theory.tex

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% Section 3: Theory — Formal Results
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\section{Theory}
\label{sec:theory}

The prior work developed thus far supplies three ingredients that are required in
the present section.  First, the abdominal analysis separates the low-frequency
flexural family from the much higher breathing mode in a fluid-filled soft shell
and identifies geometry as the quantity that controls not only resonance but
also the efficiency with which forcing projects onto those modes~\cite{BrowntoneP1}.
Secondly, the watermelon inversion provides a case in which an equivalent-sphere
model remains useful because the inferential target is deliberately narrow and
the geometry is known independently~\cite{BrowntoneP7}.  Thirdly, the
identifiability analysis shows the inverse cost of that reduction: once
geometry and material must be inferred simultaneously, the same
equivalent-radius simplification becomes structurally defective~\cite{BrowntoneP8}.  The
present section turns those ingredients into
formal statements: one proposition for excitation filtering and four proved
results for the inverse problem.

We emphasise at the outset that the statements below concern \emph{local
practical identifiability} of flexural spectra.  They do not claim global
uniqueness for arbitrary nonlinear inversions.  They ask a more disciplined
question: given a nominal operating point, do the measured modal frequencies
contain enough linearly independent information to distinguish changes in
geometry from changes in material stiffness?

\subsection{Common setup}

Consider a fluid-filled viscoelastic shell with semi-major axis $a$,
semi-minor axis $c$, Young's modulus $E$, and fixed secondary parameters
\[
  \mathbf{p}
  =
  (h,\nu,\rho_w,\rho_f,K_f,P_\mathrm{iap},\eta).
\]
For the Browntone applications, the canonical values are
$E=\SI{0.1}{\mega\pascal}$, $a=\SI{0.18}{\metre}$, $c=\SI{0.12}{\metre}$,
$h=\SI{0.01}{\metre}$, $\nu=0.45$,
$\rho_w=\SI{1100}{\kilogram\per\metre\cubed}$,
$\rho_f=\SI{1020}{\kilogram\per\metre\cubed}$,
$K_f=\SI{2.2}{\giga\pascal}$,
$P_\mathrm{iap}=\SI{1000}{\pascal}$, and $\eta=0.25$
~\cite{BrowntoneP1,BrowntoneP8}.  We restrict attention to the flexural modes
$n \geq 2$, since the breathing mode $n=0$ is fluid-compressibility dominated
and lies in the kilohertz range for the canonical abdomen rather than the
single-digit-hertz range of interest~\cite{BrowntoneP1}.  That distinction is
essential for physical consistency.

Let the unknown parameter vector be
\begin{equation}
  \boldsymbol{\theta} = (a,c,E)^\top \in \Theta \subset \mathbb{R}^3_{+},
\end{equation}
and let the measured modal set be $\mathcal{M}=\{n_1,\ldots,n_m\}$, with
$m \geq 3$.  The forward map is
\begin{equation}
  \mathbf{f}(\boldsymbol{\theta})
  =
  \bigl(
    f_{n_1}(\boldsymbol{\theta};\mathbf{p}),
    \ldots,
    f_{n_m}(\boldsymbol{\theta};\mathbf{p})
  \bigr)^\top \in \mathbb{R}^m,
  \label{eq:p10_forward_map}
\end{equation}
where $f_n$ is the natural frequency of flexural mode $n$.  The raw Jacobian is
\begin{equation}
  J_{ij}
  =
  \frac{\partial f_{n_i}}{\partial \theta_j},
  \label{eq:p10_raw_jacobian}
\end{equation}
but because the three parameter directions carry different units, the relevant
local sensitivity operator is the scaled Jacobian
\begin{equation}
  \widetilde{J}[i,j]
  =
  \frac{\partial f_{n_i}}{\partial \theta_j}
  \frac{\theta_j}{f_{n_i}}.
  \label{eq:p10_scaled_jacobian}
\end{equation}
Each entry of $\widetilde{J}$ is dimensionless and measures fractional
frequency change per fractional parameter change.  Let
\begin{equation}
  \sigma_1 \geq \sigma_2 \geq \sigma_3 \geq 0
  \label{eq:p10_singular_values}
\end{equation}
denote the singular values of $\widetilde{J}$, and define the local
conditioning metric
\begin{equation}
  \kappa(\widetilde{J}) = \frac{\sigma_1}{\sigma_3}.
  \label{eq:p10_condition_number}
\end{equation}
When $\sigma_3=0$, the inverse problem is locally rank deficient.  When
$\sigma_3>0$ but small, the inverse is formally possible but practically
fragile: small frequency errors may be amplified into large parameter errors.
Thus the smallest singular value identifies the least observable parameter
combination, while $\kappa$ measures how strongly observational errors may be
amplified in the inferred parameters.

\subsection{Excitation filtering as a geometric statement}

The inverse results concern what can be inferred once modal frequencies have
been measured.  Geometry does another job earlier in the chain: it limits which
modes can be excited efficiently in the first place.  For long-wavelength
forcing this limitation is captured by the resonant absorption cross-section.

\begin{proposition}[Breit--Wigner excitation filter]
\label{prop:p10_breit_wigner}
Consider flexural mode $n$ of a fluid-filled shell driven at resonance by an
incident acoustic field of wavelength $\lambda$.  Let $\zeta_\mathrm{rad}$ and
$\zeta_\mathrm{str}$ denote the radiation and structural damping ratios of that
mode.  Then the resonant absorption cross-section is
\begin{equation}
  \sigma_{\mathrm{abs},n}
  =
  \frac{(2n+1)\lambda^2}{4\pi}
  \frac{4\zeta_\mathrm{rad}\zeta_\mathrm{str}}
  {(\zeta_\mathrm{rad}+\zeta_\mathrm{str})^2}.
  \label{eq:p10_breit_wigner}
\end{equation}
In particular,
\begin{equation}
  0 \leq \sigma_{\mathrm{abs},n}
  \leq
  \sigma_{\max,n}
  :=
  \frac{(2n+1)\lambda^2}{4\pi},
  \label{eq:p10_sigma_max}
\end{equation}
with equality if and only if $\zeta_\mathrm{rad}=\zeta_\mathrm{str}$.  In the
Rayleigh limit for flexural radiation, $\zeta_\mathrm{rad}\propto(ka)^{2n+2}$;
hence if $ka\ll 1$ and $\zeta_\mathrm{rad}\ll\zeta_\mathrm{str}$ then
\begin{equation}
  \sigma_{\mathrm{abs},n}
  =
  \sigma_{\max,n}
  \left(
    \frac{4\zeta_\mathrm{rad}}{\zeta_\mathrm{str}}
    + O\!\left(\frac{\zeta_\mathrm{rad}^2}{\zeta_\mathrm{str}^2}\right)
  \right),
  \label{eq:p10_sigma_abs_low_ka}
\end{equation}
so long-wavelength airborne forcing is strongly filtered even when the mode
itself exists.
\end{proposition}

\begin{proof}
For a single resonant partial wave, reciprocity gives the standard
Breit--Wigner absorption law~\cite{MorseIngard1968},
\begin{equation}
  \sigma_{\mathrm{abs},n}
  =
  \sigma_{\max,n}
  \frac{4\Gamma_\mathrm{rad}\Gamma_\mathrm{str}}
       {(\Gamma_\mathrm{rad}+\Gamma_\mathrm{str})^2},
\end{equation}
where $\Gamma_\mathrm{rad}$ and $\Gamma_\mathrm{str}$ are the radiative and
structural linewidths.  Dividing by the modal natural frequency shows that the
same expression holds in terms of damping ratios
$\zeta_\mathrm{rad}=\Gamma_\mathrm{rad}/(2\omega_n)$ and
$\zeta_\mathrm{str}=\Gamma_\mathrm{str}/(2\omega_n)$, which gives
\eqref{eq:p10_breit_wigner}.  Writing
$x=\zeta_\mathrm{rad}/\zeta_\mathrm{str}\geq 0$ yields the efficiency factor
$4x/(1+x)^2 \leq 1$, with equality only at $x=1$, proving
\eqref{eq:p10_sigma_max}.  Finally, the Rayleigh-limit radiation resistance of
the $n$th flexural mode scales as $(ka)^{2n+2}$, so
$\zeta_\mathrm{rad}\propto(ka)^{2n+2}$.  Expanding the efficiency factor for
$x\ll 1$ gives $4x+O(x^2)$ and therefore \eqref{eq:p10_sigma_abs_low_ka}.  The
mode may be present in the spectrum, but geometry has already reduced how much
incident power reaches it. 
\end{proof}

\noindent\textbf{Scope of Proposition~\ref{prop:p10_breit_wigner}.}
Proposition~\ref{prop:p10_breit_wigner} is stated for the spherical
partial-wave expansion and is exact only in the spherical limit.  For the
oblate geometries considered here ($\varepsilon \approx 0.745$), the
partial-wave cross-sections acquire correction terms involving oblate
spheroidal harmonics.  However, the qualitative conclusion --- that low-order
flexural modes couple inefficiently to airborne sound in the Rayleigh regime
--- is robust: the $(ka)^{2n}$ suppression reflects the general multipole
scaling and is not an artefact of spherical symmetry.  A quantitative
treatment for oblate scattering would require the $T$-matrix formalism
~\cite{MorseIngard1968} and is beyond the present scope.

\subsection{Equivalent-radius reductions and lost geometric dimension}

Equivalent-sphere models replace the two-parameter geometry $(a,c)$ by a single
scalar descriptor, typically the equivalent radius
\begin{equation}
  R_\mathrm{eq} = (a^2c)^{1/3},
  \label{eq:p10_req}
\end{equation}
or, more generally, some smooth scalar reduction $g(a,c)$.  Such reductions can
be entirely sensible for forward estimates of resonance magnitude or scaling,
and they are used in~\cite{BrowntoneP1,BrowntoneP7} for precisely that purpose.
Their inverse implication is considerably less favourable.

\begin{theorem}[Rank collapse under scalar geometric reduction]
\label{thm:p10_scalar_reduction}
Let a reduced forward model satisfy
\begin{equation}
  f_n(a,c,E;\mathbf{p}) = \phi_n\!\bigl(g(a,c),E;\mathbf{p}\bigr)
  \qquad \text{for all } n \in \mathcal{M},
  \label{eq:p10_scalar_model}
\end{equation}
with $g:\mathbb{R}^2_{+}\to\mathbb{R}_{+}$ a smooth scalar function.  Then the
scaled Jacobian with respect to $\boldsymbol{\theta}=(a,c,E)$ satisfies
\begin{equation}
  \operatorname{rank}(\widetilde{J}) \leq 2.
  \label{eq:p10_rank_bound}
\end{equation}
Equivalently, one independent geometric direction is unobservable from modal
frequencies alone.
\end{theorem}

\begin{proof}
By the chain rule,
\begin{equation}
  \frac{\partial f_n}{\partial a}
  =
  \frac{\partial \phi_n}{\partial g}\,
  \frac{\partial g}{\partial a},
  \qquad
  \frac{\partial f_n}{\partial c}
  =
  \frac{\partial \phi_n}{\partial g}\,
  \frac{\partial g}{\partial c}.
  \label{eq:p10_chain_rule_columns}
\end{equation}
Hence the $a$- and $c$-columns of the raw Jacobian are both scalar multiples of
the same vector
$\bigl(\partial \phi_{n_i}/\partial g\bigr)_{i=1}^m$, so the geometric block of
$J$ has rank at most $1$.  Appending the $E$-column can increase the rank by at
most one, which gives $\operatorname{rank}(J)\leq 2$.

Now write the scaled Jacobian as
\begin{equation}
  \widetilde{J}
  =
  D_f^{-1} J D_\theta,
\end{equation}
where $D_f=\operatorname{diag}(f_{n_1},\ldots,f_{n_m})$ and
$D_\theta=\operatorname{diag}(a,c,E)$.  Both diagonal matrices are nonsingular
for physically admissible parameters, so left- and right-multiplication do not
change rank.  Therefore
$\operatorname{rank}(\widetilde{J})=\operatorname{rank}(J)\leq 2$, as claimed.
For the specific reduction $g=R_\mathrm{eq}=(a^2c)^{1/3}$ one obtains
$\partial R_\mathrm{eq}/\partial a=(2/3)R_\mathrm{eq}/a$ and
$\partial R_\mathrm{eq}/\partial c=(1/3)R_\mathrm{eq}/c$, so the scaled columns
satisfy $\widetilde{J}[:,a]=2\,\widetilde{J}[:,c]$ exactly. 
\end{proof}

Theorem~\ref{thm:p10_scalar_reduction} is stronger than a complaint about poor
conditioning.  It states that the missing information is \emph{structural}: no
amount of experimental precision can recover a geometric direction that the
model has collapsed into a single scalar.  One may still estimate $g(a,c)$ and
$E$, but not $a$ and $c$ separately.

\subsection{Asphericity as an identifiability-lifting perturbation}

To escape Theorem~\ref{thm:p10_scalar_reduction}, the model must retain at least
two genuinely independent geometric channels.  For an oblate shell this occurs
through curvature anisotropy.  Writing the eccentricity as
\begin{equation}
  \varepsilon = \sqrt{1 - c^2/a^2},
  \label{eq:p10_eccentricity}
\end{equation}
the spherical limit is $\varepsilon=0$, while $\varepsilon>0$ introduces
distinct meridional and circumferential curvature distributions.  In a Ritz
formulation, mode $n$ samples those curvature fields through mode-specific
weighting functions.  Schematically,
\begin{equation}
  f_n(\varepsilon)
  =
  f_n^{(0)} + \alpha_n \varepsilon^2 + \beta_n \varepsilon^4 + \cdots,
  \label{eq:p10_mode_expansion}
\end{equation}
where the coefficients $\alpha_n,\beta_n,\ldots$ depend on the mode number
because different mode shapes concentrate strain energy in different regions of
the shell.  Once those coefficients vary with $n$, the $a$- and $c$-columns of
$\widetilde{J}$ need no longer remain proportional across modes.

\begin{theorem}[Identifiability lifting by oblate asphericity]
\label{thm:p10_lifting}
Consider the oblate Ritz shell model with measured flexural modes
$\mathcal{M}\supseteq\{2,3,4\}$ and inversion parameters
$\boldsymbol{\theta}=(a,c,E)$.  Then there exists a non-empty open set
$U\subset\{\boldsymbol{\theta}\in\Theta:c<a\}$ on which
\begin{equation}
  \operatorname{rank}(\widetilde{J}(\boldsymbol{\theta})) = 3.
\end{equation}
Thus oblate asphericity lifts the scalar-reduction rank collapse and restores
local identifiability of $(a,c,E)$.
\end{theorem}

\begin{proof}
For a fixed Ritz basis and a fixed measured-mode set, the frequencies
$f_n(\boldsymbol{\theta})$ are smooth functions of $(a,c,E)$ away from modal
crossings, so the entries of $\widetilde{J}(\boldsymbol{\theta})$ are smooth as
well.  At the canonical oblate operating point
\begin{equation}
  \boldsymbol{\theta}_*
  =
  \bigl(\SI{0.18}{\metre},\SI{0.12}{\metre},\SI{0.1}{\mega\pascal}\bigr),
\end{equation}
The numerical study in~\cite{BrowntoneP8} reports
$\kappa(\widetilde{J}(\boldsymbol{\theta}_*))=69.4$ for the
five-mode inverse using modes $n=2,\ldots,6$~\cite{BrowntoneP8}.  Therefore
$\sigma_3(\widetilde{J}(\boldsymbol{\theta}_*))>0$, so
$\operatorname{rank}(\widetilde{J}(\boldsymbol{\theta}_*))=3$.

Singular values depend continuously on the matrix entries, hence continuously
on $\boldsymbol{\theta}$.  Since $\sigma_3(\widetilde{J}(\boldsymbol{\theta}_*))$
is strictly positive, there exists $\delta>0$ such that
$\sigma_3(\widetilde{J}(\boldsymbol{\theta}))>\sigma_3(\widetilde{J}(\boldsymbol{\theta}_*))/2>0$
whenever $\|\boldsymbol{\theta}-\boldsymbol{\theta}_*\|<\delta$.  Intersecting
that ball with the oblate set $\{c<a\}$ yields a non-empty open neighbourhood
$U$ on which $\operatorname{rank}(\widetilde{J})=3$.  The mechanism is the
mode-dependent curvature sampling represented in
\eqref{eq:p10_mode_expansion}: different flexural modes weight meridional and
circumferential curvature differently, so the geometric columns of
$\widetilde{J}$ separate once the shell is genuinely oblate.
\end{proof}

The theorem is existential rather than global.  It does not say that every
oblate shell is well conditioned.  It says that oblate asphericity restores the
\emph{possibility} of independent recovery by breaking the exact proportionality
imposed by scalar geometric reduction.  Conditioning then becomes a quantitative
matter rather than an algebraic impossibility.

\subsection{Near-spherical asymptotics and the corrected spherical limit}

The next question is how the restored identifiability behaves as the shell
approaches spherical symmetry.  The correct statement is subtler than the first
draft of the capstone suggested.  At the exact sphere, symmetry forces rank
collapse and therefore $\sigma_3(0)=0$.  The previously reported non-zero
intercept $\sigma_0>0$ was a low-order Ritz discretisation artefact, not a
property of the continuum inverse problem.  What survives is a regular even
expansion in the curvature parameter.

\begin{proposition}[Regular near-spherical asymptotics]
\label{prop:p10_asymptotics}
Consider a smooth one-parameter oblate family of Ritz shell models
parameterised by the eccentricity
$\varepsilon=\sqrt{1-c^2/a^2}$,
with $\varepsilon=0$ at the sphere.  Then the
smallest singular value of the scaled Jacobian satisfies
\begin{equation}
  \sigma_3(\varepsilon)
  =
  \lambda_1 \varepsilon^2 + O(\varepsilon^4),
  \qquad \varepsilon \to 0,
  \label{eq:p10_sigma_asymptotic}
\end{equation}
with $\lambda_1>0$ for the canonical oblate family considered
in~\cite{BrowntoneP8,BrowntoneP9}.  The analytic regularity of the eigenvalue
perturbation is guaranteed by Kato's perturbation theory for linear
operators~\cite{Kato1966}.  In
particular,
\begin{equation}
  \sigma_3(0)=0.
\end{equation}
\end{proposition}

\begin{proof}
Reflection symmetry implies that replacing $\varepsilon$ by $-\varepsilon$
describes the same oblate geometry.  The stiffness and mass operators of the
Ritz model are therefore even analytic functions of $\varepsilon$, so the
frequency map and the scaled Jacobian admit expansions of the form
\begin{equation}
  \widetilde{J}(\varepsilon)
  =
  \widetilde{J}_0 + \varepsilon^2 \widetilde{J}_2 + O(\varepsilon^4).
  \label{eq:p10_J_even_expansion}
\end{equation}
At $\varepsilon=0$ the shell is spherical, so geometry enters only through a
single radius and Theorem~\ref{thm:p10_scalar_reduction} applies.  Hence
$\operatorname{rank}(\widetilde{J}_0)\leq 2$ and therefore $\sigma_3(0)=0$.

Equation~\eqref{eq:p10_J_even_expansion} shows that the geometric splitting away
from the sphere enters first at order $\varepsilon^2$, because curvature
anisotropy itself is quadratic in $\varepsilon$.  Consequently the smallest
singular value cannot acquire a constant or odd-power term.  For the canonical
oblate family, Theorem~\ref{thm:p10_lifting} gives
$\sigma_3(\varepsilon)>0$ on an open set of $\varepsilon>0$, so the leading
even coefficient is non-zero and positive.  Denoting that coefficient by
$\lambda_1$ yields \eqref{eq:p10_sigma_asymptotic}.  The corrected
near-spherical analysis therefore removes the earlier fictitious intercept
$\sigma_0>0$: the apparent finite floor was a discretisation artefact of the
low-order spherical limit, not a true continuum property.
\end{proof}

\begin{corollary}
\label{cor:p10_kappa_asymptotic}
If $\sigma_1(\varepsilon)=\sigma_1(0)+O(\varepsilon^2)$ with $\sigma_1(0)>0$,
then
\begin{equation}
  \kappa(\widetilde{J}(\varepsilon))
  =
  \frac{\sigma_1(0)}{\lambda_1}\,\varepsilon^{-2} + O(1)
  \qquad \text{as } \varepsilon \to 0.
\end{equation}
\end{corollary}

\noindent The corollary has a simple interpretation.  The exact sphere is
singular, but the loss of identifiability is regular rather than exotic: the
condition number grows quadratically as spherical symmetry is restored.

\subsection{Forward adequacy and inverse adequacy are different questions}

Reduced models are often judged by forward error alone.  Let
$\mathbf{f}_\mathrm{red}$ denote a reduced model and
$\mathbf{f}_\mathrm{full}$ a geometry-resolving model.  A natural forward metric
is the relative frequency misfit
\begin{equation}
  \mathcal{E}_\mathrm{fwd}
  =
  \frac{\|\mathbf{f}_\mathrm{red}(\boldsymbol{\theta})
  - \mathbf{f}_\mathrm{full}(\boldsymbol{\theta})\|_2}
  {\|\mathbf{f}_\mathrm{full}(\boldsymbol{\theta})\|_2}.
  \label{eq:p10_forward_error}
\end{equation}
An inverse metric, by contrast, must assess the conditioning of the parameter
recovery map; locally, $\kappa(\widetilde{J})$ is the appropriate quantity.
There is no theorem forcing small $\mathcal{E}_\mathrm{fwd}$ to imply moderate
$\kappa$.  The Browntone sequence instead supplies explicit counterexamples.

\begin{proposition}[Forward adequacy does not imply inverse adequacy]
\label{prop:p10_forward_inverse}
There exist parameter domains on which a reduced equivalent-sphere model has
small forward error while remaining unusable for joint recovery of
$(a,c,E)$.  In particular, at the canonical abdominal operating point and using
modes $n=2,\ldots,6$,
\begin{equation}
  \mathcal{E}_\mathrm{fwd} < 0.1,
\end{equation}
yet the equivalent-sphere model satisfies
\begin{equation}
  \operatorname{rank}(\widetilde{J}_\mathrm{sphere}) \leq 2
  \qquad \text{and} \qquad
  \kappa(\widetilde{J}_\mathrm{sphere}) \approx 1.37\times 10^{10},
\end{equation}
whereas the oblate Ritz model on the same modal set has
\begin{equation}
  \operatorname{rank}(\widetilde{J}_\mathrm{oblate}) = 3
  \qquad \text{and} \qquad
  \kappa(\widetilde{J}_\mathrm{oblate}) = 69.4.
\end{equation}
The mechanism is a \emph{structural sensitivity gap} along the kernel of the
geometric reduction.
\end{proposition}

\begin{proof}
At the canonical abdominal point, the equivalent-sphere and oblate Ritz models
produce closely matched low-order spectra: the relative forward misfit for modes
$n=2,\ldots,6$ is $\mathcal{E}_\mathrm{fwd}=0.0936<0.1$.  Thus the reduced
model is forward-adequate in the ordinary engineering sense of reproducing the
dominant frequencies to within \SI{10}{\percent}.  By
Theorem~\ref{thm:p10_scalar_reduction}, however, the equivalent-sphere inverse
map is structurally rank deficient, so
$\operatorname{rank}(\widetilde{J}_\mathrm{sphere})\leq 2$.  The finite-difference
evaluation reported in~\cite{BrowntoneP8} gives the associated numerical condition number
$\kappa(\widetilde{J}_\mathrm{sphere})\approx 1.37\times10^{10}$.

On the same parameter domain, Theorem~\ref{thm:p10_lifting} shows that the
oblate Ritz model has full rank on an open neighbourhood of the canonical
point, and~\cite{BrowntoneP8} gives the concrete value
$\kappa(\widetilde{J}_\mathrm{oblate})=69.4$ for the five-mode inverse
~\cite{BrowntoneP8}.

The structural reason why forward and inverse adequacy decouple is made
transparent by the kernel of the projection
$\pi:(a,c,E)\mapsto(R_\mathrm{eq},E)$.  Let
$\mathbf{v}\in\ker D\pi(\boldsymbol{\theta}_0)$ with $\|\mathbf{v}\|=1$; this
is the direction that trades $a$ against $c$ while preserving
$R_\mathrm{eq}=(a^2 c)^{1/3}$.  Then
\begin{equation}
  \widetilde{J}_\mathrm{sphere}\,\mathbf{v}
  =
  DG(\pi(\boldsymbol{\theta}_0))\;
  \underbrace{D\pi(\boldsymbol{\theta}_0)\,\mathbf{v}}_{=\,\mathbf{0}}
  =
  \mathbf{0},
  \label{eq:p10_kernel_mechanism}
\end{equation}
so the reduced model is \emph{exactly insensitive} to perturbations along
$\mathbf{v}$.  By contrast, $\operatorname{rank}(\widetilde{J}_\mathrm{oblate})=3$
guarantees $\widetilde{J}_\mathrm{oblate}\,\mathbf{v}\neq\mathbf{0}$: the
full model detects parameter changes to which the reduced model is structurally
blind.

Forward adequacy \eqref{eq:p10_forward_error} constrains the \emph{values} of
the two frequency vectors to be close; it places no constraint on how their
\emph{derivatives} compare along the lost direction, because the forward error
lives in the image space $\mathbb{R}^m$ whereas the inverse problem concerns
the preimage structure in~$\Theta$.  A model can therefore reproduce measured
frequencies well yet be wholly unable to distinguish the parameter
combinations that produced them.
\end{proof}

\begin{remark}[Watermelon model as supporting example]
\label{rem:p10_watermelon_example}
An instructive companion example is provided by the watermelon model
in~\cite{BrowntoneP7}: there the equivalent-sphere reduction succeeds precisely
because geometry is fixed independently and the inverse target is effectively
one-dimensional.  This separation parallels the model-updating literature's
long-standing distinction between residual minimisation and parameter
observability~\cite{MottersheadFriswell1993,FriswellMottershead1995}.
Forward adequacy is not false; it is simply insufficient for
multivariate spectral inversion.
\end{remark}

\begin{remark}[Near-spherical divergence of the adequacy gap]
\label{rem:p10_near_spherical_gap}
In the near-spherical regime, the coexistence of forward adequacy and inverse
inadequacy becomes increasingly extreme.  The forward error shrinks as the
shell approaches spherical symmetry because the equivalent-sphere approximation
becomes exact at $\varepsilon=0$.  But the condition number diverges by
Corollary~\ref{cor:p10_kappa_asymptotic}:
$\kappa(\widetilde{J})\sim(\sigma_1(0)/\lambda_1)\,\varepsilon^{-2}$
as $\varepsilon\to 0$.  The ratio of inverse difficulty to forward discrepancy
therefore grows without bound: the sphere limit is simultaneously the most
forward-adequate and the most inverse-inadequate configuration.
Figure~\ref{fig:forward_inverse_gap} illustrates this divergence numerically.
\end{remark}

\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.85\textwidth]{figures/fig_forward_inverse_gap.pdf}
  \caption{The forward-inverse adequacy gap
    (Proposition~\ref{prop:p10_forward_inverse}).  The forward error
    $\mathcal{E}_\mathrm{fwd}$ (blue, left axis) quantifies how well the
    equivalent-sphere model reproduces oblate Ritz frequencies; the condition
    number $\kappa$ (right axis) quantifies how usable each model is for
    parameter recovery.  The oblate Ritz model (green) maintains moderate
    $\kappa$ across all eccentricities, while the equivalent-sphere model (red)
    is structurally rank deficient everywhere.  Both models share the same
    forward-error curve, yet their inverse conditioning differs by eight orders
    of magnitude.  The canonical operating point ($\varepsilon=0.745$, vertical
    dashed line) sits in a region where $\mathcal{E}_\mathrm{fwd}<10\%$ and
    $\kappa_\mathrm{sphere}/\kappa_\mathrm{oblate}\approx 2\times 10^{8}$.}
  \label{fig:forward_inverse_gap}
\end{figure}

Taken together, Theorem~\ref{thm:p10_scalar_reduction},
Theorem~\ref{thm:p10_lifting}, Proposition~\ref{prop:p10_asymptotics}, and
Proposition~\ref{prop:p10_forward_inverse} define the capstone thesis in
mathematical form.  Scalar geometric reductions collapse one inverse direction;
oblate curvature can restore it; the restoration vanishes quadratically at the
exact sphere; and forward success does not justify inverse use.  The results
section tests these formal statements against the cross-paper evidence.