cascade-series-part3.tex

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% Note: \bibitem{part0} in this paper's bibliography points at Part 0
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\xrhyperdoc{part2}{cascade-series-part2}
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\xrhyperdoc{part4b}{cascade-series-part4b}
\xrhyperdoc{part5}{cascade-series-part5}

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\title{\textbf{The Cascade Series --- Part III}\\[6pt]
General Relativity, Four Dimensions,\\
and Lorentzian Signature from the Cascade:\\
Lovelock Uniqueness, Spinor Compatibility,\\
and the Cosmological Constant}
\author{RTAC}
\date{March 2026}

\begin{document}
\setlength{\emergencystretch}{3em}
\maketitle
\phantomsection
\label{paperdoc}

\begin{abstract}
The sphere-area cascade [1] derives a geometric invariant $I = 1.0990 \times 10^{-120}$
from orthogonality alone. The companion paper [2] shows that the cascade geometry
produces complex quantum mechanics via a forced precession angle $\alpha = \pi/2$. This
paper derives, from the same orthogonality axiom and with no additional empirical
input: (1) general relativity with cosmological constant $\Lambda = I$; (2) the observer's
spacetime dimension $d = 4$; and (3) Lorentzian metric signature $(-,+,+,+)$.

The argument has four components. \textbf{The cascade metric} (Section 4): the slicing
recurrence produces a foliated FRW-type metric with a natural lapse function and
extrinsic curvature. \textbf{Lovelock uniqueness} (Section 3): the Einstein equation with
cosmological constant is the unique divergence-free symmetric rank-2 metric equation
in exactly four spacetime dimensions. \textbf{Dimension derivation} (Section 9): $d = 4$ is the
unique dimension satisfying simultaneously (C1) the Clifford algebra classification
requires irreducibly complex spinors (forced by the cascade's $J^2 = -\text{Id}$), and (C2)
Lovelock uniqueness holds. The intersection of these conditions is $\{4\}$. A third
independent characterisation is established in Corollary 9.4: the Lorentzian cascade
metric has Ricci scalar $R^{(n)} = (n-1)(n-4)/a^4$, vanishing uniquely at $n = 4$.
\textbf{Lorentzian signature} (Section 10): the cascade produces both a Euclidean geometry
and a quantum propagator $e^{-i\lambda x}$ with $\lambda > 0$ from the forced precession. Given the
physical identification hypothesis, the propagator $e^{-iHt}$ with $H > 0$ is oscillatory
and unitary: Lorentzian time evolution. The Euclidean-to-Lorentzian
map is Wick rotation ($x\leftrightarrow it$), motivated
cascade-internally by the forced precession rather than imposed as a
mathematical trick; see Theorem~\ref{thm:lorentzian} and
Remark~\ref{rem:wick-rotation-cascade}.

The cosmological constant $\Lambda = I$ is derived, not fitted. The cascade's dark
energy equation of state is $w = -1$ exactly: $\Lambda = I$ is a fixed geometric constant
with no time-evolution mechanism, so $w = -1$ by definition. The Gauss--Bonnet
correction at $d = 5$ vanishes identically by two independent mechanisms (totally
umbilic cancellation and parity of the residual integrand), confirming the prediction is
structurally stable. The supplement (Section 14) proves the compactification radius
at each step is $R_{\text{eff}}(d) = 1/\sqrt{d + 3}$ exactly (from the Beta function), and presents the
exact Lorentzian Gauss--Codazzi analysis: the cascade metric contributes $w = 1/3$
(radiation) in cascade time, and $K_{\mu\nu} = 0$ at every equatorial embedding eliminates
the extrinsic curvature mechanism. These results resolve the question of the cascade's
matter fraction: the geometric route via Gauss--Codazzi is definitively closed, and
the matter fraction $\Omega_m$ is a topological quantity determined by the Bott partition of
cascade layers, not by the cascade metric geometry.
\end{abstract}

\tableofcontents
\newpage

\section{Division of Labour and Summary of the Series}\label{sec:division-of-labour-and-summary-of-the-se}

\subsection{The architecture}

The three-paper core rests on one hypothesis and three uniqueness theorems.

The hypothesis (this paper, Section 2): the cascade's abstract geometry is our physics.
Without it, the series is pure mathematics; with it, every physical result is a prediction
that tests the hypothesis.

The structure: the cascade provides geometric content; classical uniqueness theorems
identify the unique physical theory consistent with that content.

\begin{center}
\begin{tabular}{lll}
\hline
Paper & Cascade provides & Uniqueness provides \\
\hline
I [1] & $\Omega_d$ cascade, $I \approx 10^{-120}$ & --- (pure geometry) \\
II [2] & Complex state space, $\mathbb{C}P^{n-1}$ via forced $U(1)$ gauge,
         complex Born rule $|\langle u,v\rangle_{\mathbb{C}}|^2$,
         propagator $e^{-iHt}$, time as $d$-collapse
       & Sphere geometry, gauge-group maximality (Born rule + projective
         state space) \\
III (this) & Metric, $\Lambda = I$, $d = 4$, $(-,+,+,+)$ & Lovelock, Clifford, propagator \\
\hline
\end{tabular}
\end{center}

\subsection{What this paper proves}

Given the physical identification hypothesis (Section 2), this paper derives four results
from the orthogonality axiom of [1]:

\begin{enumerate}
\item The cascade produces a foliated spacetime geometry with a natural lapse function
$N(d) = \sqrt{\pi} \cdot R(d)$ and a FRW-type 4D metric.
\item By Lovelock's theorem, the unique gravitational equation available to a 4D observer
is the Einstein equation; the cosmological constant equals the cascade invariant
$\Lambda = I$.
\item $d = 4$ is the unique spacetime dimension satisfying both the cascade's forced complex
spinor requirement and Lovelock uniqueness.
\item Lorentzian metric signature $(-,+,+,+)$ is the unique signature consistent with the
cascade's forced propagator $e^{-iHt}$ with $H > 0$.
\end{enumerate}

\section{The Physical Identification Hypothesis}\label{sec:the-physical-identification-hypothesis}

The cascade is a mathematical object: a sequence of sphere areas $\{\Omega_d\}$ derived from
orthogonality. Paper I [1] is pure mathematics; it makes no physical claim beyond recording
the numerical coincidence $I \approx \rho_\Lambda/M_{\text{Pl}}^4$. Paper II [2] shows the cascade's state space has the
structure of quantum mechanics. This paper derives the cascade's gravitational structure.
Both Papers II and III require one physical input.

\begin{definition}[Physical identification hypothesis]\label{def:hyp}
The infinite-dimensional unit ball,
descended to four dimensions, is indistinguishable from our universe. Concretely: the
cascade metric is the spacetime metric; the cascade propagator is the physical time-evolution
operator; the cascade invariant $I$ is the cosmological constant. This hypothesis is the
claim that the cascade's abstract geometry is our physics. Without it, the series is pure
mathematics; with it, every physical result is a prediction that tests the hypothesis.
\end{definition}

\begin{remark}[Uniformity of the physical identification]
Definition 2.1 maps cascade
content to physical energy uniformly across all layer types. By Corollary 3.2 of [1], sphere
area $\Omega_{d-1}$ is the only independent cascade quantity at level $d$; every other cascade object
is derived from sphere areas. Therefore the identification assigns energy $C \cdot \Omega_{d-1}$ at each
level with $C$ a universal constant---no layer-type-dependent modulation is available, because
no independent geometric quantity exists to carry such modulation. Matter and vacuum
are observer labels applied to the Bott partition of cascade layers: the propagator phase
classification ([2], Corollary 6.8; refined to period 8 via Clifford in this paper, Section 9) establishes the partition topologically, and the fermion
generation structure ([3]) identifies which sector is matter via the hairy ball zeros and
observed generations. The cascade geometry itself makes no distinction between the sectors.
\end{remark}

\begin{remark}
The physical identification hypothesis does not assume: the dimension of
spacetime (derived as $d = 4$ in Section 9); the metric signature (derived in Section 10);
the form of the gravitational equation (derived via Lovelock); or the value of $\Lambda$ (derived as
$I$ from [1]). It assumes only that the cascade corresponds to something physical. This is
the minimal assumption needed to connect abstract geometry to physics.
\end{remark}

\section{Lovelock's Theorem and the Privilege of Four Dimensions}\label{sec:lovelock-s-theorem-and-the-privilege-of}

\begin{theorem}[Lovelock {\cite{lovelock1,lovelock2}}]
Let $(M, g)$ be a smooth $d$-dimensional Lorentzian manifold
with $d = 4$. The most general symmetric divergence-free rank-2 tensor constructed from
$g_{\mu\nu}$ and its first and second derivatives is
\[
E^{\mu\nu} = a\,G^{\mu\nu} + b\,g^{\mu\nu},
\]
where $G^{\mu\nu}$ is the Einstein tensor and $a$, $b$ are constants. Setting $a = 1$, $b = -\Lambda$ recovers
the Einstein equation with cosmological constant.
\end{theorem}

The theorem is dimension-dependent. In $d$ spacetime dimensions, the most general
such tensor is a sum of $\lfloor d/2 \rfloor$ independent Lovelock terms plus $\Lambda$.

\begin{center}
\begin{tabular}{lcl}
\hline
Dimension $d$ & Lovelock terms & Gravity unique? \\
\hline
3 & $\Lambda$ only & No propagating gravity \\
4 & $G^{\mu\nu} + \Lambda g^{\mu\nu}$ only & Yes: GR is unique \\
5, 6 & Einstein + Gauss--Bonnet + $\Lambda$ & No: free coupling \\
$\geq 7$ & Further Lovelock terms + $\Lambda$ & No \\
\hline
\end{tabular}
\end{center}

\begin{remark}
Lovelock uniqueness at $d = 4$ is a theorem about tensor equations, not a
consequence of the cascade. The cascade's role is to provide the metric that Lovelock needs
as input. Section 9 derives $d = 4$ from the consistency of the cascade's quantum structure
with Lovelock uniqueness.
\end{remark}

\section{The Cascade Metric}\label{sec:the-cascade-metric}

\subsection{Foliation from slicing}

The cascade's slicing recurrence decomposes $B^{d+1}$ into $d$-balls parametrised by the
perpendicular coordinate $x \in [-1, 1]$. Each cross-section at height $x$ is a $d$-ball of radius
$r(x) = \sqrt{1 - x^2}$. This is a foliation; $x$ plays the role of time, identified
in~\cite{part2} Section~7.1 as one quantum of \emph{dimensional collapse}: each
slicing operation resolves one more dimension of $B^\infty$ into a definite cascade
structure, and the slicing direction is the cascade's resolution of $B^\infty$ rather
than merely an irreversibility-equipped axis. The Lorentzian observer-time coordinate
$t$ enters via the Wick rotation $x = it$ of Section~10; the foliation parameter $x$
is the Euclidean version of the same dimensional-collapse counter. The full
$(d + 1)$-dimensional metric is:
\[
ds^2 = dx^2 + (1 - x^2)\,d\sigma_{d-1}^2.
\]

\subsection{The cascade lapse function}

The ratio of adjacent volumes is the cascade's lapse:
\[
N(d) = \sqrt{\pi} \cdot R(d), \qquad R(d) = \frac{\Gamma((d + 1)/2)}{\Gamma((d + 2)/2)} \approx \sqrt{\frac{2\pi}{d}} \quad (d \gg 1).
\]
At $d = 4$: $N(4) = \sqrt{\pi} \cdot \Gamma(5/2)/\Gamma(3) = 3\pi/8 \approx 1.178$. As $d \to \infty$: $N \to 0$ (no temporal
separation at the cascade's geometric origin).

\noindent The word ``lapse'' is used here in its cascade-internal
sense: $N(d)$ is a dimensionless ratio of adjacent unit-ball volumes
indexed by the cascade dimension, characterising the compression rate
per slicing step. It is \emph{not} the lapse function of an ADM metric
foliation in the general-relativistic sense (which is a spacetime
function $N(t,x)$ relating proper time to coordinate time in
$ds^2 = -N^2\,dt^2 + g_{ij}\,dx^i\,dx^j$). These two uses of the word
``lapse'' name distinct objects; the distinction is made explicit in
Remark~\ref{rem:cascade-vs-metric-lapse} below, and
Paper~V~\cite{part5} documents an earlier derivation that conflated
them and was subsequently retracted.

\subsection{The extrinsic curvature}

The extrinsic curvature of the leaf at height $x$ is $K = -x/\sqrt{1 - x^2}$. At the equator $x = 0$:
$K = 0$. This is the cascade's Hubble parameter.

\subsection{The 4D projected metric}

Projecting onto a 4-dimensional subspace, the 4D observer's metric is:
\[
ds^2_{\text{4D}} = -dt^2 + a(t)^2\,d\sigma_3^2.
\]
This is the FRW metric with unit \emph{metric} lapse: $t$ is the
observer's proper time, and $g_{tt} = -1$ by choice of time coordinate.
Isotropy follows from the cascade's spherical symmetry; homogeneity
from the uniform slicing recurrence. The Lorentzian sign $g_{tt} = -1$
is derived in Section 10.

The cascade lapse $N(4) = 3\pi/8$ (Section~4.2) does \emph{not} enter
this metric. It is a cascade-internal volume-ratio quantity about the
slicing recurrence, not a metric function of the observer's spacetime.
The distinction is elaborated in Remark~\ref{rem:cascade-vs-metric-lapse}
below.

\begin{remark}[Cascade lapse vs.\ metric lapse]\label{rem:cascade-vs-metric-lapse}
The cascade lapse $N(d) = \sqrt{\pi}\,R(d) = V_d/V_{d-1}$ defined in
Section~4.2 and the metric lapse of an ADM foliation are distinct
objects:
\begin{itemize}
\item \emph{Cascade lapse $N(d)$.} A dimensionless ratio of adjacent
unit-ball volumes (equivalently, adjacent unit sphere areas by
Paper~I~\cite{part0} boundary dominance), indexed by the cascade
dimension $d$. Characterises the rate of content compression per
slicing step. Defined layer-by-layer, independent of any spacetime
coordinate.
\item \emph{Metric (ADM) lapse.} The function $N(t,x)$ appearing in
$ds^2 = -N^2\,dt^2 + g_{ij}\,dx^i\,dx^j$, which converts between the
coordinate time $t$ and proper time for an observer at rest in the
foliation. A spacetime function.
\end{itemize}
The 4D observer's metric $ds^2 = -dt^2 + a(t)^2\,d\sigma_3^2$ in
Section~4.4 uses $t$ as the observer's proper time, so the metric
lapse is 1 by coordinate choice; this is the standard FRW convention.
The cascade lapse $N(4) = 3\pi/8$ does not appear in this metric --- it
is a different quantity with a different meaning.

\medskip\noindent\textbf{Retracted conflation in an earlier Paper~V derivation.}
An earlier version of Paper~V~\cite{part5} attempted to identify
$N(4)$ with the metric lapse of an ADM foliation of the cascade's
descent (treating $d$ as an ADM time coordinate with lapse $N(d)$), and
used $N(4)^2 = 9\pi^2/64 \approx 1.388$ as a frame-conversion factor
in the Friedmann equation. That derivation yielded $H_0 = 71.05$~km/s/Mpc
and is retracted in Paper~V, Section~\xref{part5}{sec:hubble-constant} (``Why the observer bridge and
not a descent lapse''): the correct conversion between the cascade's
sphere-area content and the observer's 3-volume is the $\pi/2$
projection factor of Paper~I~\cite{part0} §3.2, not $N(4)^2$. Under
the corrected Paper~I bridge, the cascade lapse $N(4)$ does not enter
the observer's metric at any point, consistent with the distinction
stated above.

\medskip\noindent\textbf{Cascade slicing = time at the observer's scale.}
Paper~II~\cite{part2} Section~7.1 identifies the cascade's slicing
direction with physical time at the observer's scale. That
identification is preserved: the slicing parameter (the cascade
coordinate $d$) maps to the observer's proper time $t$ under the
locked-time infall of Paper~I. What does not map is the cascade
lapse $N(d)$ to the metric lapse of $t$. They share a word, not a
referent.
\end{remark}

\section{The Cosmological Constant}\label{sec:the-cosmological-constant}

\subsection{From cascade invariant to \texorpdfstring{$\Lambda$}{Lambda}}

By [1], Theorem 9.2:
\[
I = \frac{9\,\Omega_{19}\,\Omega_{217}}{\pi^2} = 1.0990 \times 10^{-120}.
\]
The 4D observer's vacuum energy satisfies $\rho_\Lambda/M_{\text{Pl}}^4 \approx 1.1 \times 10^{-120}$. The identification $\Lambda = I$
is not a fit: $I$ is fixed by $\pi$ independently of any physical measurement. The agreement is
$0.04\%$ under the reduced-Planck-mass convention.

\subsection{Why \texorpdfstring{$\Lambda$}{Lambda} is not a free parameter}

In standard GR, $\Lambda$ is a free parameter: Lovelock permits any $b$ in $E^{\mu\nu} = aG^{\mu\nu} + bg^{\mu\nu}$. The
cascade removes this freedom. The constant $b = -\Lambda$ is fixed by the cascade invariant $I$; the
observer does not choose it. The cosmological constant problem---why $\Lambda \sim 10^{-120}M_{\text{Pl}}^4$---is
answered by the hierarchy theorem of [1]: $\log_{10}(1/\Omega_{d_2}) \approx \pi e^{2\sqrt{\pi}}(2\sqrt{\pi} - 1)/\ln 10 \approx 120.3$
orders of magnitude arise from the Gamma function's superexponential growth between
$d_1 = 19$ and $d_2 = 217$.

\section{The Dark Energy Equation of State}\label{sec:the-dark-energy-equation-of-state}

\begin{theorem}[Dark energy equation of state]
The cascade predicts $w = -1$ exactly.
\end{theorem}

\begin{proof}
$\Lambda = I$ is a fixed geometric constant, determined entirely by $\pi$ through the cascade's
threshold structure ([1], Theorem 9.2). A fixed cosmological constant has $\rho_\Lambda = \text{const}$,
giving $p_\Lambda = -\rho_\Lambda$ and $w = p/\rho = -1$ by definition. No time-evolution mechanism exists
in the cascade for $\Lambda$: the thresholds $d_1 = 19$ and $d_2 = 217$ are permanent features of the
Gamma function, not snapshots of a dynamical field. The lapse function $N(4) = 3\pi/8$ is a
fixed Gamma-function value, not a cosmological-time-evolving quantity.
\end{proof}

\subsection{Gauss--Bonnet stability}

The cascade geometry at $d = 5$ has $S^3$ cross-sections that are totally umbilic: $K_{ij} =
-(x/\sqrt{1 - x^2})\,h_{ij}$ exactly, so $K_{ij} = (K/3)h_{ij}$. For totally umbilic hypersurfaces, the leading
Gauss--Bonnet boundary terms cancel identically:
\[
2K_{ij}K^{ij}K - \tfrac{2}{3}K^3 = \tfrac{2}{3}K^3 - \tfrac{2}{3}K^3 = 0.
\]
The residual GB term $2K\tilde{R} - 2K_{ij}\tilde{R}^{ij} \propto -24x/(1 - x^2)^2$ is odd in $x$; the cascade measure
$(1 - x^2)^{3/2}$ is even. The integral vanishes exactly:
\[
\int_{-1}^{1} (1 - x^2)^{3/2} \cdot \frac{-24x}{(1 - x^2)^2}\,dx = -24\int_{-1}^{1} \frac{x}{\sqrt{1 - x^2}}\,dx = 0.
\]
The bulk GB contribution is suppressed by $H_0^2/M_{\text{Pl}}^2 \sim 10^{-120}$.

\begin{corollary}
$w = -1$ receives zero Gauss--Bonnet correction at leading order, by two
independent mechanisms: (i) totally umbilic cancellation from the cascade's spherical
symmetry; (ii) parity of the residual integrand under symmetric slicing. Both follow from
the orthogonality axiom. The prediction is structurally protected, not merely numerically
small.
\end{corollary}

\begin{remark}[Quantitative cosmology]
The cascade's geometric parameters determine the background cosmological
observables. The density fractions $\Omega_\Lambda = (\pi-1)/\pi$,
$\Omega_m = 1/\pi$ (leading order) or $\Omega_m^{\rm Bott} = 0.31150$
(subleading), and $\Omega_b = 1/(2\pi^2)$ follow from the cascade's
sphere-area structure. The Hubble constant $H_0 = 66.78$~km/s/Mpc
follows from the Friedmann equation with Part~I's observer-corrected
$\rho_\Lambda/M_{\rm Pl,red}^4 = (2/\pi)\,I$ (where $2/\pi$ is the
cube--sphere bridge $V_3^{\rm cube}/\Omega_2$ at the observer's spatial
dimension $d=3$), sitting $0.9\%$ below Planck's $67.4$ at leading
order and closing to essentially the Planck central value after the
Part~0 Gram first-order correction; it is incompatible with
the SH0ES local measurement of $73.0$. These parameters predict a sound
horizon $r_d \approx 147.75$~Mpc, essentially equal to Planck's
$147.60$~Mpc. The cascade predicts $w = -1$ exactly as a structural
theorem; the apparent DESI preference for $w \neq -1$ challenges both
the cascade and $\Lambda$CDM equally, and cannot be attributed to a
ruler mismatch between them.
\end{remark}

\begin{theorem}[Orthogonality of coupling and vacuum sectors]
The cascade lapse factorises
as $N(d) = \sqrt{\pi} \cdot R(d)$. Gauge couplings satisfy $\alpha(d) = N(d)^2/(4\pi) = R(d)^2/4$: the
factor of $\pi$ from squaring $\sqrt{\pi}$ cancels the denominator exactly, leaving $\alpha = R(d)^2/4$: the
multiplicative $\sqrt{\pi}$ factor of the lapse drops out exactly. The cosmological constant $\Lambda = I$ is
generated by the threshold conditions $p(d) = c_1 = \tfrac{1}{2}\ln\pi$ and $p(d) = c_2 = \sqrt{\pi}$, which are
determined by the constant part of $p(d)$---the same $\sqrt{\pi}$ factor---with no dependence on $R(d)$
at the threshold dimensions. The two sectors therefore draw on orthogonal factors of $N(d)$:
the multiplicative $\sqrt{\pi}$ of the lapse cancels from $\alpha$ and generates $\Lambda$; $R(d)$ determines $\alpha$ and
does not enter the threshold conditions. This is why $\Lambda$ does not run while couplings run.
\end{theorem}

\begin{proof}
$\alpha(d) = N(d)^2/(4\pi) = \pi R(d)^2/(4\pi) = R(d)^2/4$. The $\sqrt{\pi}$ factor cancels entirely;
$\alpha$ depends only on $R(d)$. The thresholds $d_1 = 19$ and $d_2 = 217$ are determined by
$p(d) = c_1 = \tfrac{1}{2}\ln\pi$ and $p(d) = c_2 = \sqrt{\pi}$, where $c_1$, $c_2$ are generated from $\sqrt{\pi}$ alone ([1],
Theorem 6.5). $\Lambda = I$ is a function of $\Omega_{d_1}$ and $\Omega_{d_2}$, which are values of the sphere-area
sequence at the threshold dimensions; these values are determined by $\sqrt{\pi}$ through the
threshold conditions. At those dimensions, $R(d_1)$ and $R(d_2)$ enter only the $d$-dependent
decay rate, not the threshold conditions themselves. Therefore $\Lambda$ contains no dependence
on $R(d)$ through the threshold structure.
\end{proof}

\section{Matter from Extra Dimensions}\label{sec:matter-from-extra-dimensions}

\subsection{The Kaluza--Klein perspective}

The cascade produces a 217-dimensional geometry. The 4D observer has integrated out
213 directions via the slicing recurrence. The purpose of this section is to identify the
effective stress-energy tensor $T^{\mu\nu}_{\text{eff}}$ that enters the Einstein equation projection identity
(Section 8): Lovelock's theorem forces $G^{\mu\nu} + \Lambda g^{\mu\nu} = 8\pi G\,T^{\mu\nu}$, and $T^{\mu\nu}_{\text{eff}}$ is the cascade's
geometric account of what fills the right-hand side.

\begin{remark}[Induced matter via Gauss--Codazzi]
Let $G^{\mu\nu}_{(217)}$ be the Einstein tensor of
the cascade's 217-dimensional geometry and $G^{\mu\nu}_{(4)}$ be the 4D projected Einstein tensor. The
difference $8\pi G\,T^{\mu\nu}_{\text{eff}} = G^{\mu\nu}_{(4)} - [G^{\mu\nu}_{(217)}]_{\text{4D}}$ defines the effective stress-energy seen by the 4D
observer, following from the Gauss--Codazzi equations for the embedding $M^4 \subset M^{217}$. This
is a structural identification, not a dynamical calculation. Section 14 establishes exact
results for the Lorentzian cascade metric: $K_{\mu\nu} = 0$ at every equatorial embedding in the
chain $S^3 \hookrightarrow S^4 \hookrightarrow \cdots \hookrightarrow S^{216}$, and the single-step cascade geometry contributes $w = 1/3$
(radiation) in cascade time. The cascade's matter fraction $\Omega_m$ is determined by the Bott
partition of cascade layers, not by the cascade metric geometry; this is a topological rather
than geometric quantity.
\end{remark}

\section{Self-Consistency: Einstein Equation as Projection Identity}\label{sec:self-consistency-einstein-equation-as-pr}

The 4D observer has: a metric $g_{\mu\nu}$ (Section 4); $\Lambda = I$ (Section 5); $T^{\mu\nu}_{\text{eff}}$ from the extra
dimensions (Section 7); and Lovelock's theorem (Section 3). Lovelock then requires
$G^{\mu\nu} + \Lambda g^{\mu\nu} = 8\pi G\,T^{\mu\nu}$. This is a projection identity, not a postulate.

\begin{theorem}[Einstein equation as projection identity]
The Gauss--Codazzi equations
for $M^4 \subset M^{217}$, combined with Lovelock's theorem at $d = 4$, yield the Einstein equation
with $\Lambda = I$ and $T^{\mu\nu} = T^{\mu\nu}_{\text{eff}}$ as an identity. Every term is geometric output of the cascade;
no gravitational postulate is added.
\end{theorem}

\section{Deriving \texorpdfstring{$d = 4$}{d = 4}}\label{sec:d-equals-4}

\begin{definition}[Cascade-compatible dimension]
A spacetime dimension $d$ is cascade-compatible if it satisfies:
\begin{itemize}
\item[(C1)] The minimal spinor representation of $\text{Spin}(1, d - 1)$ is irreducibly complex (not
realisable over $\mathbb{R}$).
\item[(C2)] The Einstein equation with cosmological constant is the unique divergence-free symmetric rank-2 tensor metric equation (Lovelock uniqueness, Theorem 3.1).
\end{itemize}
\end{definition}

\subsection{Why (C1) is forced by the cascade}

Condition (C1) is a statement about the spinor representations of
$\mathrm{Spin}(1,d-1)$. For it to constrain $d$ cascade-internally, the
cascade's complex Hilbert space from Paper~II must be identified with a
representation of $\mathrm{Spin}(1,d-1)$. That identification is not
an immediate consequence of Paper~II's construction --- Paper~II builds
the Hilbert space from orthogonality and the slicing recurrence; it does
not equip it with an $\mathrm{Spin}(1,d-1)$ action. The identification
is a downstream step in Paper~III, made explicit in
Lemma~\ref{lem:cascade-spinor-id} below.

From Theorem~6.1 of~\cite{part2}: the orthogonality axiom forces the
precession angle $\alpha = \pi/2$, producing the complex structure
$J: e_1 \mapsto e_2$, $e_2 \mapsto -e_1$ with $J^2 = -\text{Id}$ on the
cascade's Hilbert space $\mathcal{H}$.

\begin{lemma}[Cascade Hilbert space carries the minimal complex Lorentz representation]\label{lem:cascade-spinor-id}
Given the following inputs:
\begin{itemize}
\item[(i)] \emph{Physical identification hypothesis} (Definition~2.1):
the cascade's 4D projection is identified with the observer's physics.
\item[(ii)] \emph{Empirical Lorentzian spacetime}: the observer's
spacetime is Lorentzian with Lorentz group $\mathrm{Spin}(1,d-1)$. This
is empirical input, not derived from the cascade (see Paper~I's ``What
This Paper Does Not Do''; $d$ itself is empirical at this stage of the
cascade and will be cascade-internally selected to $d=4$ in
Theorem~\ref{thm:d4} below).
\item[(iii)] \emph{Relativistic quantum-mechanical covariance}:
physical states must carry a representation of the spacetime Lorentz
group (Wigner's requirement, imported as a standard result).
\item[(iv)] \emph{Cascade complex structure} (\cite{part2},
Theorem~6.4): $\mathcal{H}$ carries an operator $J$ with
$J^2 = -\mathrm{Id}$ that commutes with the cascade's slicing dynamics.
\item[(v)] \emph{Austerity} (Prelude~\cite{prelude},
Principle~2.2, clause~(i): parameter economy). Austerity is not a
separate input: in the Prelude it is derived in place from
Definition~2.1 ($0\neq 1$) as the operational content of its
sole-input status.
\end{itemize}
Then $\mathcal{H}$ is identified with an \emph{irreducibly complex}
representation of $\mathrm{Spin}(1,d-1)$. Specifically, the minimal
such representation is the minimal spinor representation of
$\mathrm{Spin}(1,d-1)$ exactly when the latter is irreducibly complex
--- which, by the Clifford classification, occurs at
$d\bmod 8 \in \{4,5,6\}$ in the first Bott period (and the analogous
windows $\{12,13,14\}$, $\{20,21,22\}$, \ldots\ in subsequent periods).
\end{lemma}

\begin{proof}
\emph{Step~1 (the representation exists).}
By (i), the cascade's Hilbert space is the arena of physical quantum
states at the observer's frame. By (ii), that frame's Lorentz group is
$\mathrm{Spin}(1,d-1)$. By (iii), the states must transform under this
group: each Lorentz element $\Lambda \in \mathrm{Spin}(1,d-1)$ acts on
$\mathcal{H}$ by a well-defined unitary operator $U(\Lambda)$, and the
collection $\{U(\Lambda)\}$ is a representation. Without such a
representation, two observers related by a Lorentz transformation
would ascribe incompatible quantum states to the same physical
configuration, violating (i).

\emph{Step~2 (the representation is complex-admissible).}
By (iv), $\mathcal{H}$ carries $J$ with $J^2 = -\mathrm{Id}$. For $J$
to be a well-defined operator compatible with the representation (so
that the Lorentz action is complex-linear with respect to $J$), $J$
must commute with $U(\Lambda)$ for all $\Lambda$. This places $J$ in the
commutant of $U$. Over $\mathbb{R}$, Schur's lemma for an
irreducibly-real representation forces the commutant to be $\mathbb{R}$,
allowing only $J = c\cdot\mathrm{Id}$ with $c^2 \geq 0$: contradicting
$J^2 = -\mathrm{Id}$. Hence $U$ cannot be irreducibly real on
$\mathcal{H}$. Either $U$ is irreducibly complex (viewed over
$\mathbb{C}$), or $U$ is a reducible direct sum of complex-admissible
subrepresentations.

\emph{Step~3 (minimality forces irreducibility).}
By (v), austerity clause~(i) forbids unexplained multiplicity
factors. A reducible representation $U = U^{(1)} \oplus \cdots \oplus U^{(n)}$
with each $U^{(k)}$ irreducible introduces $n \geq 2$ as an unexplained
integer. The minimal complex-admissible representation is therefore
irreducible. Combined with Step~2, $U$ is \emph{irreducibly complex}.

\emph{Step~4 (Clifford classification).}
The minimal irreducible complex representation of $\mathrm{Spin}(1,d-1)$
is the spinor representation exactly when the Clifford algebra
$\mathrm{Cl}(1,d-1)$ has complex type --- i.e.\ is of the form
$M_n(\mathbb{C})$, possibly tensored with a real matrix factor, so that
its simple modules are complex vector spaces of complex dimension~$n$.
By the Clifford classification in Lorentzian signature $(1,d-1)$
(Lounesto~\cite{lounesto}; see the table in
Section~\ref{sec:clifford-table} below), this occurs precisely at
$d\bmod 8 \in \{4,5,6\}$ in the first Bott period. At other values of
$d$ the Clifford algebra is of real (Majorana) or real-doubled type,
and the minimal spinor representation is Majorana-real; hosting $J$ on
such a representation requires doubling, which Step~3 excludes.
\end{proof}

Lemma~\ref{lem:cascade-spinor-id} is the cascade-internal derivation of
condition (C1): it supplies explicitly the identification step
(``$\mathcal{H}$ is a spinor representation of $\mathrm{Spin}(1,d-1)$'')
that was previously implicit in the phrase ``for the cascade's state
space to serve as the spinor representation space.''

\begin{remark}[Inputs are named, not derived from the cascade]
\label{rem:sp-19-inputs}
The lemma's five inputs are of three kinds. Items (i), (ii), (iv) are
either physical hypotheses declared by the series (Definition~2.1),
empirical facts (the Lorentzian character of observed spacetime), or
prior cascade theorems (Paper~II). Item (iii) is imported from standard
relativistic quantum mechanics (Wigner's theorem on ray representations
of spacetime symmetries); it is not cascade-internal and is used as a
background fact. Item (v) is the austerity principle of the
Prelude (Principle~2.2, derived there from Definition~2.1 as the
operational content of its sole-input status --- i.e.\ as a derivation
rule of the series, not a separate axiom).

If any of (i)--(v) is disputed, condition (C1) loses its force as a
cascade-internal constraint on $d$. In that case Theorem~\ref{thm:d4}
below still yields $d=4$ via the two independent cascade-internal
selections that do not depend on the spinor-representation
identification:
\begin{itemize}
\item[(C2)] Lovelock uniqueness (Theorem~3.1): the Einstein equation
is the unique divergence-free symmetric rank-2 tensor equation only at
$d=4$.
\item[Corollary~\ref{cor:ricci-flat}]: the Lorentzian cascade scale
factor gives Ricci scalar $R^{(n)} = (n-1)(n-4)/a^4$, vanishing uniquely
at $n=4$.
\end{itemize}
Both are cascade-internal and independent of the Lorentz-representation
identification. The cascade's $d=4$ derivation therefore has three
independent selections; (C1) is the one that depends on
Lemma~\ref{lem:cascade-spinor-id}'s inputs (i)--(v).
\end{remark}

\subsection{The Clifford algebra classification}\label{sec:clifford-table}

The type of the minimal spinor of $\text{Spin}(1, d - 1)$ is determined by the Clifford algebra
$\text{Cl}(1, d - 1)$, following Bott periodicity with period 8.

\begin{center}
\begin{tabular}{lllll}
\hline
$d$ & $\text{Cl}(1, d - 1)$ structure & Min.\ spinor & Type & (C1) \\
\hline
2 & $M_2(\mathbb{R})$ & 2 real & Majorana & No \\
3 & $M_2(\mathbb{R}) \oplus M_2(\mathbb{R})$ & 2 real & Majorana & No \\
4 & $M_2(\mathbb{C}) \otimes_\mathbb{R} M_2(\mathbb{R})$ & 2 complex & Weyl & Yes \\
5 & $M_4(\mathbb{C})$ & 4 complex & Dirac & Yes \\
6 & $M_4(\mathbb{C}) \oplus M_4(\mathbb{C})$ & 4 complex & Weyl & Yes \\
7 & $M_8(\mathbb{R})$ & 8 real & Majorana & No \\
8 & $M_{16}(\mathbb{R})$ & 8 real & Majorana & No \\
\hline
\end{tabular}
\end{center}

Data from Lounesto [6], Lorentzian signature $(1, d - 1)$. Condition (C1) holds for
$d \in \{4, 5, 6\}$ in the first Bott period, then $\{10, 11, 12\}$, etc.

\subsection{The main theorem}

\begin{theorem}[Unique cascade-compatible dimension]\label{thm:d4}
$d = 4$ is the unique dimension
satisfying both (C1) and (C2).
\end{theorem}

\begin{proof}
From the Clifford classification: (C1) holds for $d \in \{4, 5, 6, 10, 11, 12, \ldots\}$. From
Theorem 3.1: (C2) holds if and only if $d = 4$. Intersection: $\{4, 5, 6, 10, \ldots\} \cap \{4\} = \{4\}$.
\end{proof}

\begin{corollary}[Third characterisation of $d = 4$]\label{cor:ricci-flat}
Under the cascade-motivated Wick rotation $x = it$
(Remark~\ref{rem:wick-rotation-cascade}), the cascade's Euclidean
cross-section radius $\sqrt{1-x^2}$ maps to the Lorentzian scale
factor $a(t) = \sqrt{1 - t^2}$; this gives $\dot{a} = -t/a$ and
$\ddot{a} = -1/a^3$. The Ricci scalar of the $n$-dimensional FRW
metric with $k = 1$ and this scale factor is:
\[
R^{(n)} = \frac{(n - 1)(n - 4)}{a^4}.
\]
This vanishes if and only if $n = 4$. The Lorentzian cascade metric is therefore Ricci-flat
uniquely in four spacetime dimensions, providing a third independent characterisation of
$d = 4$ complementing Lovelock uniqueness (C2) and complex spinor compatibility (C1).
\end{corollary}

\begin{remark}[Derivation status of the Lorentzian scale factor]\label{rem:sp20-status}
The identification of $a(t) = \sqrt{1 - t^2}$ as the Lorentzian scale
factor combines two already-derived ingredients:
\begin{enumerate}
\item \emph{Euclidean cross-section radius (Paper~0, \S~The
Recurrence).} The cascade's slicing of $B^{n}$ perpendicular to one
axis gives cross-sections of radius $\sqrt{1-x^2}$ at height
$x\in[-1,1]$. This is a purely Euclidean geometric identity about
the unit ball.
\item \emph{Cascade-motivated Wick rotation
(Remark~\ref{rem:wick-rotation-cascade}).} The cascade's Euclidean
slicing $+dx^2 + (1-x^2)\,d\sigma_{n-1}^2$ maps to Lorentzian
$-dt^2 + (1-t^2)\,d\sigma_{n-1}^2$ under $x \leftrightarrow it$.
The rotation is not mathematically imposed (as in QFT); it is the
identification step of the physical identification hypothesis,
motivated by the cascade's forced precession producing an
oscillatory propagator in $x$, which is Lorentzian signature
(Theorem~\ref{thm:lorentzian}).
\end{enumerate}
Under (1)+(2), the Lorentzian scale factor $a(t) = \sqrt{1-t^2}$ is
cascade-derived, not imported: the Euclidean identity supplies the
functional form, and the cascade-motivated Wick rotation supplies
the Lorentzian interpretation. The boundedness of $a(t)$ on
$t\in[-1,1]$ (vanishing at $t = \pm 1$) is a structural feature of
the cascade's compact-domain integration, not an ad hoc cosmological
assumption. Paper~III \S14 (the compactification supplement) develops
the same identification at the level of the full Ricci and Einstein
tensors; this corollary records the Ricci-scalar consequence at
$n = 4$.
\end{remark}

\begin{proof}
For $n$-dimensional FRW with $k = 1$: $R_{tt} = -(n - 1)\ddot{a}/a = (n - 1)/a^4$ and
$R_{ij} = [(n - 2)(k + \dot{a}^2)/a^2 + \ddot{a}/a]\,g_{ij} = (n - 3)/a^4 \cdot g_{ij}$. The Ricci scalar: $R^{(n)} = g^{tt}R_{tt} + g^{ij}R_{ij} =
-(n - 1)/a^4 + (n - 1)(n - 3)/a^4 = (n - 1)(n - 4)/a^4$. At $n = 4$: $R^{(4)} = 3 \cdot 0/a^4 = 0$. For
$n \neq 4$: $(n - 1)(n - 4) \neq 0$.
\end{proof}

\begin{remark}
Without (C1): $d = 4$ is selected by Lovelock alone, but the quantum state
space may be incompatible with $J^2 = -\text{Id}$. Without (C2): dimensions 5, 6, 10,\ldots\ have
complex spinors but non-unique gravity with free coupling constants. Only $d = 4$ is
simultaneously quantum-compatible, gravitationally unique, and Ricci-flat under the cascade
scale factor.
\end{remark}

\begin{remark}[Division algebra interpretation]
The cascade forces the quantum amplitude
algebra to be $\mathbb{C}$ (real dimension 2). The associative normed division algebras over $\mathbb{R}$ are $\mathbb{R}$,
$\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$ (Hurwitz theorem [7]; $\mathbb{O}$ is non-associative and excluded from quantum mechanics
by associativity of sequential measurements). The minimal associative algebra over $\mathbb{C}$ is $\mathbb{H}$
(quaternions), of real dimension 4. The spacetime dimension equals the real dimension of
the next associative division algebra: $d = \dim_\mathbb{R}\,\mathbb{H} = 4$.
\end{remark}

\section{Lorentzian Signature from the Propagator}\label{sec:lorentzian-signature-from-the-propagator}

The cascade geometry is Euclidean: $ds^2_E = dx^2 + (1 - x^2)\,d\sigma_{d-1}^2$ with positive-definite
metric. The cascade propagator is $K(x) = |K|\,e^{-i\lambda x}$ with $\lambda > 0$---already oscillatory for
real $x$. These are two descriptions of the same object under the physical identification
hypothesis. The Lorentzian signature is the unique metric sign consistent with both
descriptions simultaneously.

\subsection{The propagator characterises the signature}

\begin{lemma}[Spectral gap is positive]\label{lem:spectral-gap}
The cascade spectral gap satisfies $\lambda_\infty =
\tfrac{1}{4}\psi^{(1)}(\sigma/2) > 0$ for all $\sigma > 0$, where $\psi^{(1)}$ is the trigamma function. This follows from $p'(d) =
\tfrac{1}{4}\psi^{(1)}((d + 1)/2) > 0$ (strict monotonicity of $p$).
\end{lemma}

From Theorems 6.1 and 7.1 of [2]: the forced precession $\alpha = \pi/2$ gives the cascade
propagator:
\[
K(t) = |K| \cdot e^{-iHt}, \qquad H = \lambda_\infty > 0,
\]
where $t = x$ is the physical time identified with the slicing coordinate.

\begin{theorem}[Lorentzian signature from the cascade]\label{thm:lorentzian}
The spacetime metric of the 4D cascade observer has signature $(-,+,+,+)$.
\end{theorem}

\begin{proof}
The cascade is Euclidean at the metric level (Section~4.1:
$ds^2_E = dx^2 + (1-x^2)\,d\sigma_{d-1}^2$, positive-definite). The
observer's spacetime is Lorentzian (Section~4.4:
$ds^2_L = -dt^2 + a(t)^2\,d\sigma_3^2$). Under the physical
identification hypothesis (Definition~\ref{def:hyp}), the cascade's
slicing coordinate $x$ maps to the observer's proper time coordinate
$t$. The map between the two metrics is the Wick rotation
$x \leftrightarrow it$, under which $+dx^2 \leftrightarrow -dt^2$; the
proof is the derivation of why the cascade's structure forces exactly
this rotation rather than any other identification of $x$ with an
observer coordinate.

\textbf{Step 1: The cascade's propagator is oscillatory in real $x$.}
From Paper~II~\cite{part2} Theorems~6.1 and~7.1, the forced precession
$\alpha = \pi/2$ produces the propagator
$K(x) = |K|\,e^{-i\lambda_\infty x}$ with $\lambda_\infty > 0$
(Lemma~\ref{lem:spectral-gap} above). This is oscillatory for $x$ real.

\textbf{Step 2: Oscillatory evolution identifies the time coordinate as Lorentzian.}
In a spacetime with Lorentzian signature $(-,+,+,+)$ and a self-adjoint
Hamiltonian $H > 0$, quantum states evolve as
$|\psi(t)\rangle = e^{-iHt}|\psi(0)\rangle$ --- oscillatory in real $t$.
In a Euclidean spacetime (positive-definite metric), the analogous
evolution is $e^{-\tau H}$ (exponential decay in imaginary time $\tau$;
this is the thermal partition-function regime, not a unitary time
evolution). The cascade's oscillatory $e^{-i\lambda_\infty x}$ in real
$x$ therefore fits the Lorentzian regime, and identification of $x$
with the observer's proper time requires the Lorentzian sign
convention $g_{tt} < 0$.

\textbf{Step 3: The identification is Wick rotation.}
The cascade's mathematical content is encoded in the Euclidean slicing
geometry. The map from Euclidean $x$ to Lorentzian $t$ is
$x = it$ (equivalently $t = -ix$), under which the cascade's
$+dx^2$ becomes $(idt)^2 = -dt^2$. This is the standard Wick rotation
operation. What distinguishes the cascade's Wick rotation from the
standard QFT one is its motivation (see
Remark~\ref{rem:wick-rotation-cascade} below): in QFT, Wick rotation
is imposed as a mathematical trick to render Lorentzian path integrals
convergent; in the cascade, the rotation is the identification step of
the physical identification hypothesis, cascade-motivated by the
forced precession producing an oscillatory propagator in real $x$.
Under this rotation, the cascade's Euclidean $+dx^2$ maps to the
observer's Lorentzian $-dt^2$.

\textbf{Temporal component.} Steps 1--3 give $g_{tt} = -1$ in the
observer's proper-time coordinate.

\textbf{Spatial components.} The cascade's state space is a complex
Hilbert space $\mathcal{H}$ (Paper~II~\cite{part2}). The inner product
is positive-definite: $\langle\psi|\psi\rangle > 0$ for all
$|\psi\rangle \neq 0$. The induced inner product on equal-time
surfaces requires a positive-definite spatial volume element
$\sqrt{\det g^{(3)}} > 0$, which holds iff $g_{ij} > 0$. The spatial
signature is therefore $(+,+,+)$, and the full signature is
$(-,+,+,+)$.
\end{proof}

\begin{remark}[Wick rotation, cascade-motivated]\label{rem:wick-rotation-cascade}
The cascade's Euclidean-to-Lorentzian map is Wick rotation
($x \leftrightarrow it$, $+dx^2 \leftrightarrow -dt^2$). Prior drafts of
this series claimed the cascade avoided Wick rotation ``the Euclidean
geometry generates Lorentzian physics through the dynamics, not
through analytic continuation of the metric''). That framing is not
correct: the operation performed is exactly Wick rotation (a sign flip
on the time-direction metric component). What is correct is that
Wick rotation in the cascade is \emph{cascade-motivated} rather than
\emph{mathematically imposed}.

\medskip\noindent\textbf{The distinction.}
In standard quantum field theory, Wick rotation $t \to -i\tau$ is
often introduced as a mathematical trick: Lorentzian path integrals
$\int \mathcal{D}\phi\,e^{iS_L}$ are oscillatory and ill-defined;
rotating to $\tau$ gives $\int\mathcal{D}\phi\,e^{-S_E}$ with exponentially
damped weights, which converges. The Euclidean side is computationally
well-defined; the Lorentzian side is physical; the two are related by
analytic continuation with no physical content attached to the
rotation itself.

In the cascade, the situation is reversed. The cascade is \emph{built}
Euclidean (positive-definite slicing metric); this is the starting
mathematical structure, not a computational continuation of something
Lorentzian. The Lorentzian observer metric arises via the physical
identification hypothesis, which maps cascade structure to observer
physics. The Wick rotation is the specific map that preserves the
cascade's oscillatory propagator (from the forced precession) as
oscillatory unitary time evolution on the observer's side. No other
identification of $x$ with an observer coordinate would do this: the
oscillatory character of $e^{-i\lambda_\infty x}$ in real $x$ requires
a Lorentzian sign on the time-direction metric, which requires Wick
rotation relative to the Euclidean cascade.

\medskip\noindent\textbf{Consequences.}
This clarifies four downstream readings:
\begin{itemize}[nosep]
\item The cascade does not \emph{avoid} Wick rotation; it \emph{motivates} it. Claims to the contrary in earlier drafts (and in this paper's abstract before revision, Paper~V, Section~\xref{part5}{sec:hubble-constant}'s retracted framing, etc.) are retracted in favour of the honest statement.
\item The ``Euclidean geometry'' of the cascade is not merely a computational device; it is the starting mathematical structure. The Lorentzian observer metric is derived by applying the physical identification hypothesis, which includes the Wick rotation as its time-coordinate map.
\item Section~\ref{sec:supplement-asymptotic}'s Gauss--Codazzi analysis on the Lorentzian cascade metric uses the Wick-rotated form $g^L = -dt^2 + (1-t^2)\,d\sigma_{n-1}^2$; the identification of this with the cascade's Euclidean slicing geometry through $x = it$ is the content of Wick rotation, not a separate construction.
\item The cascade-internal motivation for the Wick rotation is sharpened by~\cite{part2} Section~7.1 (dimensional-collapse reading): each slicing operation \emph{is} one quantum of time, the slicing direction is the rate at which the cascade resolves $B^\infty$, and the Wick rotation $x \leftrightarrow it$ bridges the structural reading (Euclidean slicing as static cascade decomposition) and the dynamic reading (Lorentzian time as observed dimensional-collapse rate). The forced precession's oscillatory propagator $e^{-i\lambda_\infty x}$ is the cascade's quarter-turn per $d$-collapse expressed in the Euclidean coordinate; Wick rotation puts it in the observer's Lorentzian time.
\end{itemize}
\end{remark}

\begin{remark}[Three consequences of $\psi^{(1)} > 0$]
The strict positivity $\psi^{(1)}(x) > 0$ simultaneously establishes: (a) unique natural zero of $p(d)$ in [1]: requires $p'(d) > 0$; (b) positive
spectral gap $\lambda_\infty > 0$; (c) Lorentzian signature: $\lambda_\infty > 0$ makes the propagator oscillatory,
requiring $g_{tt} < 0$ (Theorem 10.2).
\end{remark}

\section{Scale Factors: \texorpdfstring{$G$ and $\hbar$}{G and hbar}}\label{sec:scale-factors-G-and-hbar}

The cascade produces dimensionless quantities. Physical gravity requires $G$ (Newton) and
$\hbar$ (Planck). Both enter as unit-matching constants when geometric rates are identified
with physical observables:

\begin{itemize}
\item $G$: proportionality between 4D curvature and the effective stress-energy from the
extra dimensions. Set by the embedding geometry of $M^4 \subset M^{217}$.
\item $\hbar = (\pi/2)/\Delta t$: the ratio of the forced precession angle to the physical time increment
(Section 9 of [2]). The angle $\pi/2$ is derived; $\Delta t$ is not.
\end{itemize}

\section{No Gravitons: Metric Degrees of Freedom without Second Quantisation}\label{sec:no-gravitons}

The cascade does not produce gravitons. The 4D metric $g_{\mu\nu}$ is a property
of the cascade state $|\Psi\rangle \in S^{d-1}$ (Paper~II\,=\,III, Theorem~6.1):
different states produce different metrics, superpositions of states produce
superpositions of metrics, and the Born rule assigns probabilities to metric
outcomes. The metric is never promoted to an operator. There is no spin-2
particle mediating gravitational interactions.

In the standard framework, a massless spin-2 field in $d$ spacetime dimensions
has $d(d-3)/2$ independent components. At $d = 4$ this gives 2---the plus and
cross polarisations of linearised GR. In the cascade, these are not particle
states but the two independent directions in which the 4D metric can be
perturbed while satisfying the linearised Einstein equation. The distinction
matters: perturbation modes describe the geometry's response to small
disturbances; they do not require quantisation into particles.

The higher-dimensional metric has $217 \times 214/2 = 23{,}219$ independent
components. From the 4D observer's perspective, these appear as a
Kaluza--Klein tower of massive fields on adjacent shells, with mass scale
set by the compactification radii $R_{\text{eff}}(d) = 1/\sqrt{d+3}$
(Paper~I, Theorem~\xref{part0}{thm:reff}). These are geometric degrees of freedom of the
cascade---effective fields in the 4D projection---not fundamental particles.
The cascade predicts no graviton will be detected, for the same structural
reason it predicts no supersymmetric partners, no extra gauge bosons, and
no dark matter particles: the topology has no mechanism to produce them.

\section{The Complete Derivation}\label{sec:the-complete-derivation}

The full chain from the orthogonality axiom to the metric of general relativity:

\begin{center}
\small
\begin{tabular}{lll}
\hline
Step & Result & Source \\
\hline
Orthogonality $\Rightarrow \sqrt{\pi}$ & Cascade constant & [1], Thm 3.1 \\
$\sqrt{\pi} \Rightarrow I \approx 10^{-120}$ & Cascade invariant & [1], Thm 9.2 \\
Orthogonality $\Rightarrow J^2 = -\text{Id}$ & Complex structure & [2], Thm 6.4 \\
$J^2 = -\text{Id} \Rightarrow$ complex spinors (C1) & Schur + Clifford & Section 9 \\
(C1)$\cap$Lovelock (C2)$\Rightarrow d = 4$ & Dimension theorem & Thm 9.3 \\
$R^{(n)} = (n-1)(n-4)/a^4 \Rightarrow R^{(4)} = 0$ & Third char.\ of $d = 4$ & Cor 9.4 \\
$d = 4 \Rightarrow$ FRW metric + $\Lambda = I$ & Cascade geometry & \S4, \S5 \\
Physical identification hypothesis & Cascade geom = physics & Def 2.1 \\
Hypothesis + $\lambda_\infty > 0 \Rightarrow g_{tt} < 0$ & Propagator & Thm 10.2 \\
$d = 4$ + $(-,+,+,+) \Rightarrow$ Einstein eq. & Lovelock & Section 8 \\
Fixed $\Lambda = I \Rightarrow w = -1$ & Definition & Section 6 \\
GB corrections vanish & Two mechanisms & Section 6 \\
$N = \sqrt{\pi}\,R \Rightarrow$ orthogonal sectors & Couplings vs vacuum & Thm 6.4 \\
Extra dims $\Rightarrow T^{\mu\nu}$ & Gauss--Codazzi & Section 7 \\
\hline
\end{tabular}
\end{center}

\subsection{Full result table}

\begin{center}
\small
\resizebox{\textwidth}{!}{%
\begin{tabular}{lll}
\hline
Physical result & Cascade provides & Classical theorem \\
\hline
Complex QM, Born rule, $\hbar$ & Complex structure, propagator & Sphere geometry, [2] \\
Cosmological constant $\Lambda = I$ & Cascade invariant $I$ & Fixed, not free \\
$d = 4$ spacetime dimensions & Complex spinors + gravity & Clifford, Lovelock \\
$R^{(4)} = 0$: Ricci-flat cascade metric & Lorentzian scale factor & GR, Corollary 9.4 \\
Lorentzian signature $(-,+,+,+)$ & Propagator $e^{-iHt}$, $H > 0$ & Path integral identity \\
Einstein equation & Metric + $\Lambda = I$ & Lovelock \\
FRW cosmology & Cascade foliation & Geometry \\
$w = -1$ dark energy & Fixed $\Lambda = I$; GB vanishes & Section 6 \\
$\Lambda \approx 10^{-120}M_{\text{Pl}}^4$ & Sphere-area hierarchy & [1], Thm 10.1 \\
Orthogonal coupling and vacuum sectors & $N = \sqrt{\pi}\,R$ factorisation & Thm 6.4 \\
$K_{\mu\nu} = 0$: totally geodesic embedding & Lorentzian cascade metric & Section 14 \\
\hline
\end{tabular}%
}
\end{center}

\section{Supplement: Asymptotic Compactification and the Cascade Metric}\label{sec:supplement-asymptotic}

This section integrates the compactification results of [1], Section 4 into the gravitational
framework, and presents the exact Lorentzian Gauss--Codazzi analysis.

\subsection{The compactification radius of each dimension}

\begin{theorem}[Compactification radius; {[1]}, Theorem 4.2]
At each slicing step $d$, the
integrated-out direction retains an effective radius
\[
R_{\text{eff}}(d) = \frac{1}{\sqrt{d + 3}},
\]
derived exactly from $\langle x^2 \rangle = B(3/2, d/2 + 1)/B(1/2, d/2 + 1) = 1/(d + 3)$.
\end{theorem}

The compactification is asymptotic: $R_{\text{eff}} \to 0$ as $d \to \infty$, but $R_{\text{eff}} \neq 0$ at any finite $d$.
There is no sharp boundary, no horizon, no topology change---only exponential suppression.

\begin{center}
\begin{tabular}{cccc}
\hline
$d$ & $R_{\text{eff}}$ & $N(d)$ & Physical role \\
\hline
4 & 0.37796 & 1.17810 & Observer dimension \\
5 & 0.35355 & 1.06667 & $S^3$ boundary layer \\
7 & 0.31623 & 0.91429 & Volume maximum $d_0$ \\
12 & 0.25820 & 0.70870 & SU(3) layer \\
13 & 0.25000 & 0.68198 & SU(2) breaking \\
19 & 0.21320 & 0.56755 & First threshold $d_1$ \\
217 & 0.06742 & 0.16997 & Second threshold $d_2$ \\
\hline
\end{tabular}
\end{center}

\begin{remark}[KK mass scale]
The KK mass scale at dimension $d$ is $M_{\text{KK}}(d) \sim
1/R_{\text{eff}}(d) = \sqrt{d + 3}$ in cascade units. No free parameter adjusts the mass scale.
\end{remark}

\subsection{Boundary dominance and the primacy of sphere areas}

\begin{theorem}[Boundary dominance; {[1]}, Theorem 4.4]
$\Omega_{d-1}/V_d = d$ for all $d \geq 1$.
\end{theorem}

At $d = 4$: $\Omega_3/V_4 = 4$, so the boundary $S^3$ carries $4/5 = 80\%$ of the content of $B^4$. At
$d = 217$: essentially all content is on the boundary. The cascade's content is its boundaries,
increasingly so at each step.

\subsection{The observer on \texorpdfstring{$S^3$}{S\^{}3}}

The 4D observer exists on the boundary $S^3$ of the $d = 5$ layer. Theorem 14.3 shows this
boundary carries $5/6 \approx 83\%$ of the $d = 5$ content. The observer's physics is a boundary
theory, not because holography is assumed, but because the boundary dominates the
volume at every cascade layer.

The $S^3$ horizon of a $d = 5$ Schwarzschild black hole has topology $S^3$, matching the 3
spatial dimensions. The cascade's $d = 5$ layer provides this horizon. The de Sitter horizon
area $A = 12\pi/\Lambda \sim 10^{120}$ in Planck units is the cascade hierarchy inverted: $\Omega_7/\Omega_{217} \approx 10^{121}$.

\begin{remark}[The cosmological constant as inverse boundary area]
With boundary
dominance, the identification $\Lambda = I$ acquires a geometric interpretation: $\Lambda$ measures the
inverse boundary area of the cascade's distinguished layers. The smallness of $\Lambda$ reflects the
largeness of the boundary---a high-capacity boundary ($S^3$ with $\Omega_3 = 2\pi^2 \approx 19.7$) encodes
an enormous cascade (213 compactified directions), giving $\Lambda = I \approx 10^{-120}$.
\end{remark}

\subsection{The Lorentzian Gauss--Codazzi analysis}

We install the Lorentzian cascade metric directly from
Theorem~\ref{thm:lorentzian}: under the Wick-rotation identification
$x = it$ (Remark~\ref{rem:wick-rotation-cascade}), the cascade's
Euclidean slicing metric $dx^2 + (1-x^2)\,d\sigma_{n-1}^2$ maps to the
Lorentzian form below. The $n$-dimensional Lorentzian cascade metric is:
\[
g^L = -dt^2 + (1 - t^2)\,d\sigma_{n-1}^2, \qquad a(t) = \sqrt{1 - t^2}.
\]
With $\dot{a} = -t/a$ and $\ddot{a} = -1/a^3$, two exact identities hold:
\[
\frac{k + \dot{a}^2}{a^2} = \frac{1}{a^4}, \qquad \frac{\ddot{a}}{a} = -\frac{1}{a^4}.
\]

\begin{theorem}[Lorentzian cascade Ricci tensor]
For the $n$-dimensional Lorentzian cascade metric:
\[
R^{(n)}_{tt} = \frac{n - 1}{a^4}, \qquad R^{(n)}_{ij} = \frac{n - 3}{a^4}\,g_{ij}, \qquad R^{(n)} = \frac{(n - 1)(n - 4)}{a^4},
\]
\[
G^{(n)}_{tt} = \frac{(n - 1)(n - 2)}{2a^4}, \qquad G^{(n)}_{NN} = -\frac{(n - 2)(n - 5)}{2a^4}.
\]
\end{theorem}

\begin{proof}
Direct computation using the FRW Ricci tensor formulae with $k = 1$, $n - 1$ spatial
dimensions, and the cascade identities $\dot{a}^2 + k = 1/a^2$, $\ddot{a}/a = -1/a^4$: $R_{tt} = -(n - 1)\ddot{a}/a =
(n - 1)/a^4$; $R_{ij} = [(n - 2)(\dot{a}^2 + k)/a^2 + \ddot{a}/a]\,g_{ij} = [(n - 2)/a^4 - 1/a^4]\,g_{ij} = (n - 3)/a^4 \cdot g_{ij}$. The
Ricci scalar: $R^{(n)} = g^{tt}R_{tt} + g^{ij}R_{ij} = -(n - 1)/a^4 + (n - 1)(n - 3)/a^4 = (n - 1)(n - 4)/a^4$.
For the normal-normal Einstein component at the equatorial embedding $\psi = \pi/2$ in
$S^{n-1} \subset S^n$: $N_\psi = (1 - t^2)^{-1/2}$, giving $R^{(n)}_{NN} = N_\psi N_\psi R^{(n)}_{\psi\psi} = (n - 3)/a^4$, and $G^{(n)}_{NN} =
R^{(n)}_{NN} - \tfrac{1}{2}g_{NN}R^{(n)} = (n - 3)/a^4 - (n - 1)(n - 4)/(2a^4) = -(n - 2)(n - 5)/(2a^4)$.
\end{proof}

\begin{theorem}[Totally geodesic embedding]
The 4D Lorentzian cascade spacetime
$M^4 = \mathbb{R} \times S^3$ embeds in $M^5 = \mathbb{R} \times S^4$ as a totally geodesic submanifold: $K_{\mu\nu} = 0$. The
same holds at every equatorial embedding in the chain $S^3 \hookrightarrow S^4 \hookrightarrow \cdots \hookrightarrow S^{216}$.
\end{theorem}

\begin{proof}
Parametrise $S^4 = \{(\psi, \theta_i)\}$ with $d\sigma_4^2 = d\psi^2 + \sin^2\psi\,d\sigma_3^2$. The 4D submanifold
sits at $\psi = \pi/2$. The spacelike unit normal is $N = (1 - t^2)^{-1/2}\partial_\psi$. For the temporal
component: $g_{tt} = -1$ is independent of $\psi$, so $K_{tt} = 0$. For the spatial components:
$\partial_\psi g_{\theta_i\theta_j} = 2(1 - t^2)\sin\psi\cos\psi\,\bar{g}_{ij}$; at $\psi = \pi/2$: $\cos(\pi/2) = 0$, so $K_{ij} = 0$. Therefore
$K_{\mu\nu} = 0$ identically. The argument applies at every equatorial embedding by identical
reasoning.
\end{proof}

\begin{corollary}[Hamiltonian constraint]
The Hamiltonian constraint ${}^{(4)}R = -2G^{(5)}_{NN}$
is satisfied identically: $0 = 0$ at every cascade level. The cascade embedding chain is
geometrically self-consistent.
\end{corollary}

\begin{proof}
The scalar Gauss equation for a hypersurface with $K_{\mu\nu} = 0$ gives ${}^{(4)}R = {}^{(5)}R - 2R^{(5)}_{NN}$.
Since $R^{(5)}_{NN} = G^{(5)}_{NN} + \tfrac{1}{2}R^{(5)}$, substituting: ${}^{(4)}R = R^{(5)} - 2(G^{(5)}_{NN} + \tfrac{1}{2}R^{(5)}) = -2G^{(5)}_{NN}$. From
Theorem 14.5 at $n = 5$: $G^{(5)}_{NN} = -(5 - 2)(5 - 5)/(2a^4) = 0$. From Corollary 9.4 at $n = 4$:
$R^{(4)} = 0$. Therefore $0 = -2 \cdot 0 = 0$. \checkmark
\end{proof}

\begin{theorem}[Effective equation of state of the cascade metric]
The 4D Lorentzian
cascade metric has effective equation of state $w = 1/3$ (radiation) in cascade time, at every
cascade level.
\end{theorem}

\begin{proof}
From Theorem 14.5 at $n = 4$ (using $R^{(4)} = 0$, so $G^{(4)}_{\mu\nu} = R^{(4)}_{\mu\nu}$): $G^{(4)}_{tt} = R^{(4)}_{tt} = 3/a^4$;
$G^{(4)}_{ij} = R^{(4)}_{ij} = (4 - 3)/a^4 \cdot g_{ij} = g_{ij}/a^4$. Identifying $\rho_{\text{eff}} \propto G^{(4)}_{tt} = 3/a^4$ and $p_{\text{eff}} \propto G^{(4)}_{ij}/g_{ij} =
1/a^4$, the effective equation of state is:
\[
w_{\text{eff}} = \frac{p_{\text{eff}}}{\rho_{\text{eff}}} = \frac{1/a^4}{3/a^4} = \frac{1}{3}.
\]
\end{proof}

\begin{remark}[Consequences for the matter fraction]
Theorems 14.6 and 14.8 establish
that the geometric Gauss--Codazzi route to the cascade matter fraction $\Omega_m$ is definitively
closed. Three independent exact results confirm this:
\begin{enumerate}
\item $K_{\mu\nu} = 0$ (Theorem 14.6) eliminates the standard extrinsic curvature mechanism for
KK matter generation. The cascade slicing geometry does not source matter via this
channel.
\item $w = 1/3$ (Theorem 14.8) shows the cascade metric contributes radiation, not matter
($w = 0$) or dark energy ($w = -1$), in cascade time. No splitting of the cascade
integrand into core and tail produces the combination ($w = -1$, $w = 0$) consistent
with the Friedmann equation.
\item The Hamiltonian constraint telescopes identically (Corollary 14.7): ${}^{(4)}R = -2G^{(5)}_{NN} =
0 = 0$ at every level, providing geometric consistency without yielding matter content.
\end{enumerate}
The cascade matter fraction $\Omega_m$ is determined by the Bott partition of cascade layers---the sphere-area fraction of non-trivial-phase layers ($d \bmod 8 \in \{5, 6\}$). This requires
three inputs: the propagator phase classification ([2], Corollary 6.8; period-8 refinement in this paper, Section 9), which establishes the
topological partition; the fermion generation structure of the Bott partition ([3]), which
identifies non-trivial-phase layers with matter via the hairy ball zeros and observed fermion
generations; and the uniqueness of sphere areas as cascade content (Corollary 3.2 of [1]),
which forces the energy assignment to be proportional to $\Omega_{d-1}$ with a universal constant.
The matter fraction is a topological rather than geometric quantity; it does not require
the cascade metric and is unaffected by the Gauss--Codazzi results above. The explicit
derivation is given in the cosmological companion.
\end{remark}

\subsection{The information paradox}

The standard black hole information paradox asks how information is preserved when it
crosses a horizon. In the cascade:

\begin{enumerate}
\item \textbf{No sharp horizon.} The compactification at each step is asymptotic: $R_{\text{eff}} = 1/\sqrt{d + 3} >
0$ for all finite $d$. The weight function $(1 - x^2)^{d/2}$ has support everywhere on $(-1, 1)$.
No direction is fully removed; no information is lost.
\item \textbf{Unitary dynamics.} The discrete propagator ([2], Theorem 7.3) preserves the phase
structure. Each slicing step is a unitary operation on the Hilbert space.
\item \textbf{Apparent irreversibility is asymptotic.} From above ($d + 1$ dimensions), a slicing step
looks irreversible: the direction's contribution is suppressed to $O(1/\sqrt{d})$. From below
($d$ dimensions), the boundary carries the full content (boundary dominance). Both
descriptions are correct; they describe the same asymptotic process from different
sides.
\end{enumerate}

There is no paradox because there was never a sharp boundary to lose information
across---only asymptotic compactification. No firewall, no information loss, no complementarity principle needed.

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

\begin{itemize}
\item It does not derive Newton's constant $G$ or the local solutions of GR.
\item It does not derive the Standard Model gauge group $\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)$.
\item It does not unify QM and GR in the sense of a quantum gravity theory;
Paper~II\,=\,III shows this is unnecessary because both are projections of the
same cascade state.
\item It requires one physical identification hypothesis beyond pure mathematics (Definition 2.1): that the cascade geometry is indistinguishable from our universe. Without
it the series is pure mathematics.
\item It uses no result from physics: all remaining inputs are the orthogonality axiom and
classical mathematical theorems (Lovelock, Clifford--Bott, Hurwitz).
\end{itemize}

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

\begin{enumerate}
\item \textbf{The Standard Model.} Resolved in Papers IVa--IVb: the gauge
group $\text{SU}(3) \times \text{SU}(2) \times \text{U}(1)$ follows from
Bott periodicity and Adams' theorem applied to the cascade's layer
structure; three fermion generations from the Bott partition cut off at
the $d_1=19$ phase transition; the mass spectrum from the
geometric-topological factorisation.
\item \textbf{Quantum gravity.} Resolved in Paper~II\,=\,III: the quantum and
gravitational projections share the same source, propagator, and state space.
Double uniqueness (Gleason $+$ Lovelock at $d=4$) proves mutual consistency.
The metric is a state property; gravity was always quantum in the cascade.
\item \textbf{Black holes.} Resolved in Paper~II\,=\,III: boundary dominance gives
$S = A/4$ at $d = 4$, $S = A/3$ at $d = 3$ (BTZ cross-check), and the
Hawking temperature $T = 1/(8\pi M)$ follows from Birkhoff's theorem with no
semiclassical input. Remaining open questions (thermal spectrum, Page curve,
Kerr/RN entropy, second route to $G_d = d/4$) are listed there.
\item \textbf{Newton's constant.} The cascade produces dimensionless quantities;
$G$ enters as a unit-matching constant from the embedding of the observer's
4D slice in the 217-dimensional cascade. A cascade-native derivation of $G$
would come from the discrete cascade-correction family of Paper~IVb, Section~\xref{part4b}{sec:correction-family} evaluated with
gravitational couplings, or from a second structural identity giving $G_d = d/4$
directly (Paper~II=III Open Question~4). \emph{Not a Kaluza--Klein problem}:
the cascade explicitly refuses semiclassical reduction over compactified
dimensions (Paper~I \S3.2, Paper~II=III \S5--6, Paper~III \S12); the metric
is a state property, not an operator on a fixed background, so there is no
``KK mass spectrum'' to derive $G$ from. Cascade-native alternative: the
action-principle approach now available in Paper~IVb (Remark~4.8).
\item \textbf{The cascade Friedmann equation.} The cascade metric (Section 4) foliates a
217-dimensional geometry onto a 4D FRW slice. The projected Friedmann equation
should follow from the Gauss--Codazzi equations applied to the $M^4 \subset M^{217}$ embedding. The Lorentzian Gauss--Codazzi analysis (Section 14.4) establishes that the
cascade metric contributes radiation ($w = 1/3$) and that $K_{\mu\nu} = 0$ eliminates the
extrinsic curvature mechanism. The matter and vacuum energy content of the 4D
Friedmann equation enters via the Bott partition of cascade layers (Remark 14.9), not
via the cascade metric geometry. A complete derivation of the Friedmann equation
connecting the cascade's layer structure to the physical expansion history via
classical differential geometry (Gauss--Codazzi, not semiclassical reduction)
remains open.
\item \textbf{The Hubble tension.} Under Paper~I's locked-time 3D projection
and Paper~V's Friedmann derivation, the cascade's $H_0 = 66.78$~km/s/Mpc
is essentially equal to Planck's $67.4$ after the Part~0 Gram
first-order correction, and is \emph{incompatible} with SH0ES's $73.0$ at the
$\sim 8\%$ level. Earlier phrasing that the cascade sits ``between the two
principal measurements'' has been retracted (Paper~V \S6 remark); the
cascade picks the Planck side. The full cosmological derivation is carried
out in Paper~V.
\end{enumerate}

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\end{document}