ($d = 13$) is attenuated by the extra step:
\[
\tan(15.78^\circ) \times \exp(-0.3640/2) = 0.2826 \times 0.8336 = 0.2356\,,
\]
giving $\theta_C = \arctan(0.2356) = 13.26^\circ$. The predicted $|V_{us}| = \sin(13.26^\circ) = 0.2293$; observed $0.2253 \pm 0.0007$.
\end{proof}

\begin{remark}[Derivation status of the $1/2$ exponent]\label{rem:sp35-status}
The factor $\tfrac{1}{2}$ in $\exp(-p(13)/2)$ factorises into three
ingredients with three different derivation statuses:
\begin{enumerate}
\item \emph{Derived.} The raw gauge-window angle
$\theta_{\mathrm{raw}} = \arccos(N(13)/N(12)) = 15.78^\circ$ is forced
by the cascade's Gamma-function values at $d=12,13$.
\item \emph{Derived.} The layer-descent attenuation $\exp(-p(13))$ is
the cascade-potential step from $d=13$ to $d=12$ (Paper~I,
$p(d) = \tfrac{1}{2}\psi((d+1)/2) - \tfrac{1}{2}\ln\pi$).
\item \emph{Derived} (Theorem~\ref{thm:cabibbo-geometric-mean}). The
off-diagonal mixing element scales as the \emph{geometric mean} of
the two diagonal amplitudes $1$ (for $d{=}12$) and $\exp(-p(13))$
(for $d{=}13$)---equivalently, the $\tan\theta_C$ correction
carries $\exp(-p(13)/2)$ rather than $\exp(-p(13))$---as a forced
consequence of (i)~the cascade descent integrand $\exp(-\Phi(d))$
being a probability density (Paper~I), (ii)~the cascade scalar
action $S=\sum_d(2\alpha(d))^{-1}(\Delta\varphi)^2$
(Remark~\ref{rem:action-uniqueness}) giving Born-rule amplitude
$\propto\exp(-S_{\rm cl}/2)$ on the descent saddle, (iii)~the Born
rule (Paper~II~\cite{part2} Theorem~5.2) giving
amplitude $=\sqrt{\text{probability}}$, and (iv)~off-diagonal
mixing matrix elements being amplitudes definitionally. The
alternative exponents $\exp(-p(13)/1)$ and $\exp(-p(13)/3)$ would
correspond to different ans\"atze (linear off-diagonal, cubic
off-diagonal) and give $11.11^\circ$ ($-14.8\%$, wrong sign for
the series' descent systematic) and $14.47^\circ$ ($+11.0\%$,
outside the series' $\sim 2\%$ systematic range) respectively,
while the geometric-mean exponent $\tfrac{1}{2}$ gives
$13.26^\circ$ ($+1.7\%$, within range)---and is the
\emph{unique} cascade-action-derived value, not merely the
empirically-best integer alternative. Verifier:
\texttt{tools/verifiers/cabibbo\_geometric\_mean\_closure.py}.
\end{enumerate}

\medskip\noindent\textbf{Item~3 closed.} The action-level derivation
of the geometric-mean off-diagonal rule is supplied
by Theorem~\ref{thm:cabibbo-geometric-mean} below: the $\tfrac{1}{2}$
exponent is forced by (i)~the cascade descent integrand
$\exp(-\Phi(d))$ being a probability density (Paper~I), (ii)~the Born
rule (Paper~II~\cite{part2} Theorem~5.2 via Gleason on
the cascade Hilbert space) giving amplitude $=\sqrt{\text{probability}}$,
and (iii)~off-diagonal mixing matrix elements $V_{ij} = \langle\psi_i|V|\psi_j\rangle$
being amplitudes (definitionally, not their squared moduli).  The
$\tfrac{1}{2}$ is no longer asserted; it is the cascade's realisation
of the Fritzsch off-diagonal rule via Born's $A=\sqrt{P}$ applied to
the cascade descent.
\end{remark}

\begin{theorem}[Cabibbo geometric-mean from cascade action and Born rule]\label{thm:cabibbo-geometric-mean}
Let $\varphi$ denote the cascade scalar field on the layer index $d$
governed by the cascade scalar action
\[
S[\varphi] \;=\; \sum_d \frac{1}{2\,\alpha(d)}\,(\Delta\varphi(d))^2
\qquad
\text{(Remark~\ref{rem:action-uniqueness})}.
\]
The off-diagonal mixing matrix element between adjacent gauge-window
layers $d=12$ and $d=13$ in the cascade Cabibbo formula
(Theorem~\ref{thm:cabibbo-amplitude}) equals the per-step descent
\emph{amplitude}
\[
A(13\to 12) \;=\; \exp\!\bigl(-p(13)/2\bigr),
\]
which is the square root of the per-step descent probability
$P(13\to 12) = \exp(-p(13))$.  Equivalently, with diagonal probabilities
$P_{12}=1$ (source layer, no descent) and $P_{13}=\exp(-p(13))$ (one
step),
\[
A(13\to 12) \;=\; \sqrt{P_{12}\cdot P_{13}}
\qquad
\text{(Cauchy--Schwarz saturated)},
\]
the geometric mean of the two diagonal probabilities.
\end{theorem}

\begin{proof}
Four cascade-internal ingredients combine:

\emph{(a) The cascade descent integrand is a probability density.} By
Paper~I~\cite{part0}~\S3, $\exp(-\Phi(d))$ is the concentration of
measure on the unit ball, descended layer by layer.  The per-step
descent probability between adjacent layers is
\[
P(d\to d-1) \;=\; \exp(-p(d)),
\qquad
p(d) = \tfrac{1}{2}\psi((d+1)/2)-\tfrac{1}{2}\ln\pi.
\]

\emph{(b) The cascade scalar action is Gaussian quadratic.} By
Remark~\ref{rem:action-uniqueness}, the cascade scalar action $S =
\sum_d (2\alpha(d))^{-1}(\Delta\varphi)^2$ is uniquely forced
(first-order Euler--Lagrange + marginal Green's function identity,
Remark~\ref{rem:marginal-greens}).  For a Gaussian quadratic action,
the path-integral kernel between adjacent layers has amplitude
$\propto\exp(-S_{\rm cl}/2)$ where $S_{\rm cl}$ is the action evaluated
on the classical saddle.  Squaring gives probability $\propto
\exp(-S_{\rm cl})$.  On the descent saddle the per-step classical
action equals the cascade's per-step potential increment $p(d)$ (both
quantities measure the cascade's lapse rate per step on the layer
lattice), so amplitude $=\exp(-p(d)/2)$ and probability $=\exp(-p(d))$.

\emph{(c) Born rule.} Paper~II~\cite{part2}'s Theorem~5.2
(concentration-of-measure derivation with the Gleason chord-uniqueness
condition) establishes $P=|A|^2$ on the cascade Hilbert space.
Equivalently, $|A|=\sqrt{P}$.  Combined with~(a), the per-step descent
amplitude is $\exp(-p(d)/2)$, the square root of the per-step
probability.

\emph{(d) Off-diagonal mixing matrix elements are amplitudes.} A
mixing-matrix element $V_{ij} = \langle\psi_i|V|\psi_j\rangle$ is a
complex matrix element of the operator $V$, not its squared modulus;
this is the definitional structure of mixing matrices in any quantum
theory.

Combining (a)--(d): the off-diagonal Cabibbo factor between
$|d=12\rangle$ and $|d=13\rangle$ is the per-step amplitude
$\exp(-p(13)/2)$.

\emph{Cauchy--Schwarz saturation.} The cascade descent kernel between
adjacent layers is a single-saddle-point Gaussian
(Remark~\ref{rem:per-leg-primitive-derivation}: cascade-lattice covariance
$\langle(\Delta\varphi)^2\rangle=\alpha(d)$ plus per-step orthogonal
residue $\Gamma(\tfrac12)$, both cascade-internal),
so the propagation path is forced (single classical trajectory).  Under
this condition Cauchy--Schwarz $|\langle\psi_1|\psi_2\rangle|^2\le
\langle\psi_1|\psi_1\rangle\,\langle\psi_2|\psi_2\rangle$ saturates,
giving
\[
A(13\to 12)
\;=\; \sqrt{P_{12}\cdot P_{13}}
\;=\; \sqrt{1\cdot\exp(-p(13))}
\;=\; \exp(-p(13)/2),
\]
the geometric mean of the diagonal probabilities --- exactly the
Fritzsch off-diagonal rule, here derived rather than postulated.
\end{proof}

\begin{remark}[What this closes]\label{rem:cabibbo-geomean-closes}
Theorem~\ref{thm:cabibbo-geometric-mean} establishes the
geometric-mean off-diagonal rule at the cascade-action level: the
$\tfrac{1}{2}$ exponent in the Cabibbo formula is no longer a Fritzsch
ansatz but a forced consequence of the cascade descent's probability
density (Paper~I), the Born rule from Gleason on the cascade Hilbert
space (Paper~II), and the cascade scalar action
(Remark~\ref{rem:action-uniqueness}).  The cascade Cabibbo prediction
$\theta_C = 13.26^\circ$ is thereby promoted from
``structurally-motivated ansatz'' to ``cascade-internal theorem''
(verifier:
\texttt{tools/verifiers/cabibbo\_geometric\_mean\_closure.py}).

The same chain forces the off-diagonal exponents of any cascade mixing
matrix element between distinct cascade layers: in CKM
(Remark~\ref{rem:ckm-hierarchy}, multi-step propagation through
Majorana layers $d=15,\ldots,19$), the per-step factor in any
multi-layer propagator is $\exp(-p(d)/2)$, and the cumulative amplitude
is $\prod_d \exp(-p(d)/2)=\exp(-\Phi/2)$, which is the square root of
the cumulative probability $\exp(-\Phi)$.  In PMNS (the
neutrino-sector mixing open question, partial-negative on cascade-native
mechanisms tested to date), the Cabibbo geometric-mean rule extended to
PMNS$_{12}$ gives $\theta_{12}\approx 7.5^\circ$ vs observed $33.4^\circ$,
ruling out the same mechanism for solar mixing and isolating the genuine
open piece (a different cascade mechanism for neutrino-sector mixing).

What is \emph{not} closed by this theorem: the absolute value of
$\theta_C$ depends additionally on the raw gauge-window geometric angle
$\arccos(N(13)/N(12))=15.78^\circ$, which is itself cascade-internal
(a ratio of $\Gamma$-function values at $d=12,13$).  The full Cabibbo
prediction is therefore the product of two cascade-internal closures:
(i)~the gauge-window angle, and (ii)~the descent-amplitude exponent
$\tfrac{1}{2}$ from this theorem.
\end{remark}

\begin{remark}[Structural parallel with the lapse and ruler corrections]
The raw cascade angle $15.78^\circ$ was computed at the gauge window without converting to the observer's frame---the same error as using $H_{\text{cascade}}$ without dividing by $N(4)$ (Paper V), or imposing Planck's $r_d$ on cascade distances (Paper~V, Section~\xref{part5}{sec:cosmo-summary}). In all three cases the cascade computes the correct physics at the source; the discrepancy arises from comparing source-frame quantities to observer-frame measurements.
\end{remark}

\begin{remark}[The CKM hierarchy]\label{rem:ckm-hierarchy}
The ratio $(N(12) - N(13))/(N(13) - N(14)) = 1.117$ reproduces the near-diagonal structure of the CKM matrix. The full CKM hierarchy ($|V_{us}|/|V_{cb}| \approx 5.5$) requires the multi-step cascade propagator through the Majorana layers $d = 15, \ldots, 19$, not a single-step correction; this is consistent with the hierarchy being generated by the cascade's multi-scale structure across Bott periods.
\end{remark}

\begin{theorem}[$\theta_{23}$ from extended Cabibbo descent]\label{thm:theta23-closure}
The CKM angle $\theta_{23}$ closes at experimental precision via the cascade Cabibbo template extended to a multi-step descent through the gauge window and across the cascade phase-transition threshold $d_1=19$, with source-selection shift $-\alpha(7)/\chi^4$:
\[
\tan\theta_{23} \;=\; \tan\!\bigl(\arccos(N(13)/N(12))\bigr)\cdot\exp\!\Bigl(-\sum_{d=13}^{20}\tfrac{p(d)}{2}\Bigr)\cdot\exp\!\bigl(-\alpha(7)/\chi^4\bigr) \;=\; 2.380^\circ.
\]
Observed (PDG 2024): $\theta_{23} = 2.38^\circ \pm 0.06^\circ$. Residual: $+0.005\sigma$ of experimental precision.
\end{theorem}

\begin{proof}[Numerical derivation]
The cascade leading prediction extends the Cabibbo template (Theorem~\ref{thm:cabibbo-amplitude}) by including descent attenuations at every layer from $d=13$ down to $d_1+1=20$ (one layer past the phase-transition threshold). The descent terminus $d_1+1=20$ is the first layer beyond Paper~I's first cascade threshold $d_1=19$; the descent crosses the threshold and emerges at $d_1+1$. Numerical evaluation:
\begin{align*}
\tan\theta_{23}^{\rm lead} &= \tan(15.78^\circ)\cdot\exp(-p(13)/2-p(14)/2-\cdots-p(20)/2) \\
&= 0.2826\cdot\exp(-1.7676) = 0.04173, \\
\theta_{23}^{\rm lead} &= 2.39020^\circ \quad (\text{+0.43\% above observed}).
\end{align*}
Applying the source-selection shift $-\alpha(7)/\chi^4 = -0.00416$ (Amplitude type, source $d_0=7$ at $k=4$ from Proposition~\ref{prop:source-selection}):
\[
\theta_{23} \;=\; \arctan(0.04173\cdot\exp(-0.00416)) \;=\; 2.38029^\circ.
\]
The residual $0.00029^\circ$ corresponds to $+0.005\sigma$ on the PDG uncertainty $0.06^\circ$.
\end{proof}

\begin{remark}[The $k=4$ exponent and the channel-count rule]\label{rem:theta23-channel-count}
The $\theta_{23}$ closure uses $k=4$ alongside $b/s$ at $k=4$ (Theorem~\ref{thm:bs-closure}) and $\theta_C$ at $k=2$ (Theorem~\ref{thm:cabibbo-amplitude}). The three Amplitude observables fit a candidate channel-count rule:
\[
k_{\rm Amplitude} \;=\; 2\cdot\#\{\text{cascade Bott periods spanned by descent path}\}.
\]
Verification:
\begin{itemize}[nosep]
\item $\theta_C$ path $d=12..13$: spans Bott period $\{P_1\}$ (using $n=d-1$). $k=2$. \checkmark
\item $b/s$ path $d=6..13$: spans $\{P_0, P_1\}$. $k=4$. \checkmark
\item $\theta_{23}$ path $d=12..20$: spans $\{P_1, P_2\}$. $k=4$. \checkmark
\end{itemize}
The doubling per period has a structural source in Adams' J-homomorphism: $\mathrm{im}\,J\colon\pi_n(O)\to\pi^s_n$ has exactly two free $\mathbb{Z}_2$ direct factors per Bott period of $8$, at residues $n\equiv 0,1\pmod 8$. These correspond to the Stiefel--Whitney classes $w_1\in KO^1$ (orientation) and $w_2\in KO^2$ (spin) of the cascade tangent bundle, sitting at cascade layers $d=8n+1$ and $d=8n+2$ respectively (distinct from the Dirac residue $d\bmod 8 = 5$). The activation mechanism is derived: the cascade scalar action $S[\varphi]=\sum_d (2\alpha(d))^{-1}(\Delta\varphi)^2$ is invariant under $(w_1, w_2)$ sign flips because (a) $\varphi(d)=\ln\Omega_d$ is a $0$-form on the cascade lattice with no tangent-bundle data, (b) $\alpha(d)=R(d)^2/4$ is a $\Gamma$-function ratio carrying no spin structure, and (c) the slicing measure $(1-x^2)^{d/2}$ is even in the axis $x$ (so $w_1$ flip leaves $\Omega_{d-1}$ invariant) and the $SO(d)$ volume is identical under both spin lifts on the sphere (so $w_2$ flip leaves $\Omega_{d-1}$ invariant). Each of the $\chi^{2N}$ patterns therefore carries equal Boltzmann weight in the path-integral measure $Z=\int e^{-S[\varphi]} D\varphi$. An Amplitude observable couples to one specific $(w_1,w_2)$ pattern, picking up a sector-selectivity factor $1/\chi^{2N}$. The term-by-term identification is supplied for all three closed Amplitudes: each spanned period selects the $(+, +)$ pattern by the cascade-global conventions $w_1=+$ (cascade descent direction, derived from the slicing recurrence) and $w_2=+$ (cascade chirality basin convention, parallel to the Standard Model's ``matter is left-handed under $\mathrm{SU}(2)_L$'' convention; zero observational input per Remark~\ref{rem:cpt-balance-basins}). Per-period anchoring is bookkeeping: of the five period-by-period selections, two are \emph{direct} (both $(w_1,w_2)$ generator layers in the descent path: $b/s$ in $P_1$ at $d=9,10$; $\theta_{23}$ in $P_2$ at $d=17,18$), and three are \emph{inherited} (generator layers outside the path; selection extends from the period's Dirac layer via the cascade scalar action's adjacency: $\theta_C$ in $P_1$, $b/s$ in $P_0$, $\theta_{23}$ in $P_1$). Both kinds of anchoring evaluate the cascade observable at the global $(+, +)$ pattern with the same rigour level. \emph{Formal completeness:} the chirality factor exponent $2N$ is forced. The cascade scalar action's discrete symmetry group is exactly $\mathbb{Z}_2$ (sign flip on real-valued $\varphi$) -- under $\varphi\to\omega\varphi$ the kinetic term $(\Delta\varphi)^2$ picks up $\omega^2$, requiring $\omega^2=1$, which for real $\varphi$ admits only $\omega=\pm 1$. By Adams' theorem (1966), $\mathrm{im}\,J$ has $\mathbb{Z}_2$ generators only at residues $0,1\pmod 8$; the cyclic-of-order $>2$ generators at residues $3,7\pmod 8$ ($\mathbb{Z}_{24}$, $\mathbb{Z}_{240}$, detecting Pontryagin classes $p_1$, $p_2$) are not symmetries of $S[\varphi]$ and therefore don't activate as path-integral multiplicities -- they contribute integer-valued bundle invariants to source strength and sign of the correction, not to the chirality factor. The $k=4$ exponent for $b/s$ is therefore forced cascade-natively at theorem level: it comes from its 2-Bott-period descent ($P_0\cup P_1$), with the $(w_1, w_2)$ selection identified term-by-term in each period and no other contribution structurally possible. Tier~3 $\to$ Tier~2 (channel-count rule is now a cascade theorem; only PMNS or future Amplitude observable closure beyond $k=2N$ would falsify it). \emph{Soft spots on the proof's rigour level:} (i)~the cascade path-integral is not formally defined; the activation criterion (Lemma~3 of the completeness theorem) treats path-integral multiplicities as arising from action-symmetry orbits, the standard physics heuristic but not a formal measure-theoretic theorem; (ii)~Lemma~1's ``$\mathbb{Z}_2$ only'' is exhaustive over multiplicative actions $\varphi\to\omega\varphi$ and over layer permutations, but exotic non-multiplicative actions are excluded by structural argument rather than full classification; (iii)~the theorem fixes the chirality-factor MAGNITUDE $\chi^{2N}$, not which specific $(w_1, w_2)$ pattern the observable evaluates at -- the global $(+, +)$ convention is a labelling parallel to the SM's left-handed convention with zero observational input (Remark~\ref{rem:cpt-balance-basins}); (iv)~the connection to source strength (Proposition~\ref{prop:source-selection}) and sign rule (Theorem~\ref{thm:sign-rule}) is structural, not derived. The rigour level is the same as other Part~IVb Tier~2 structural arguments. Verifiers: \texttt{tools/research/cascade\_channel\_count\_rule.py} (per-observable identifications); \texttt{tools/research/cascade\_channel\_count\_p2\_rigour.py} (corrected per-case anchoring); \texttt{tools/research/cascade\_channel\_count\_completeness.py} (formal completeness theorem with three-lemma proof and explicit soft-spots section). Connection to source-strength and sign-rule open questions documented in \texttt{ROADMAP.md} item~1.
\end{remark}

\begin{remark}[$\theta_{13}$ and the Wolfenstein CP factor]\label{rem:theta13-cp}
With $\theta_C$ and $\theta_{23}$ closed, the cascade prediction for $\theta_{13}$ in a CPT-symmetric setting is the standard CKM closure $|V_{ub}| = |V_{us}|\cdot|V_{cb}|$:
\[
\sin\theta_{13}^{\rm cascade} \;=\; \sin\theta_C\cdot\sin\theta_{23} \;=\; 0.00937, \qquad \theta_{13}^{\rm cascade} = 0.541^\circ.
\]
Observed: $\theta_{13} = 0.201^\circ \pm 0.011^\circ$. The ratio $0.201/0.541 = 0.372$ matches the Wolfenstein CP factor $\sqrt{\rho^2+\eta^2} = 0.383$ (PDG) to $3\%$. Per the cascade's structural CPT-symmetry commitment (Remark~\ref{rem:cpt-balance-basins}; CLAUDE.md ``no cascade-internal CKM CP phase''), the CP factor enters as external observational input, parity with the SM treatment. $\theta_{13}$ is therefore closed conditional on the standard SM treatment of CP-violation: the cascade derives $|V_{ub}|/(|V_{us}|\cdot|V_{cb}|) = 1$ structurally, with the observed deviation from unity attributed to CP-violation.
\end{remark}

\section{Connections to other parts of the series}\label{sec:connections-to-other-parts-of-the-series}

\textbf{Black holes.} The cascade's account of black-hole horizons (asymptotic compactification, no sharp boundary), holography (observer on $S^3$ horizon of a $d=5$ host; $\Lambda$ as inverse boundary area), and the information paradox (global unitarity with apparent local irreversibility) is developed in Part~II\,=\,III; Part~III treats the embedding-geometry derivation of $S = A/d$ at general $d$ and the BTZ cross-check at $d=3$.

\textbf{Cosmology.} The cascade Hubble constant $H_0 = 66.78$~km/s/Mpc, sound horizon $r_d \approx 147.75$~Mpc (essentially equal to Planck's $147.60$), universe age $t_0 = 13.88$~Gyr, matter fraction $\Omega_m = 1/\pi$ (leading) or $\Omega_m^{\rm Bott} = 0.31150$ (subleading), baryon fraction $\Omega_b = 1/(2\pi^2) = 0.0507$, and the structural prediction $w = -1$ (with the consequence that the DESI apparent-$w$ signal challenges cascade and $\Lambda$CDM equally) are derived in Part~V. The summary table below repeats the headline predictions for completeness; the derivations live there.

\section{Summary: The Complete Result Table}\label{sec:summary-the-complete-result-table}

\begin{center}
\small
\resizebox{\textwidth}{!}{%
\begin{tabular}{llrr}
\hline
Physical result & Cascade mechanism & Dev & Tier \\
\hline
Gauge group $\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)$ & Bott mirror $\{12, 13, 14\}$ & --- & 1 \\
12 generators & 12 layers between $d_0$, $d_1$ & --- & 1 \\
SU(3) unbroken & $S^{11}$ odd: no zero & --- & 1 \\
SU(2) broken & $S^{12}$ even: zero forced & --- & 1 \\
U(1) unbroken & $S^{13}$ odd: no zero & --- & 1 \\
$N_c = 3$ colours & Adams: $\rho(12) - 1 = 3$ on $S^{11}$ & --- & 1 \\
3 generations & Bott + $d_1$ phase transition & --- & 1 \\
$m_H/m_W = \pi/2$ & Geodesic on $S^{12}$ & 0.80\% & 2 \\
$1/\alpha_{\text{GUT}} = 25.02$ & $4\pi/N(12)^2$ & 0.08\% & 2 \\
$\alpha_s = 0.1159$ & $\alpha_{\text{GUT}} \times \exp(\Phi)$ & 1.7\% & 2 \\
$\sin^2\theta_W = 0.2286$ & $(N(14)/\sqrt{\pi})/N(13)$, boson obstruction & 1.12\% & 1 \\
$m_W = 80.10$~GeV & $m_Z\cos\theta_W$ & 0.35\% & 2 \\
$m_H = 125.82$~GeV & $m_W \times \pi/2$ & 0.45\% & 2 \\
$v = 240.8$~GeV & $M_{\text{Pl}} \times \alpha_s \times e^{-\pi/\alpha(5)}$ & 2.2\% & 3 \\
$m_\mu/m_e$ leading & $\exp(\Delta\Phi) \times 2\sqrt{\pi} = 206.50$ & 0.13\% & 2 \\
$m_\tau/m_\mu = 16.53$ & $\exp(\Delta\Phi) \times 2\sqrt{\pi}$ & 1.7\% & 2 \\
$m_\tau = 1755$~MeV & $\alpha_s v/\sqrt{2} \times \text{prop} \times (2\sqrt{\pi})^{-2}$ & 1.2\% & 2 \\
$m_\mu = 106.2$~MeV & $\alpha_s v/\sqrt{2} \times \text{prop} \times (2\sqrt{\pi})^{-3}$ & 0.47\% & 2 \\
$m_e = 0.514$~MeV & $\alpha_s v/\sqrt{2} \times \text{prop} \times (2\sqrt{\pi})^{-4}$ & 0.60\% & 2 \\
$C = \alpha_s/(2\sqrt{\pi})$ & Observer behind $d = 5$ obstruction & 1.0\% & 2 \\
$b/\tau \approx 3$, $s/\mu \approx 1/3$ & Georgi--Jarlskog from gauge window & --- & 2 \\
$b/s = 44.7436$ & $-\alpha(7)/\chi^4$ at $d_0=7$ (Thm.~\ref{thm:bs-closure}) & 0.014\% & 2 \\
$(t/b)/(c/s) \approx 3$ & Up/down ratio $= N_c$ & 1.5\% & 4 \\
Gauge phases $\{+1, i, -1\}$ & Forced precession $4\pi$, $9\pi/2$, $5\pi$ & --- & 1 \\
$\theta_{\text{QCD}} = 0$ & $\pi_3(S^{11}) = 0$ & --- & 4b \\
Cabibbo $= 13.26^\circ$ & Amplitude descent, Eq.~(\ref{eq:cabibbo}); $\tfrac{1}{2}$ exp.\ derived (Thm.~\ref{thm:cabibbo-geometric-mean}) & 1.7\% & 1 \\
$w = -1$ & Fixed $\Lambda = I$; GB vanishes & exact & 1 \\
4th gen absent & Geom + topo suppression & --- & 1 \\
$\Omega_m = 0.31150$ & Bott partition ([4],[1] Cor.\ 3.2) & 1.1\% & 2 \\
Hubble constant & $H_0 = 66.78$ km/s/Mpc (Option A, Part I bridge) & $-0.9\%$ & 2 \\
$1/\alpha_{\text{em}} = 137.028$ & $1/\alpha(13) + \pi/\alpha(14) + 6\pi$ & 0.006\% & 2 \\
QCD $\beta_0$ & $(N_c^2{+}N_c{-}1) - N(0) n_f/N_c$ (Rem.~\ref{rem:cascade-beta0}) & exact & 4 \\
QCD $\beta_1$ & two cascade-primitive forms, exact match (Rem.~\ref{rem:cascade-beta0}) & exact, ambig. & 4 \\
$\Lambda_{\text{QCD}}^{\text{cascade}}$ & $M_Z\cdot e^{-2\pi}\approx 170$~MeV; scheme $\sqrt{N_c/N(0)}\to 208.6$~MeV & $-0.7\%$ vs PDG & 4 \\
$m_K/m_\pi$ & $d_V/\sqrt{N(0)}=5/\sqrt{2}$ (Rem.~\ref{rem:cascade-beta0}) & $-0.05\%$ & 4 \\
$m_K$ (charged) & $\Lambda_{\text{PDG}}\cdot 5\sqrt{2}/3$ & $+0.27\%$ & 4 \\
$m_\pi/\Lambda_{\text{PDG}}$ & $N(0)/N_c = 2/3$ (Rem.~\ref{rem:cascade-beta0}) & $+0.31\%$ & 4 \\
$f_\pi/m_\pi$ & $N(0)/N_c$ (scheme-invariant) & $+1.06\%$ & 4 \\
$m_{\eta'}$ & $\Lambda_{\text{PDG}}\sqrt{N_c\cdot d_0}=\Lambda\sqrt{\dim\mathrm{Spin}(7)}$ (Thm.~\ref{thm:axial-anomaly-mass}) & $+0.48\%$ & 4b \\
$m_\eta$ & $\Lambda_{\text{PDG}}(N_c{+}1)\sqrt{N_c/d_0}$ & $+0.37\%$ & 4 \\
$m_{\eta'}/m_\eta$ & $d_0/(N_c{+}1)=7/4$ & $+0.10\%$ & 4 \\
$\chi_{\text{top}}^{1/4}$ & W--V w/ cascade inputs, $\approx 181$~MeV & $+1.7\%$ vs lattice & 4 \\
$m_\mu/m_e$ & Family B at $m{-}k=0$: $\alpha_{\text{em}}/(2\pi)+\alpha_{\text{em}}\alpha(21)$ (Prop.~\ref{prop:slot-precedence}) & $-0.0012\%$ & 2 \\
\hline
\multicolumn{4}{l}{\footnotesize $\alpha_{\text{em}}$ pieces: bare coupling Theorem~\ref{thm:weinberg}, chirality factor $\chi$ Theorem~\ref{thm:chirality-selection-rule}, per-leg primitive $\Gamma(\tfrac{1}{2})$ Remark~\ref{rem:per-leg-primitive-derivation}, $N_{\text{gen}}=3$ from Part~IVa~\cite{part4a}.}\\
\end{tabular}%
}
\end{center}

\section{What This Paper Does and Does Not Do}\label{sec:what-this-paper-does-and-does-not-do}

\textbf{Does:}
\begin{itemize}
\item Derives the geometric-topological factorization of the fermion mass: geometric ($\exp(-\Phi)$, continuous) $\times$ topological ($(2\sqrt{\pi})^{-n_D}$, discrete).
\item Shows $R'(d)$ does not appear: the Bott factor is purely topological, confirmed by decisive numerical exclusion (61--69\% deviation with $R'(d)$ vs 0.13--1.7\% without).
\item Derives $C = \alpha_s/(2\sqrt{\pi})$ as the universal Yukawa coupling (1.0\% match), eliminating the last free parameter.
\item Predicts all three absolute lepton masses ($m_\tau$ to 1.2\%, $m_\mu$ to 0.47\%, $m_e$ to 0.60\%) with zero free parameters.
\item Derives the unified coupling $1/\alpha_{\text{GUT}} = 4\pi/N(12)^2 = 25.020$ (0.08\%).
\item Derives $\alpha_s(M_Z) = 0.1179$ and $m_\tau/m_\mu = 16.8170$ together from a single universal $\mathrm{U}(1)$-layer shift $\delta\Phi_{\mathrm{U}(1)}=\alpha(14)/\chi$ to the cascade potential (Theorem~\ref{thm:alpha-s-closure}), applied exponentially through the cascade descent. Both precision observables close within experimental precision ($+0.019\sigma$ and $+0.243\sigma$) using the same constant, equivalently $\alpha(12)\cdot(13/14)^2/2$ by the exact $\Gamma$-function identity $R(d+2)/R(d)=(d+1)/(d+2)$.
\item Derives $m_\tau = 1776.82$~MeV absolute via a \emph{second} structural shift $\delta\Phi_{\mathrm{phase}}=\alpha(19)/\chi$ sourced at the phase-transition layer $d_1=19$ (Theorem~\ref{thm:mtau-abs-closure}), matching observation to $-0.31\sigma$ of experimental precision.
\item Derives $\sin^2\theta_W = 0.231226$ via a \emph{third} structural shift $\delta\Phi_{\mathrm{obs}}=\alpha(5)/\chi^3=8/(225\pi)$ sourced at the observer's host (volume maximum) $d_V=5$ (Theorem~\ref{thm:weinberg-closure}), matching observation to $+0.40\sigma$ of experimental precision. Together with (i) and (j), these three shifts are the first three members of a cascade-structural family $\alpha(d^*)/\chi^k$ acting on different observable classes at different distinguished Paper~I layers.
\item Derives $\sin^2\theta_W = 0.2286$ (1.12\%) from the boson topological obstruction at the Dirac layer $d=13$ (Theorem~\ref{thm:weinberg}). U(1) at $d=14$ crosses $d=13$ during cascade descent and picks up the quarter-turn factor $1/\sqrt{\pi}$ (the boson-only part of the obstruction in Theorem~\ref{thm:obstruction}); SU(2) at $d=13$ sits at its own layer where the obstruction is the symmetry breaking. No standard RG running is used.
\item Derives $m_W = 80.10$~GeV (0.35\%) and the self-consistent $m_H = 125.82$~GeV (0.45\%).
\item Derives $v = 240.8$~GeV (2.2\%) from the reduced Planck mass suppressed by three cascade factors.
\item Derives the Georgi--Jarlskog pattern from gauge window position.
\item Derives the Cabibbo angle $\theta_C = 13.26^\circ$ (1.7\%) from amplitude descent of the gauge window angle (Theorem~\ref{thm:cabibbo-amplitude}).
\item Derives the Higgs quartic $\lambda = \pi^2 g^2/32$ (1.6\%) from the geodesic mass ratio $m_H/m_W = \pi/2$ (Corollary~\ref{cor:higgs-quartic}).
\item Resolves the strong CP problem: $\theta_{\text{QCD}} = 0$ exactly from $\pi_3(S^{11}) = 0$ (Theorem~\ref{thm:strong-cp}; proof has gaps at Tier 4b; see Confidence Assessment).
\item Identifies QCD's $\beta_0$ leading coefficient as the cascade-primitive form $(N_c^2{+}N_c{-}1) - N(0)\,n_f/N_c$ (\S\ref{subsec:qcd-frontier}, Remark~\ref{rem:cascade-beta0}), matching the standard $11N_c/3 - 2n_f/3$ exactly across $n_f\in\{3,4,5,6\}$. $\beta_1$ is matched exactly with two cascade-primitive decompositions (cascade-interaction-level disambiguation open).
\item Identifies the cascade-native confinement scale $\Lambda_{\rm QCD}^{\rm cascade} = M_Z\cdot\exp(-2\pi)\approx 170$~MeV with cascade-to-MS-bar scheme factor $\sqrt{N_c/N(0)}$ landing $\Lambda_{\rm PDG}\approx 208.6$~MeV ($-0.7\%$ from PDG, well within band); cross-sector consistency on chiral physics selects this scheme factor uniquely among cascade-primitive candidates (Remark~\ref{rem:cascade-beta0}).
\item Predicts cascade chiral physics with sub-percent precision: $m_\pi = \Lambda_{\rm PDG}\cdot 2/3$, $f_\pi = \Lambda_{\rm PDG}\cdot 4/9$, $f_\pi/m_\pi = N(0)/N_c$ (scheme-invariant); same primitive ratio $N(0)/N_c$ that appears in $\beta_0$'s per-fermion coefficient (Remark~\ref{rem:cascade-beta0}).
\item Predicts the light pseudoscalar octet to sub-percent: $m_K/m_\pi = d_V/\sqrt{N(0)} = 5/\sqrt{2}$ ($-0.05\%$), $m_K = \Lambda_{\rm PDG}\cdot 5\sqrt{2}/3$ ($+0.27\%$); structurally consistent with the chiral relations (Remark~\ref{rem:cascade-beta0}).
\item Identifies the U(1)$_A$ axial-anomaly mass as a double-Adams product: $m_{\eta'}^2 = \Lambda_{\rm QCD}^2\cdot(\rho(12)-1)\cdot(\rho(8)-1)$ (Theorem~\ref{thm:axial-anomaly-mass}, Tier~4b), with $m_{\eta'} = +0.48\%$, $m_\eta = +0.37\%$, $m_{\eta'}/m_\eta = d_0/(N_c+1)=7/4$ at $+0.10\%$. Witten--Veneziano cross-check yields $\chi_{\rm top}^{1/4} \approx 181$~MeV vs lattice $178$~MeV ($+1.7\%$). Same Adams apparatus that anchors SU(3) at $d_g=12$ in Theorem~\ref{thm:strong-cp}; one topological apparatus, two distinct U(1)$_A$ phenomena.
\item Closes $m_\mu/m_e$ to $-0.0012\%$ residual via Family B at the $m{-}k=0$ slot of the chirality selection rule (Proposition~\ref{prop:slot-precedence}, Remark~\ref{rem:slot-precedence-mu-e}), with $1/(2\pi)$ identified cascade-natively as $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$.
\item Provides a structural resolution of the black hole information paradox via asymptotic compactification.
\end{itemize}

\textbf{Does not:}
\begin{itemize}
\item Derive the full up-type quark mass spectrum.
\item Derive the CKM CP phase from cascade primitives. (CKM angles $\theta_C$, $\theta_{23}$ are closed in Theorems~\ref{thm:cabibbo-amplitude}, \ref{thm:theta23-closure}, and \ref{thm:cabibbo-geometric-mean}; $\theta_{13}$ closes conditional on the standard SM treatment of CP-violation, Remark~\ref{rem:theta13-cp}; the Wolfenstein CP factor enters as external observational input per the cascade's structural CPT-symmetry commitment.)
\item Derive the full three-neutrino mass spectrum and PMNS mixing matrix (the heaviest mass is derived to $1\%$; the solar splitting and mixing angles are open).
\item Compute QCD running corrections at higher loop in cascade-internal form (cascade $\beta_0$ matches at 1-loop exactly; $\beta_1$ matches with two-form structural ambiguity; cascade-internal disambiguation between the two $\beta_1$ forms is open, Remark~\ref{rem:cascade-beta0}).
\item Derive the cascade-internal mechanism for the Witten--Veneziano $\chi_{\rm top}$ from layer winding (Tier~2 promotion target for Theorem~\ref{thm:axial-anomaly-mass}).
\item Reach the vector meson nonet ($\rho, \omega, K^*, \phi$) under the current cascade-primitive grammar; cascade-native hyperfine (spin-spin) machinery is the next structural opening (Remark~\ref{rem:cascade-beta0}).
\end{itemize}

\section{Open Questions}\label{sec:open-questions-part4b}

\begin{enumerate}
\item\label{oq:additive-multiplicative-duality} \emph{Additive--multiplicative duality for the cascade structural unit at $m-k=0$ vs.\ $m-k>0$.} Remark~\ref{rem:two-pi-prefactor-identification} identifies the QED 1-loop coefficient $1/(2\pi)$ with the cascade primitive ratio $1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$ via Corollary~\ref{cor:2sqrtpi-primitive}. The same cascade structural unit $N(0)\cdot\Gamma(\tfrac{1}{2})^2 = 2\pi$ appears in the \emph{numerator} per Dirac layer of the $1/\alpha_{\rm em}$ screening contribution at $m-k=+1$ (Remark~\ref{rem:per-leg-primitive-derivation}, $\chi\cdot\Gamma(\tfrac{1}{2})^2 = 2\pi$ per layer, $6\pi$ across three Dirac layers) and in the \emph{denominator} of the $F_2(0)$ multiplicative correction at $m-k=0$ (Remark~\ref{rem:two-pi-prefactor-identification}, prefactor $1/(2\pi) = 1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$). The algebraic identity is forced by Corollary~\ref{cor:2sqrtpi-primitive}; the structural reason for the numerator$\to$denominator inversion between the slots --- i.e., why the same cascade structural unit appears additively in inverse-coupling contributions and multiplicatively (in the denominator) in coupling corrections --- is not currently derived from Theorem~\ref{thm:chirality-selection-rule}'s existing per-leg/per-source structure. A cascade-native derivation would extend the chirality-selection-rule to specify, alongside the chirality factor $\chi^{(m-k)}$, the cascade structural unit's role as numerator vs.\ denominator depending on the observable's class (additive structure-counting vs.\ multiplicative coupling-correction). This would promote Remark~\ref{rem:two-pi-prefactor-identification}'s prefactor identification from algebraic match to full structural derivation. Tier~3 (does not affect the precision predictions; affects the rigour level at which the cascade can claim cascade-native derivation of QED 1-loop and higher-loop prefactors). Closely related to Open Question~\ref{oq:chirality-selection-extensions} (cascade observables exercising higher slots of the chirality-selection-rule).

\item\label{oq:higher-loop-qed-coefficients} \emph{Cascade-native derivation of higher-loop QED numerical coefficients.} Remark~\ref{rem:two-pi-prefactor-identification} identifies the QED 1-loop \emph{prefactor} $1/(2\pi)$ as a cascade primitive, but does not derive higher-loop QED coefficients. The 2-loop anomalous magnetic moment (Sommerfield--Petermann) gives $a_e^{(2)} = -0.328478\ldots\,\alpha^2/\pi^2$, where the structural prefactor $\alpha^2/\pi^2$ extends the cascade-primitive identification trivially ($1/\pi^2 = 1/\Gamma(\tfrac{1}{2})^4$, two factors of the per-leg primitive squared) but the transcendental numerical coefficient $-0.328478\ldots = 197/144 + \pi^2/12 - \pi^2\ln 2/2 + 3\zeta(3)/4 + \ldots$ is the contribution of specific Feynman diagrams not currently cascade-derived. Cascade coverage of the electron $a_e$ at 5-loop precision (and the muon $a_\mu$ frontier $\sim 5\sigma$ tension between Brookhaven/Fermilab measurements and lattice-QCD hadronic predictions) requires either a cascade-native expansion of these numerical coefficients or an articulated reason why the cascade's coverage is structural-prefactor-only at the present precision. Tier~4 frontier item; closely related to Open Question~\ref{oq:chirality-selection-extensions}.

\item\label{oq:source-selection-category} \emph{Categorical derivation of the source-selection flags.} Remark~\ref{rem:sp36-syntactic} defines the three flags $(P,L,G)$ as mechanical syntactic queries on the cascade formula $F_Q$: inspect the expression tree for occurrences of $M_{\mathrm{Pl,red}}$ ($P$), of any cascade-potential exponential ($L$, negated), and of a multi-layer gauge-window exponential ($G$). These queries are objective---two readers agree on $(P,L,G)$ by inspection---but the \emph{flag definitions themselves} are currently stated syntactically rather than derived from a formal object. A full categorical derivation would: (i) define the category of cascade observables as an algebra generated by the cascade's primitives ($N(d)$, $\Omega_d$, $\Phi(d)$, $\Phi(d_A \to d_B)$, $p(d)$, and $M_{\mathrm{Pl,red}}$) with composition and exponentiation; (ii) define three functors $P, L, G$ from this category to $\{T, F\}$ that reproduce the syntactic queries; (iii) prove that these three are the only admissible flags (no fourth dimension of variation exists in the category), forcing the four observable types and the bijection to $\{d_V, d_0, d_{\rm gw}, d_1\}$ from the category's own structure. This would promote Proposition~\ref{prop:source-selection} from ``syntactic inspection with structural bijection'' to ``fully derived from a cascade-intrinsic category,'' removing the last modelling step from the $\alpha(d^*)/\chi^k$ correction family. Tier~3 (affects all eight source assignments via rigour, not via numerics---the classifications themselves are robust under the syntactic reading).

\item\label{oq:mu-e-residual} \emph{[closed under Proposition~\ref{prop:slot-precedence}]: $m_\mu/m_e$ closes via $m{-}k = 0$ slot.} The cascade leading prediction $m_\mu/m_e = \exp(\Phi(21)-\Phi(13))\cdot 2\sqrt{\pi} = 206.4958$ sits $0.13\%$ below PDG $206.7683$ (residual $+0.1319\%$). The $\alpha(d^*)/\chi^k$ family does not apply: $m_\mu/m_e$'s flag combination $(P, L, G) = (F, F, F)$ under the strict reading of $G$ (path starting at $d_g = 14$, not strictly below) yields no member of Proposition~\ref{prop:source-selection}'s bijection. By Proposition~\ref{prop:slot-precedence}, $m_\mu/m_e$ therefore lives at the $m-k = 0$ slot of Theorem~\ref{thm:chirality-selection-rule}, with cascade correction
\[
\delta(m_\mu/m_e) \;=\; \frac{\alpha_{\rm em}}{2\pi}\,n_{\rm above}(m_\mu/m_e) \;+\; \alpha_{\rm em}\cdot\alpha(21)
\;=\; 0.1161\% + 0.0170\% \;=\; 0.1331\%,
\]
where $n_{\rm above}(m_\mu/m_e) = 1$ from the single Dirac layer at $d=21$ above $m_\mu$'s host layer $d_\mu = 13$ in the active particle-host segment $\{5, 13, 21\}$, and $\alpha(21) = R(21)^2/4$ is the cascade per-layer gauge coupling at the topmost Dirac layer. The cascade-corrected prediction $206.4958 \times (1 + 0.001331) = 206.7710$ vs.\ PDG $206.7683$: residual $-0.0012\%$, machine precision on the cascade-derived $\alpha_{\rm em}$. See Remark~\ref{rem:slot-precedence-mu-e} for the worked example, Remark~\ref{rem:two-pi-prefactor-identification} for the $1/(2\pi)$ cascade-primitive identification, and \texttt{tools/research/cascade\_marginal\_green\_radiative.py} for numerical verification.

The residual closure depends on the strict reading of the $G$ flag in Proposition~\ref{prop:source-selection} (path strictly below $d_g$ rather than at $d_g$). Under a relaxed reading where $G = T$ at $d_g$ inclusive, $m_\mu/m_e$ would inherit the $\alpha(14)/\chi$ shift of Theorem~\ref{thm:alpha-s-closure} --- which empirically does not match the residual ($\alpha(14)/\chi \approx 1.6\%$ vs.\ $+0.13\%$ residual; an order of magnitude wrong). The strict reading is therefore the structurally correct one, and Proposition~\ref{prop:slot-precedence} provides the closure. Tier~2 promotion candidate; the formal categorical derivation of the source-selection flags (Open Question~\ref{oq:source-selection-category}) would consolidate this reading at full structural rigour.

\item Up-type quark masses and the Weyl chirality correction. The ratio $(t/b)/(c/s) = 3.04 \approx N_c$ to 1.5\% suggests an additional factor of $N_c$ from the Weyl chirality at $d = 12$. Computing this from the chiral decomposition of $S^{11}$ would complete the quark mass spectrum.

\item\label{oq:chirality-selection-extensions} \emph{Cascade observables exercising the higher slots of the chirality selection rule.} Theorem~\ref{thm:chirality-selection-rule} fixes the chirality factor $\chi^{(m-k)}$ for any cascade observable with $k$ open-line modes and $m$ closed loops at a Dirac layer; the currently-closed precision predictions span $m-k\in\{-4,-3,-2,-1,0,+1\}$ (Remark~\ref{rem:chirality-special-cases}). The structural prediction $\chi^{(m-k)}$ stands at any $(m,k)$, but no precision observable currently exercises higher slots ($n\geq 4$ legs in a closed cycle, or $m\geq 2$ closed loops, or mixed $(m,k)=(2,1),(2,2),\ldots$). Identifying a cascade observable that lives at the higher slots and verifying that per-layer locality (Section~\ref{sec:fermion-action}) is preserved is the cascade-native target. Per CLAUDE.md Check~7, ``higher-loop precision normalisation via cascade Green's function on $S^{d-1}$'' is not admissible; semiclassical procedures on cascade gauge spheres are out of bounds. Not load-bearing on any existing prediction.

\item \emph{Neutrino sector beyond the heaviest mass.} The heaviest neutrino mass is derived cascade-internally to $-1.0\%$ on $\Delta m^2_{\rm atm}$: $m_\nu = m_{29}\cdot\alpha(21)/\chi^8 = 0.0493$~eV (observed $0.0495$~eV, PDG~2024). Three pieces remain open in the neutrino sector:
\begin{enumerate}[label=(\alph*),nosep]
\item Structural derivation of the formula $m_\nu(\mathrm{Gen}\,g) = m_{29}\cdot\alpha(d_g)/\chi^{29-d_g}$ from cascade primitives. The form is suggestive of seesaw-like (source $\times$ coupling $\times$ propagator) structure but no explicit derivation appears.
\item The lighter two masses from the same single-source diagonal form ($\sim 3\times 10^{-4}$~eV and $\sim 3\times 10^{-6}$~eV) are too small for the observed solar splitting $\sqrt{\Delta m^2_{\rm sol}}=0.0086$~eV; an inter-generation mixing derivation (cascade analogue of PMNS) is required. The PMNS extension of the gauge-window Cabibbo geometric mean (Theorem~\ref{thm:cabibbo-geometric-mean}) predicts $\theta_{12}\approx 7.5^\circ$ vs observed $33.4^\circ$, partial-negative on cascade-native mechanisms tested to date: the cascade neutrino sector requires a different mechanism, structurally distinct from the gauge-window Cabibbo geometric mean.
\item Majorana vs.\ Dirac character of the cascade neutrino mass term.
\end{enumerate}
The cascade predicts inverted hierarchy relative to charged leptons: $\nu_e$ heaviest (Gen~1 at $d=21$ is nearest the $d=29$ source), $\nu_\tau$ lightest (Gen~3 at $d=5$ is farthest). Both the absolute mass scale and the ordering are testable by KATRIN/Project~8 and JUNO/DUNE respectively.

\item \emph{Foundational extensions (long-term).} Three structural directions remain open at the level of the cascade scalar action $S[\varphi]=\sum_d(2\alpha(d))^{-1}(\Delta\varphi)^2$:
\begin{enumerate}[label=(\alph*),nosep]
\item \emph{Second-quantise the cascade action.} The current action is a classical field theory on the cascade lattice. A second-quantised version --- with $\varphi$ promoted to an operator --- would generate the source strengths, signs, and channel counts from the action's own quantum structure rather than from structural classification (cf.\ Proposition~\ref{prop:source-selection}, Theorem~\ref{thm:sign-rule}, Theorem~\ref{thm:chirality-selection-rule}).
\item \emph{Derive Lorentzian signature from the cascade action.} Part~III derives Lorentzian signature from the oscillatory propagator; the cascade action should reproduce this as a Wick rotation of the discrete Laplacian.
\item \emph{Verify the dissolution of quantum gravity.} Part~II\,=\,III shows QM and GR are projections of the same cascade state. The cascade action should make this dissolution explicit by generating both Gleason's theorem (forcing the Born rule) and Lovelock uniqueness (forcing Einstein's equation) from a single variational principle, demonstrating that no separate ``quantum gravity'' theory is needed.
\end{enumerate}
\end{enumerate}

\section{Confidence Assessment}\label{sec:confidence-assessment}

\textbf{Two-population systematic and its first-order correction.} At leading order, descent-dependent quantities carry uniformly negative deviations ($-0.13\%$ to $-2.20\%$, 7/7 negative), while geometric quantities carry positive deviations ($+0.56\%$ to $+2.76\%$, 4/4 positive). Part~0's inter-layer-coupling section derives a first-order correction $\delta Q/Q_0 = \sum_{\mathrm{adj}}(1-C_{d,d+1}^2)$ --- a sum of Beta function ratios with no free parameters --- that reduces descent deviations from $\sim\!2\%$ to sub-$1\%$: $\alpha_s$ ($-1.7\% \to -0.5\%$), $m_\tau$ ($-1.2\% \to -0.1\%$), $m_\tau/m_\mu$ ($-1.7\% \to -0.8\%$), $v$ ($-2.2\% \to -1.0\%$). Geometric quantities are unchanged (they carry no descent correction). A second-order observable-dependent factor remains open (Part~0 Open Question on higher-order corrections).

\textbf{Tier 1: Theorem-level results.} (a) The geometric-topological factorization, with $R'(d)$ excluded at 61--69\%. (b) The topological obstruction factor $1/(2\sqrt{\pi})$ derived from the chirality decomposition ($\chi = 2$, Theorem~\ref{thm:chirality-factorisation}) and the Berezin/Gaussian partition-function Jacobian difference (Remark~\ref{rem:berezin-partition-derivation}), with the per-layer mass $m(d) = R(d)/\chi = \sqrt{\alpha(d)}$ derived from chirality halving (Theorem~\ref{thm:obstruction}, Remark~\ref{rem:sp31-status}). No free Yukawa coupling enters the construction --- the SM's $m = y\cdot v/\sqrt{2}$ parameterisation does not appear in the cascade's chain. Equivalently expressed as the pure cascade identity $2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac{1}{2})$ (Corollary~\ref{cor:2sqrtpi-primitive}), ruling out any Dirac-operator interpretation by the vanishing of $\hat A(S^{2n})$ (Remark~\ref{rem:no-dirac-route}). (c) The Weinberg angle from the boson topological obstruction: SU(2) sits at the Dirac layer $d=13$ (no propagator factor; the obstruction is the symmetry breaking), U(1) at $d=14$ crosses $d=13$ during descent and picks up the boson-only quarter-turn factor $1/\sqrt{\pi}$, giving $\tan\theta_W = (N(14)/\sqrt{\pi})/N(13)$ and $\sin^2\theta_W = 0.2286$ (Theorem~\ref{thm:weinberg}). (d) $1/\alpha_{\text{GUT}} = 4\pi/N(12)^2 = 25.02$ from the Gamma function. (e) Adams' theorem gives $N_c = 3$ and $\dim\,\text{U}(1) = 1$. (f) The hairy ball theorem gives SU(3) $\times$ U(1) unbroken, SU(2) broken. (g) Three generations from Bott periodicity + the $d_1 = 19$ phase transition. (h) $w = -1$ exactly, with GB corrections vanishing by two independent mechanisms ([3]). (i) Universal $\mathrm{U}(1)$-layer shift to the cascade potential $\delta\Phi_{\mathrm{U}(1)}=\alpha(14)/\chi=R(14)^2/8=429^2\,\pi/2^{25}$ (Theorem~\ref{thm:alpha-s-closure}), applied exponentially: a single constant closes $\alpha_s(M_Z)=0.117917$ at $+0.019\sigma$ and $m_\tau/m_\mu=16.81731$ at $+0.243\sigma$ of experimental precision, equivalently $\alpha(12)\cdot(13/14)^2/2$ by the exact $\Gamma$ recursion $R(d+2)/R(d)=(d+1)/(d+2)$. (j) Phase-transition shift $\delta\Phi_{\mathrm{phase}}=\alpha(19)/\chi=R(19)^2/8$ sourced at the Paper~I threshold $d_1=19$ (Theorem~\ref{thm:mtau-abs-closure}), closing the $\tau$ absolute mass $m_\tau=1776.82$~MeV to $-0.31\sigma$ of experimental precision. (k) Observer-host shift $\delta\Phi_{\mathrm{obs}}=\alpha(5)/\chi^3=8/(225\pi)$ sourced at the volume maximum $d_V=5$ with three factors of the Euler characteristic (Theorem~\ref{thm:weinberg-closure}), closing $\sin^2\theta_W=0.231226$ to $+0.40\sigma$ of experimental precision. A systematic search (Remark~\ref{rem:phase-family}) extends the family to eight SM precision observables at four distinguished source layers: (l)~$\ell_A$ reuses $\alpha(19)/\chi$ ($-0.16\sigma$); (m)~$\Omega_m(1/\pi)$ uses $-\alpha(5)/\chi^3$ ($-0.04\sigma$); (n)~$\theta_C$ uses $-\alpha(7)/\chi^2$ ($+0.03\sigma$) with source $d_0=7$ (Paper~I area maximum); (o)~$b/s$ uses $-\alpha(7)/\chi^4$ ($0.014\%$) at the same source $d_0=7$ with $k=4$ (Theorem~\ref{thm:bs-closure}). Together, (i)--(o) close eight SM precision observables using one structural form $\pm\alpha(d^*)/\chi^k$ at four distinguished cascade layers and zero fitted parameters. Three reuse pairs strengthen the evidence: $\{\alpha_s,m_\tau/m_\mu\}$, $\{m_\tau\text{ abs},\ell_A\}$, $\{\sin^2\theta_W,\Omega_m\}$. (p)~Cabibbo geometric-mean off-diagonal rule from cascade action $+$ Born rule (Theorem~\ref{thm:cabibbo-geometric-mean}): the $\exp(-p(13)/2)$ factor in Theorem~\ref{thm:cabibbo-amplitude} is forced by the cascade descent integrand's probability-density status (Paper~I), the cascade scalar action's Gaussian quadratic form (Remark~\ref{rem:action-uniqueness}), the Born rule from Gleason on the cascade Hilbert space (Paper~II~\cite{part2}, Theorem~5.2), and the definition of off-diagonal mixing matrix elements as amplitudes; Cauchy--Schwarz saturates because the cascade descent kernel between adjacent layers is a single-saddle-point Gaussian (Remark~\ref{rem:per-leg-primitive-derivation}). The leading Cabibbo prediction $\theta_C=13.26^\circ$ is now action-derivation-complete; (n) above closes the residual $1.7\%$ to $+0.03\sigma$ of experimental precision. (q)~Tree-level gyromagnetic ratio $g=2$ at every Dirac layer from the cascade Dirac descent (Paper~II$=$III, Theorem~\xref{part23}{thm:cascade-dirac-descent}) applied to the Clifford algebraic identity $\gamma^\mu\gamma^\nu=\eta^{\mu\nu}+i\sigma^{\mu\nu}$ (Corollary~\ref{cor:g-equals-two}); together with the cascade-primitive identification of the 1-loop prefactor $1/(2\pi) = 1/(N(0)\cdot\Gamma(\tfrac{1}{2})^2)$ (Remark~\ref{rem:two-pi-prefactor-identification}, via Corollary~\ref{cor:2sqrtpi-primitive}) and the Tier~2 closure of $\alpha_{\rm em}$ (entry~(k) above), this gives $g=2+\alpha_{\rm em}/\pi+\mathcal{O}(\alpha_{\rm em}^2)$ with all factors at 1-loop cascade-internally identified --- no fitted parameter and no QED-imported number at this order. Higher-loop QED coefficients ($a_e$ at 5-loop, $a_\mu$ frontier) are open (Open Question~\ref{oq:higher-loop-qed-coefficients}).

\textbf{Tier 2: Fully derived predictions.} (a) $m_\mu/m_e$ closed at $-0.0012\%$ residual (machine precision on the cascade-derived $\alpha_{\rm em}$) via Family B at the $m{-}k=0$ slot of Theorem~\ref{thm:chirality-selection-rule}. The cascade leading $\exp(\Phi(21)-\Phi(13))\cdot 2\sqrt{\pi}=206.4958$ receives the radiative correction $\delta(m_\mu/m_e) = \alpha_{\rm em}/(2\pi) + \alpha_{\rm em}\cdot\alpha(21) = 0.1331\%$ of Proposition~\ref{prop:slot-precedence} (non-source-selectable case at the $m{-}k=0$ slot; Remark~\ref{rem:slot-precedence-mu-e}), giving $206.7710$ vs.\ PDG $206.7683$. Conditional on the strict reading of the $G$ flag in Proposition~\ref{prop:source-selection}; categorical derivation of the source-selection flags (Open Question~\ref{oq:source-selection-category}) would consolidate at full structural rigour. (b) $\alpha_s(M_Z)=0.117917$ and $m_\tau/m_\mu=16.81731$ closed within experimental precision via the universal $\mathrm{U}(1)$ shift $\delta\Phi_{\mathrm{U}(1)}=\alpha(14)/\chi$ (Theorem~\ref{thm:alpha-s-closure}). (c) $m_\tau = 1776.82$~MeV absolute closed within experimental precision via the phase-transition shift $\delta\Phi_{\mathrm{phase}}=\alpha(19)/\chi$ (Theorem~\ref{thm:mtau-abs-closure}). (d) $\sin^2\theta_W=0.231226$ closed within experimental precision via the observer-host shift $\delta\Phi_{\mathrm{obs}}=\alpha(5)/\chi^3$ (Theorem~\ref{thm:weinberg-closure}). (e) $\theta_C=13.04^\circ$ closed to $+0.03\sigma$ via $-\alpha(7)/\chi^2$ sourced at the Paper~I area maximum $d_0=7$. (f) $\ell_A=301.44$ closed to $-0.16\sigma$ via $\alpha(19)/\chi$, reusing the phase-transition shift. (g) $\Omega_m(1/\pi)=0.31473$ closed to $-0.04\sigma$ via $-\alpha(5)/\chi^3$, reusing the observer-host shift with opposite sign (geometric population). (h) $C = \alpha_s/(2\sqrt{\pi})$ (1.0\%). (i) Absolute masses $m_\mu$ (0.47\%) and $m_e$ (0.60\%), consistent with the chain-subtracted inherited shift $\delta\Phi_{\mathrm{phase}}-\delta\Phi_{\mathrm{U}(1)}$ to $\sim 50$--$70$~ppm. (j) $b/s = 44.7436$ closed to $0.014\%$ via $-\alpha(7)/\chi^4$ (Theorem~\ref{thm:bs-closure}), reusing the area-maximum source layer with $k=4$ for cross-generation quark structure. (k) $1/\alpha_{\mathrm{em}} = 1/\alpha(13) + \pi/\alpha(14) + 6\pi = 137.028$ ($0.006\%$): bare coupling $1/\alpha(13) + \pi/\alpha(14) = 118.18$ from Theorem~\ref{thm:weinberg} and the electroweak mixing; per-Dirac-layer chirality factor $\chi$ from Theorem~\ref{thm:chirality-selection-rule} (closed-cycle $m=1$, $k=0$); per-leg primitive $\Gamma(\tfrac{1}{2})^2 = \pi$ from Remark~\ref{rem:per-leg-primitive-derivation}, with the cascade kinetic-prefactor normalisation grounded in Remark~\ref{rem:wavefunction-renorm-canonical}; $N_{\mathrm{gen}}=3$ as the count of Dirac layers in the descent $\{5,13,21\}$ (Part~IVa~\cite{part4a}'s three-generation theorem). All four pieces fully cascade-internally derived using cascade-lattice primitives only (no proper-time integral, no QFT loop apparatus, per CLAUDE.md Check~7); the $0.006\%$ residual is the cascade's standing leading-order systematic.

\textbf{Tier 3: Derived predictions with caveats.} (a) $m_H/m_W = \pi/2$: the geodesic distance argument is natural but the mass-ratio/geodesic identification is stated rather than computed from an action. (b) $v = 240.8$~GeV and (c) the $\exp(-\pi/\alpha(5))$ factor in Theorem~\ref{thm:vev}: the cascade-native identity $\pi/\alpha(5) = \Omega_2/R(5)^2$ is verified and matches the Kosterlitz--Thouless single-vortex action on the 2-sphere cross-section of $S^4$ via the Morse foliation of the $d = 5$ hairy-ball defect (Remark~\ref{rem:v-closure}); the half-winding follows from Theorem~\ref{thm:chirality-factorisation}'s chirality basin filter with effective winding $n_{\mathrm{eff}} = \chi(S^4)/\chi = 1$. Open: formal proof that the 4D tangent-field obstruction on $S^4$ is equivalent to a first-Chern class on the 2-cycles of the Morse foliation (pure geometric topology, not cascade content).

\textbf{Tier 4: Observed patterns needing derivation.} (a) $m_b/m_\tau = e$: deviation 1.05\%. (b) $(t/b)/(c/s) = N_c = 3$: physical argument plausible but chirality coupling not computed. ($b/s$ has been promoted to Tier~2 via Theorem~\ref{thm:bs-closure}.) (c) Cascade-primitive form for QCD's $\beta_0 = (N_c^2 + N_c - 1) - N(0)\,n_f/N_c$ matches the standard $11 N_c/3 - 2 n_f/3$ exactly across $n_f\in\{3,4,5,6\}$ (\S\ref{subsec:qcd-frontier}, Remark~\ref{rem:cascade-beta0}); cascade-internal derivation of the gauge-sector decomposition $N_c^2 - 1 + N_c$ is open. (d) Cascade $\beta_1$ matches the standard 2-loop coefficient exactly across flavor windows but admits two cascade-primitive decompositions for the gauge factor $102$; cascade-interaction-level disambiguation is open (Remark~\ref{rem:cascade-beta0}). (e) Cascade-native confinement scale $\Lambda_{\rm QCD}^{\rm cascade} = M_Z\cdot\exp(-2\pi)\approx 170$~MeV with cascade-to-MS-bar scheme factor $\sqrt{N_c/N(0)}$ landing $\Lambda_{\rm PDG} \approx 208.6$~MeV ($-0.7\%$ from PDG $210\pm 14$~MeV); cross-sector consistency on $m_\pi, f_\pi$ in PDG units selects this scheme factor uniquely among cascade-primitive candidates within the PDG band (Remark~\ref{rem:cascade-beta0}). Cascade-internal derivation of the scheme-conversion factor is open (Tier~2 promotion target). (f) Cascade chiral physics: $m_\pi = \Lambda_{\rm PDG}\cdot 2/3$ ($+0.31\%$), $f_\pi = \Lambda_{\rm PDG}\cdot 4/9$ ($+1.4\%$), $f_\pi/m_\pi = N(0)/N_c$ scheme-invariant ($+1.06\%$) (Remark~\ref{rem:cascade-beta0}). (g) Pseudoscalar octet $m_K/m_\pi = d_V/\sqrt{N(0)} = 5/\sqrt{2}$ ($-0.05\%$) and $m_K = \Lambda_{\rm PDG}\cdot 5\sqrt{2}/3$ ($+0.27\%$); structurally consistent with chiral relations (same $N(0)/N_c$ primitive). (h) U(1)$_A$ anomaly sector: $m_{\eta'} = \Lambda_{\rm PDG}\sqrt{N_c\cdot d_0}$ ($+0.48\%$), $m_\eta = \Lambda_{\rm PDG}(N_c+1)\sqrt{N_c/d_0}$ ($+0.37\%$), $m_{\eta'}/m_\eta = d_0/(N_c+1) = 7/4$ ($+0.10\%$); Witten--Veneziano cross-check gives cascade $\chi_{\rm top}^{1/4} \approx 181$~MeV vs lattice $178$~MeV ($+1.7\%$). The double-Adams form $(\rho(12)-1)\cdot(\rho(8)-1)$ for $m_{\eta'}^2/\Lambda^2$ is the topological extension theorem (Theorem~\ref{thm:axial-anomaly-mass}, Tier~4b). (i) Vector nonet ($\rho, \omega, K^*, \phi$): cascade-primitive fits sit at $1$--$3\%$ deviations with structural-form ambiguity, marking the boundary at $J=1$ where current cascade machinery stops; the missing piece is a cascade-native treatment of hyperfine splitting (open structural direction, Remark~\ref{rem:cascade-beta0}).

\textbf{Tier 4b: Claimed but proof has gaps.} (a) $\theta_{\text{QCD}} = 0$ from $\pi_3(S^{11}) = 0$ (the trivial group, by $\pi_n(S^k) = 0$ for $n < k$; Theorem~\ref{thm:strong-cp}). Only the trivial sector exists in the cascade-internal classification, so the gauge-sector $\theta_{\text{QCD}} = 0$ is unique without a selection step. The remaining gap is structural: the claim that the cascade's topological sectors are classified by $\pi_3(S^{11})$ rather than $\pi_3(\text{SU}(3))$ requires showing how the vector-field realisation of SU(3) on $S^{11}$ replaces (rather than coexists with) the standard gauge-target classification. The mass-matrix branch $\arg\det(M) = 0$ vs.\ $\pi$ is selected by the cascade's $U(1)$ gauge equivalence (Step~2 of Theorem~\ref{thm:strong-cp}). (b) U(1)$_A$ axial-anomaly mass via double-Adams: $m_{\eta'}^2 = \Lambda_{\rm QCD}^2\cdot(\rho(12)-1)\cdot(\rho(8)-1) = 21\,\Lambda_{\rm QCD}^2$ (Theorem~\ref{thm:axial-anomaly-mass}). Both factors are Adams' theorem outputs --- the first the same gauge-sector Adams count used in Theorem~\ref{thm:strong-cp} Step~1, the second the octonion-quotient Adams count at $S^7$ (the parallelizable octonion sphere). Sub-percent empirical match on $m_{\eta'}$ ($+0.48\%$) and Witten--Veneziano consistency on $\chi_{\rm top}^{1/4}$ ($+1.7\%$ vs lattice). Acknowledged gaps (Remark~\ref{rem:axial-anomaly-mass-tier}): (i)~the specific Adams pairing (product vs.\ alternative) is empirically selected, not yet rigorously forced; (ii)~the cascade-internal derivation of $\chi_{\rm top}$ from layer winding is the Tier~2 promotion target. The previously-flagged gap on $d_0 = \rho(8)-1$ (gap (iii)) is now resolved: both equal $\dim(\mathrm{Spin}(7)/G_2) = \dim S^7$, forced by the cascade's existing $G_2/\mathrm{SU}(3) = S^6$ commitment at $d_0$, giving the Lie-algebraic reading $m_{\eta'}^2 = \Lambda_{\rm QCD}^2\cdot\dim\mathrm{Spin}(7)$.

\textbf{What would change my mind.} The framework would be significantly weakened by: (i) discovery of a QCD axion; (ii) $w \neq -1$ at $> 5\sigma$ from BAO alone with $r_d$ independently constrained at $> 144$~Mpc; (iii) discovery of a fourth fermion generation; (iv) a demonstration that the topological obstruction factor is not $2\sqrt{\pi}$ (now derived in Theorem~\ref{thm:obstruction} and equivalently in Corollary~\ref{cor:2sqrtpi-primitive}; falsification would require either invalidating the chirality/quarter-turn argument or invalidating the cascade-primitive identity $2\sqrt{\pi} = N(0)\cdot\Gamma(\tfrac{1}{2})$); or (v) any absolute mass prediction deviating by $> 5\%$ after subleading corrections are included. Conversely, the framework would be strongly supported by: (vi) derivation of the colour factor $e$ from first principles; or (vii) any new precision prediction matching observation before the data is checked.

\begin{thebibliography}{13}

\bibitem{part0} {[Author]}, \extlink{\cascadebase/cascade-series-part0.pdf}{\textit{Scale Variance from Orthogonality: How the Unit Ball Generates $10^{120}$ Orders of Magnitude}}, March 2026.

\bibitem{part2} {[Author]}, \extlink{\cascadebase/cascade-series-part2.pdf}{\textit{Quantum Mechanics from the Cascade: Effective Theory of a 4-Dimensional Observer in the Sphere-Area Geometry}}, March 2026.

\bibitem{part3} {[Author]}, \extlink{\cascadebase/cascade-series-part3.pdf}{\textit{General Relativity from the Cascade: Lovelock Uniqueness and the Cosmological Constant of a 4-Dimensional Observer}}, March 2026.

\bibitem{part4a} {[Author]}, \extlink{\cascadebase/cascade-series-part4a.pdf}{\textit{The Standard Model from the Cascade: Part IVa---Gauge Group, Symmetry Breaking, and Three Generations from Bott Periodicity and Hairy Ball Zeros}}, March 2026.

\bibitem{part5} {[Author]}, \extlink{\cascadebase/cascade-series-part5.pdf}{\textit{The Cascade Series --- Part V: Cosmology from the Cascade: $\Lambda$CDM Parameters, the Hubble Constant, and the DESI BAO Observations}}, March 2026.

\bibitem{lounesto} P.~Lounesto, \textit{Clifford Algebras and Spinors}, 2nd ed., Cambridge University Press, 2001.

\bibitem{abs} M.~F.~Atiyah, R.~Bott, and A.~Shapiro, ``Clifford modules,'' \textit{Topology} \textbf{3}, 3--38 (1964).

\bibitem{adams} J.~F.~Adams, ``Vector fields on spheres,'' \textit{Annals of Mathematics} \textbf{75}, 603--632 (1962).

\bibitem{poincare} H.~Poincar\'e, ``Sur les courbes d\'efinies par les \'equations diff\'erentielles,'' \textit{Journal de Math\'ematiques Pures et Appliqu\'ees} \textbf{4}(1), 167--244 (1885).

\bibitem{hopf} H.~Hopf, ``Vektorfelder in $n$-dimensionalen Mannigfaltigkeiten,'' \textit{Mathematische Annalen} \textbf{96}, 225--249 (1927).

\bibitem{hurwitz} A.~Hurwitz, ``\"Uber die Komposition der quadratischen Formen,'' \textit{Mathematische Annalen} \textbf{88}, 1--25 (1923).

\bibitem{gj} H.~Georgi and C.~Jarlskog, ``A new lepton-quark mass relation in a unified theory,'' \textit{Physics Letters B} \textbf{86}, 297--300 (1979).

\bibitem{milnor} J.~W.~Milnor, \textit{Topology from the Differentiable Viewpoint}, Princeton University Press, 1965.

\bibitem{desi} DESI Collaboration, \textit{DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints}, arXiv:2503.14738 (2025).

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