% Paper 29: Geometric Exactness and the Training-Exposure Distinction — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: current canon; the training/exposure distinction stands. Two theorem-hypothesis caveats are recorded; corpus use unaffected.
% Registry: P029_1 (retired, closed by A278); P029_2_c (scope), P029_3_c (scope) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P029.py (historical series; includes expected-fail checks that document recorded gaps).

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\title{Geometric Exactness and the Training--Exposure Distinction \\[0.5em]
\large Why Architectures Encoding Proven Identities\\
Are Degraded by Gradient Descent}

\author{L\'{e}on Fernando Vlegels \\
\small Independent Researcher}
\date{2025-12}

\begin{document}
\maketitle
\tableofcontents
\newpage

\begin{abstract}
We formalize a distinction between two modes of computation:
\emph{training}, in which free parameters are adjusted by gradient descent
to minimize a loss function, and \emph{exposure}, in which a
fixed-architecture system receives inputs that select trajectories through
an already-correct geometric structure.  We prove that when a computational
graph exactly implements a mathematical identity (with zero
approximation error at its design point), any gradient step generically
increases error.  The architecture sits at an isolated global minimum of
the loss landscape, and gradient descent cannot improve it.  We apply this
framework to the identities established in the geometric Theory of
Everything: the Hopf projection identity $v = \cos(2\beta)$, the
Lorentz lapse $m = \sqrt{1-v^2}$, the axis-resolved map, and the
measure-reconciliation constant $\Omega_0 = \pi^3/4$.  These identities
define a class of \emph{geometrically exact} computational architectures
for which the correct operating mode is exposure, not training.  We
establish necessary and sufficient conditions for a computational graph to
belong to this class and discuss implications for the relationship between
mathematical structure and machine learning.
\end{abstract}


%==========================================================================
\section{Introduction}\label{sec:intro}
%==========================================================================

There are two ways a computational system can come to compute a function
correctly.  It can search for the function, adjusting internal parameters
until its outputs match observed data; or it can be built as the
function, with every internal constant fixed by construction.  The first
way is the default culture of modern machine learning.  The second is the
natural mode for any system whose target is a proven mathematical
identity.  A trained system carries residual error that optimization can
in principle reduce; an identity-implementing system carries none at its
design point, so the same machinery, applied to it, can only move it away
from correctness.  This paper makes the distinction precise and draws the
operational consequence: a system that implements an identity should be
operated by \emph{exposure}, by feeding it inputs and reading its
outputs, with nothing inside it ever adjusted.

\subsection{The Standard Paradigm}

Modern machine learning operates by parameterizing a function class
$\{f_\theta : \theta \in \Theta\}$ and optimizing
\begin{equation}\label{eq:erm}
\theta^* = \arg\min_{\theta \in \Theta}\;
\mathcal{L}(\theta) := \frac{1}{N}\sum_{i=1}^N
\ell\bigl(f_\theta(x_i),\, y_i\bigr)
\end{equation}
via gradient descent: $\theta_{t+1} = \theta_t - \eta\,\nabla_\theta
\mathcal{L}(\theta_t)$.  This assumes that the correct parameters are
\emph{unknown} and must be discovered from data.

The assumption is usually justified.  Where the target function exists
only as a statistical regularity in a corpus, search is the only
available strategy, and the apparatus that has grown around
equation~\eqref{eq:erm}, train--test splits, regularization,
learning-rate schedules, early stopping, exists to manage the
consequences of that ignorance.  Every element of it presupposes a gap
between $f_\theta$ and the target, to be closed by adjustment.

\subsection{The Geometrically Exact Case}

Consider instead a system where the target function $f^*$ is known
\emph{exactly}: not as an approximation, not as an asymptotic limit, but
as a proven mathematical identity.  If the architecture can implement
$f^*$ with specific parameter values $\theta^*$ such that
$f_{\theta^*} \equiv f^*$, then:
\begin{enumerate}[label=(\roman*)]
\item The loss at $\theta^*$ is exactly zero: $\mathcal{L}(\theta^*) = 0$.
\item Any perturbation $\theta^* + \delta$ yields
  $\mathcal{L}(\theta^* + \delta) \geq 0$, with equality only if
  $f_{\theta^* + \delta} \equiv f^*$.
\item Gradient descent from $\theta^*$ cannot reduce loss further.
\end{enumerate}

In this case the training apparatus loses its purpose.  There is no gap
to close: the parameters are known quantities to be written down, not
unknown quantities to be estimated.  And there is no generalization
question in the statistical sense: an identity holds for every input by
the meaning of the word identity, not with high probability over a
distribution.

The question is: under what conditions is this minimum \emph{isolated},
so that gradient descent from a nearby initialization cannot reliably
reach $\theta^*$, while direct construction can?  Isolation is what turns
the observation above into a design constraint: if the zero-loss point is
isolated, every displacement costs accuracy, and any procedure that
displaces parameters degrades the system.

\subsection{Connection to the Geometric TOE}

Paper~27 \cite{Paper27} established a collection of exact identities
governing dual-observer dynamics on $S^3$:
\begin{align}
v(q) &= \cos(2\beta), \label{eq:v-identity} \\
m(q) &= \sqrt{1 - v(q)^2}, \label{eq:m-identity} \\
v &= n_\parallel^2 + (1 - n_\parallel^2)\cos\eta, \label{eq:axis-identity} \\
\Omega_0 &= \pi^3/4. \label{eq:omega0-identity}
\end{align}

These are not regression targets.  They are theorems.  A computational
system that implements them does not benefit from optimization; it
benefits from correct \emph{construction}.  This paper makes that claim
precise.

The stakes for the corpus are direct.  Every constant in the chain of
derivations is fixed by geometry: $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$
from the $(B^4, S^3)$ volume structure \cite{Paper1}, $\Omega_0 =
\pi^3/4$ from measure reconciliation, the Hopf coefficients from the
fibration itself.  Operated as a trainable model, each derivation would
be quietly converted back into a fitting problem, and the central claim
of the corpus, that the constants are derived rather than fitted, would
be undone at the implementation layer.

\subsection{Outline}

Section~\ref{sec:framework} fixes the formal vocabulary and proves the
zero-loss theorem.  Section~\ref{sec:isolation} proves the isolation
theorem, records two theorem-scope caveats explicitly, and works through
a closed-form scalar example.  Section~\ref{sec:exposure} defines
exposure as an operating mode and develops the resonance analogy as a
worked case.  Section~\ref{sec:toe} applies the framework to the TOE
identity catalog, Section~\ref{sec:general} states when each paradigm is
appropriate, and Section~\ref{sec:implications} draws the implications.


%==========================================================================
\section{Formal Framework}\label{sec:framework}
%==========================================================================

\subsection{Computational Graphs and Parameter Spaces}

The definitions below are deliberately minimal; nothing in the argument
depends on the graph being a neural network.

\begin{definition}[Parameterized Computational Graph]\label{def:graph}
A parameterized computational graph is a tuple $\mathcal{G} = (V, E,
\theta)$ where $V$ is a set of nodes (operations), $E$ is a directed
acyclic graph of data flow, and $\theta \in \Theta \subseteq \R^p$ is
a vector of adjustable parameters.  The graph computes a function
$f_\theta: \mathcal{X} \to \mathcal{Y}$.
\end{definition}

\begin{definition}[Geometric Exactness]\label{def:exact}
A computational graph $\mathcal{G}$ is \emph{geometrically exact} for a
target function $f^*$ if there exists $\theta^* \in \Theta$ such that
\begin{equation}\label{eq:exact}
f_{\theta^*}(x) = f^*(x) \qquad \forall\, x \in \mathcal{X}.
\end{equation}
We call $\theta^*$ the \emph{geometric parameter}.
\end{definition}

The qualifier \emph{geometric} signals where such exactness comes from.
Universal-approximation results guarantee approximation to any tolerance;
they do not produce a $\theta^*$ at which the error is identically zero.
Exact implementation is available only when the target can be written
directly into the graph: a polynomial with its own coefficients, an
algebraic operation as that operation, a constant as that constant.  The
identities of Section~\ref{sec:toe} are of exactly this kind.

\begin{definition}[Loss Landscape]\label{def:loss}
Given a distribution $\mu$ on $\mathcal{X} \times \mathcal{Y}$ with
$y = f^*(x)$, the population loss is
\begin{equation}\label{eq:pop-loss}
\mathcal{L}(\theta) = \mathbb{E}_{(x,y) \sim \mu}\bigl[
\|f_\theta(x) - f^*(x)\|^2\bigr].
\end{equation}
\end{definition}

Note that the labels are noiseless: $y = f^*(x)$ exactly, because the
target is an identity, not a measurement.  The loss floor is exactly
zero, and it is attained.

\subsection{The Zero-Loss Theorem}

\begin{theorem}[Zero Loss at Geometric Parameter]\label{thm:zero-loss}
If $\mathcal{G}$ is geometrically exact for $f^*$ with geometric
parameter $\theta^*$, then $\mathcal{L}(\theta^*) = 0$ and
$\nabla_\theta \mathcal{L}(\theta^*) = 0$.  That is, $\theta^*$ is a
critical point of $\mathcal{L}$.
\end{theorem}

\begin{proof}
By Definition~\ref{def:exact}, $f_{\theta^*}(x) = f^*(x)$ for all $x$,
so $\|f_{\theta^*}(x) - f^*(x)\|^2 = 0$ for all $x$.  Therefore
$\mathcal{L}(\theta^*) = 0$.  Since $\mathcal{L} \geq 0$ everywhere
and $\mathcal{L}(\theta^*) = 0$, the point $\theta^*$ is a global
minimum, hence $\nabla_\theta \mathcal{L}(\theta^*) = 0$.
\end{proof}

The proof is elementary by design, and it is worth noticing what it does
not use: it does not differentiate through the graph, and it does not
depend on $\mu$ at all, since the integrand vanishes pointwise.
Everything distribution-sensitive enters later, in the isolation
question, where one must show that points \emph{near} $\theta^*$ have
strictly positive loss; there the choice of $\mu$ matters, and the
remarks in Section~\ref{sec:isolation} record exactly how.


%==========================================================================
\section{Isolation and Training Incompatibility}\label{sec:isolation}
%==========================================================================

\subsection{When Is the Minimum Isolated?}

Zero loss at $\theta^*$ says training cannot improve the system.
Isolation says training can damage it: any process that moves the
parameters moves them uphill.

\begin{definition}[Isolated Geometric Minimum]\label{def:isolated}
The geometric parameter $\theta^*$ is \emph{isolated} if there exists
$\epsilon > 0$ such that for all $\delta \in \R^p$ with
$0 < \|\delta\| < \epsilon$,
\begin{equation}\label{eq:isolated}
\mathcal{L}(\theta^* + \delta) > 0.
\end{equation}
\end{definition}

The obstruction to isolation is parameter redundancy: if distinct
parameter vectors near $\theta^*$ compute the same function, the loss is
zero along a continuum.  The hypothesis that rules this out is local
injectivity of the parameter-to-function map.

\begin{theorem}[Isolation from Rigidity]\label{thm:isolation}
Let $\mathcal{G}$ be geometrically exact for $f^*$ with geometric
parameter $\theta^*$.  Suppose $f^*$ is not identically zero and the
map $\theta \mapsto f_\theta$ is locally injective at $\theta^*$ (i.e.,
no redundant parameters near $\theta^*$).  Then $\theta^*$ is isolated.
\end{theorem}

\begin{proof}
Local injectivity means: for all $\delta$ with $\|\delta\|$ sufficiently
small and $\delta \neq 0$, there exists $x \in \mathcal{X}$ such that
$f_{\theta^* + \delta}(x) \neq f_{\theta^*}(x) = f^*(x)$.  Therefore
$\|f_{\theta^* + \delta}(x) - f^*(x)\|^2 > 0$ for that $x$, giving
$\mathcal{L}(\theta^* + \delta) > 0$.
\end{proof}

The argument is a two-step transfer: local injectivity supplies one
input $x$ at which the perturbed function disagrees with the target, and
the loss is then claimed positive because the integrand is positive at
that $x$.  The second step is where an unstated hypothesis enters: one
point of disagreement forces positive \emph{population} loss only if
$\mu$ assigns weight near that point.  The following remarks record this
and a second scope limit; the honest statement of the theorem is the
theorem together with them.

\begin{remark}[TBS]\label{rmk:tbs:P029_2_c}
The proof of Theorem~\ref{thm:isolation} does not establish that $\mathcal{L}(\theta^*+\delta)>0$ for an \emph{arbitrary} data distribution $\mu$. Local injectivity guarantees a single $x$ where the perturbation creates a nonzero residual, but if $\mu$ does not charge that point (e.g.\ if $\mu$ is supported on a lower-dimensional set), the population loss may remain zero despite the local non-injectivity of $\theta^*+\delta$. A full proof requires $\mu$ to have full topological support or an equivalent regularity condition not stated here.
\end{remark}

\paragraph{Status.} Recorded in the registry as P029\_2\_c (scope), confirmed by A295. The theorem requires a full-support hypothesis on the data distribution that the proof does not state; this is a theorem-hypothesis caveat, not a defect in the architecture. Corpus use of the training/exposure distinction is unaffected. Ledger: Paper 40.

A concrete instance of the gap: for $f_\theta(x) = \theta x$ on $[0,1]$
with target $f^*(x) = x$, the map $\theta \mapsto f_\theta$ is injective,
yet if $\mu$ is a point mass at $x = 0$ then $\mathcal{L}(\theta) = 0$
for every $\theta$.  Full support restores the conclusion, as the worked
example below shows.

\begin{remark}[TBS]\label{rmk:tbs:P029_3_c}
The isolation of $\theta^*$ (Definition~\ref{def:isolated}) establishes only that no \emph{nearby} points achieve zero loss, not that $\theta^*$ is the \emph{unique} global zero-loss point. Other isolated minima elsewhere in $\Theta$ are not precluded. Uniqueness of the global minimum would require a global convexity argument or a proof that $\mathcal{L}(\theta)=0 \Rightarrow \theta=\theta^*$, neither of which is provided.
\end{remark}

\paragraph{Status.} Recorded in the registry as P029\_3\_c (scope), confirmed by A295. The isolation result is local; global uniqueness of the zero-loss point is a stronger statement that the paper does not claim to need. Corpus use is unaffected. Ledger: Paper 40.

This second limit is likewise illustrated by a one-line model: for
$f_\theta(x) = (\theta - 2)^2\, x$ with target $f^*(x) = x$, the loss
vanishes at $\theta = 1$ and at $\theta = 3$; each zero is locally
isolated, neither globally unique.  Operational use does not require
global uniqueness: the system is constructed at a specific $\theta^*$,
and the relevant question, whether motion away from that point costs
accuracy, is the local statement.

\begin{corollary}[Training Incompatibility]\label{cor:incompatible}
At an isolated geometric minimum, any gradient step from
$\theta^*$ either:
\begin{enumerate}[label=(\alph*)]
\item remains at $\theta^*$ (zero gradient), or
\item moves to $\theta^* + \delta$ with
  $\mathcal{L}(\theta^* + \delta) > 0$.
\end{enumerate}
Gradient descent from $\theta^*$ cannot improve performance.  If
stochastic perturbation or regularization displaces $\theta$ from
$\theta^*$, subsequent gradient steps may or may not return to
$\theta^*$ depending on the basin geometry, but they cannot find a
\emph{better} point.
\end{corollary}

The corollary is the operational content of the paper.  Exact gradient
descent at $\theta^*$ is a fixed point, but practical training is never
exact gradient descent: minibatch sampling, finite precision, and
regularizers all inject displacements, and at an isolated minimum every
displacement has strictly positive cost.  A training loop wrapped around
a geometrically exact graph is a mechanism whose only possible effects
are nothing or harm.

\subsection{A Scalar Worked Example}

Every quantity in the framework can be computed in closed form for the
simplest nontrivial model.  Take $\mathcal{X} = [-1, 1]$ with $\mu$
uniform, target $f^*(x) = x$, and the one-parameter graph
$f_\theta(x) = \theta x$.  The population loss is
\[
\mathcal{L}(\theta)
= \mathbb{E}\bigl[(\theta x - x)^2\bigr]
= (\theta - 1)^2\,\mathbb{E}[x^2]
= \frac{(\theta - 1)^2}{3},
\]
with geometric parameter $\theta^* = 1$.  Then $\mathcal{L}(1) = 0$ and
$\mathcal{L}'(\theta) = 2(\theta - 1)/3$ vanishes at $\theta^*$
(Theorem~\ref{thm:zero-loss}); the second derivative
$\mathcal{L}'' = 2/3 > 0$ makes the minimum strict
(Theorem~\ref{thm:isolation}); a nearby perturbation gives positive
loss, $\mathcal{L}(1.1) = 0.01/3 \approx 0.0033$; and an exact gradient
step from $\theta^*$ at any learning rate returns $\theta^*$, the
fixed-point behavior of Corollary~\ref{cor:incompatible}.  The same
model with $\mu$ a point mass exhibits the failure recorded in
Remark~\ref{rmk:tbs:P029_2_c}; the contrast between the two choices of
$\mu$ is the content of the full-support hypothesis.

\subsection{Hessian Structure}

\begin{proposition}[Positive-Definite Hessian]\label{prop:hessian}
Under the conditions of Theorem~\ref{thm:isolation}, if $f_\theta$ is
twice differentiable in $\theta$ and $\mu$ has full support, then the
Hessian
\begin{equation}\label{eq:hessian}
H_{ij} = \frac{\partial^2 \mathcal{L}}{\partial \theta_i
\partial \theta_j}\bigg|_{\theta^*}
= 2\,\mathbb{E}\biggl[\sum_k \frac{\partial f_k}{\partial \theta_i}
\frac{\partial f_k}{\partial \theta_j}\biggr]_{\theta^*}
\end{equation}
is positive semi-definite, and is strictly positive-definite if the
Jacobian $\partial f / \partial \theta|_{\theta^*}$ has full column
rank.
\end{proposition}

\begin{proof}
At the exact minimum, the second-order expansion of $\mathcal{L}$ reduces
to the Gauss--Newton approximation (the residual term vanishes since
$f_{\theta^*} = f^*$).  The result follows from the rank condition on
the Jacobian.
\end{proof}

\begin{remark}
A strictly positive-definite Hessian means $\theta^*$ is a
\emph{strict local minimum with quadratic basin}.  Gradient descent
from nearby points will converge toward $\theta^*$, but from distant
initializations there is no guarantee: the loss landscape may have
other critical points.  Direct construction bypasses this landscape
entirely.
\end{remark}

The Gauss--Newton reduction in the proof is special to the exact case:
the residual-weighted curvature term, the usual source of saddle points
in trained networks, is identically zero, and only the positive
semi-definite Jacobian term survives.  The landscape around $\theta^*$
is a clean quadratic bowl.  In the scalar example above the formula
gives $H = 2\,\mathbb{E}[x^2] = 2/3$, matching the direct computation.


%==========================================================================
\section{The Exposure Paradigm}\label{sec:exposure}
%==========================================================================

\subsection{Definition}

\begin{definition}[Exposure]\label{def:exposure}
Let $\mathcal{G}$ be a geometrically exact computational graph with
fixed parameter $\theta^*$.  \emph{Exposure} is the process of
providing inputs $x \in \mathcal{X}$ and reading outputs
$f_{\theta^*}(x) \in \mathcal{Y}$, with no modification of $\theta^*$.
\end{definition}

In the exposure paradigm:
\begin{enumerate}[label=(\roman*)]
\item Parameters are \emph{constructed}, not learned.
\item Inputs set \emph{initial conditions}, not training signals.
\item The forward pass \emph{computes} the target identity, not an
  approximation to it.
\item No loss function, no backward pass, no weight update.
\end{enumerate}

Operationally, an exposure run has a different shape from a training run.
There is no dataset to assemble: a single input is a complete unit of
work, and its output is the exact value of the identity at that input.
There is no held-out set, because the function computed on unseen inputs
is the same function, by construction.  What replaces all of this is
verifying that the construction is correct.  A wrong output from an
exposure-mode system is not a fitting error to be reduced by more data;
it is a construction error, repaired by reading the derivation, not by
running an optimizer.  Wrongness in exposure mode is a bug, not a
residual.

\subsection{Formal Comparison}

\begin{center}
\begin{tabular}{lll}
\toprule
\textbf{Property} & \textbf{Training} & \textbf{Exposure} \\
\midrule
Parameters & Unknown, discovered & Known, constructed \\
Input role & Training signal & Initial condition \\
Output role & Prediction (approximate) & Computation (exact) \\
Error source & Parameter mismatch & None (at design point) \\
Gradient & Informative & Zero (at $\theta^*$) \\
Data requirement & Large corpus & Single instance suffices \\
Generalization & Statistical & Guaranteed (identity holds $\forall\, x$) \\
\bottomrule
\end{tabular}
\end{center}

Each row of the table is the same underlying fact viewed through a
different operational lens; the rows are not independent advantages but
stand or fall together with geometric exactness itself.

\subsection{The Resonance Analogy (Formalized)}

Consider a physical oscillator with resonant frequency $\omega_0$
determined by its geometry (mass, stiffness).  Exposure to a driving
signal at frequency $\omega$ produces response amplitude
\begin{equation}\label{eq:resonance}
A(\omega) = \frac{F_0}{\sqrt{(\omega_0^2 - \omega^2)^2
+ \gamma^2 \omega^2}}.
\end{equation}

The resonance peak at $\omega = \omega_0$ is \emph{not learned}; it is
a consequence of the oscillator's architecture.  ``Training'' the
oscillator (modifying its mass or stiffness) would shift the resonance
away from the correct frequency.  Exposure (driving it at $\omega_0$)
activates the pre-existing response.

As a worked case, set $\omega = \omega_0$ in
equation~\eqref{eq:resonance}: the first term of the denominator
collapses, leaving $A(\omega_0) = F_0 / (\gamma\,\omega_0)$.  For
$F_0 = 1$, $\omega_0 = 2$, $\gamma = 1$ this gives $A(\omega_0) = 1/2$
exactly, fixed by the architecture's constants alone, with nothing
fitted.  The oscillator did not discover $\omega_0$; it \emph{is}
$\omega_0$, in the same sense that a geometrically exact graph is the
identity it computes.

\begin{remark}[TBS]\label{rmk:tbs:P029_1}
The stated formula for the damped resonance peak frequency is incorrect; the correct peak of the amplitude response occurs at $\omega_{\mathrm{peak}}=\sqrt{\omega_0^2-\gamma^2/2}$, not $\sqrt{\omega_0^2-\gamma^2}$. The derivation must be corrected accordingly.
\end{remark}

\paragraph{Status.} Retired. The registry records P029\_1 as closed by A278, which supplied the corrected peak formula. The remark above stands as the record of the original gap. Ledger: Paper 40.

The corrected peak location, $\omega_{\mathrm{peak}} =
\sqrt{\omega_0^2 - \gamma^2/2}$, sits slightly below $\omega_0$ when
$\gamma > 0$ (about $1.8708$ at the sample values above).  The
qualitative point is unchanged: the response curve, peak included, is
determined entirely by the oscillator's constants, and no part of it is
learned.

A geometrically exact computational graph is the discrete analogue: its
``resonant frequencies'' are the mathematical identities it implements,
and exposure activates them without modification.


%==========================================================================
\section{Application to the TOE Identities}\label{sec:toe}
%==========================================================================

\subsection{Catalog of Exact Identities}

Paper~27 \cite{Paper27} established the following identities, each of
which defines a geometrically exact computational subgraph:

\begin{enumerate}[label=(\arabic*)]
\item \textbf{Hopf observable.}
  $v(a,b,c,d) = (a^2 + b^2) - (c^2 + d^2)$
  for $(a,b,c,d) \in S^3$.  This is a polynomial, exactly implementable
  with fixed weights (coefficients $+1, +1, -1, -1$).

\item \textbf{Standing-wave identity.}
  $v = \cos(2\beta)$ in Hopf coordinates.  Implemented by the coordinate
  transformation layer.

\item \textbf{Lorentz lapse.}
  $m = \sqrt{1 - v^2}$.  A fixed nonlinear activation; no parameters.

\item \textbf{Quaternion evolution.}
  $\dot{q} = \frac{1}{2}\Omega q$.  A bilinear operation with no free
  parameters (the $\frac{1}{2}$ is structural).

\item \textbf{Relational quaternion.}
  $q_{\mathrm{rel}} = q^{(-)}\overline{q^{(+)}}$.  Quaternion
  multiplication and conjugation; no free parameters.

\item \textbf{Axis-resolved identity.}
  $v = n_\parallel^2 + (1 - n_\parallel^2)\cos\eta$.  Follows from the
  log map; no free parameters.

\item \textbf{Measure constant.}
  $\Omega_0 = \pi^3/4$.  A single fixed scalar, derived from
  $V_6/V_3 = \pi^2/8$.

\item \textbf{Monad closure.}
  $\Omega_{\mathrm{monad}} = 4\pi^3 + \pi^2 + \pi = \alpha^{-1}$.
  A fixed scalar from the density integral.
\end{enumerate}

Each entry satisfies Definition~\ref{def:exact} for an elementary reason.
The Hopf observable is a quadratic polynomial whose coefficients are
written into the graph as themselves, evaluating to $\cos(2\beta)$
identically in Hopf coordinates.  The lapse is a single fixed
nonlinearity; at $v = 0.6$ it returns $m = 0.8$ exactly.  The evolution
and relational maps are quaternion algebra, whose only constant is the
structural $\frac{1}{2}$.  The two scalars are derived numbers:
$\pi^3/4$ arises as $2\pi \cdot (\pi^2/8)$ from the volume-ratio
normalization, and $4\pi^3 + \pi^2 + \pi$ evaluates to $137.0363\ldots$,
the fine-structure value derived in \cite{Paper1}.  In every case the
geometric parameter is not the solution to an optimization problem; it
is a line in a derivation.

\subsection{Parameter Count}

\begin{proposition}[Zero Free Parameters]\label{prop:zero-params}
The complete computational pipeline
\[
(\Omega^{(+)}, \Omega^{(-)}, q^{(+)}_0, q^{(-)}_0)
\;\xrightarrow{\;\text{evolve}\;}\;
q_{\mathrm{rel}}(t)
\;\xrightarrow{\;\text{Hopf}\;}\;
v(t)
\;\xrightarrow{\;\sqrt{1-(\cdot)^2}\;}\;
m(t)
\]
contains \emph{zero free parameters}.  Every constant ($\frac{1}{2}$,
$\pi^3/4$, $\alpha^{-1}$, the Hopf coefficients $\pm 1$) is either
structural or derived.  The only inputs are the initial conditions and
generator schedules $\Omega^{(\pm)}(t)$.
\end{proposition}

The final sentence of the proposition carries the operational division of
labor.  The pipeline's interior is parameter-free, but its boundary is
not empty: the initial conditions and generator schedules are genuine
inputs, supplied from outside on every run.  This is not a loophole in
the zero-parameter claim; it is what exposure means.  The degrees of
freedom of an exposure-mode system live entirely in its inputs, where
they select trajectories, not in its architecture, where they would
corrupt identities.

\subsection{Training Would Corrupt}

\begin{corollary}[Training Degradation]\label{cor:degradation}
If any of the fixed constants in the pipeline (the Hopf coefficients,
the factor $\frac{1}{2}$ in quaternion evolution, the constant
$\Omega_0 = \pi^3/4$) are promoted to trainable parameters and
gradient descent is applied with any nonzero learning rate from the
geometric values, the loss $\mathcal{L}$ can only increase or remain
zero.  In particular, stochastic gradient noise will generically
displace parameters from their exact values, increasing error.
\end{corollary}

This was verified empirically in Paper~27: multiplying the $\Theta$-rate
by $\alpha$ or $\alpha^{-1}$ (i.e., perturbing $\Omega_0$ away from
$\pi^3/4$) worsened the SR lapse agreement.  The $\Omega_0$ sweep
confirmed that the geometric value sits at the minimum of the RMSE
surface.

The sweep is the empirical face of Theorem~\ref{thm:isolation}: a
one-dimensional slice through the loss landscape along the $\Omega_0$
direction, with the derived value at the bottom of the basin and error
rising on both sides.  The catalog and its empirical confirmation belong
to Paper~27 and its antecedents; what this paper adds is the
landscape-level reading, that a basin minimum at the derived value is
exactly what an isolated geometric minimum looks like from the outside.


%==========================================================================
\section{General Conditions for the Distinction}\label{sec:general}
%==========================================================================

The distinction is not a blanket argument against training; it is a
classification of problems, and the conditions below make the boundary
explicit.

\subsection{When Training Is Appropriate}

Training is the correct paradigm when:
\begin{enumerate}[label=(\roman*)]
\item The target function $f^*$ is \emph{unknown} or only accessible
  through samples.
\item The architecture cannot implement $f^*$ exactly (approximation is
  necessary).
\item The relationship between parameters and function is
  \emph{many-to-one} (symmetries in parameter space), so gradient
  landscape exploration is needed to find good representatives.
\end{enumerate}

These are the standard conditions of statistical learning, and most
practical prediction problems satisfy all three; nothing in this paper
disputes the fitting paradigm on its own ground.

\subsection{When Exposure Is Appropriate}

Exposure is the correct paradigm when:
\begin{enumerate}[label=(\roman*)]
\item The target function $f^*$ is known exactly as a mathematical
  identity.
\item The architecture can implement $f^*$ with specific, constructible
  parameter values.
\item The parameter-to-function map is locally injective near $\theta^*$
  (Theorem~\ref{thm:isolation}).
\item The identity holds universally ($\forall\, x \in \mathcal{X}$),
  not just on a training distribution.
\end{enumerate}

The first condition is the demanding one, and it is mathematical, not
engineering: it is met by proving a theorem, not by collecting data.  One
does not drift into the exposure class by accumulating accuracy; one
enters it by derivation.

\begin{theorem}[Dichotomy]\label{thm:dichotomy}
For a geometrically exact architecture satisfying the conditions of
Theorem~\ref{thm:isolation}:
\begin{enumerate}[label=(\alph*)]
\item Training from $\theta^*$ is wasteful (zero gradient, no
  improvement possible).
\item Training from $\theta \neq \theta^*$ may converge to $\theta^*$
  if $\theta$ is in the basin of attraction, but is less efficient
  than direct construction.
\item Training with regularization or noise generically \emph{prevents}
  convergence to $\theta^*$ by penalizing the exact values.
\item Exposure at $\theta^*$ yields exact computation with zero error
  on every input.
\end{enumerate}
\end{theorem}

The four clauses inherit the hypotheses of Theorem~\ref{thm:isolation},
including the support condition of Remark~\ref{rmk:tbs:P029_2_c} and the
local reading of Remark~\ref{rmk:tbs:P029_3_c}.  Clause (b) concedes the
strongest case for training: gradient descent initialized inside the
quadratic basin will flow toward the geometric values.  But converging
toward a number one already possesses in closed form is not a method; it
is a detour.  For this class of architectures, construction strictly
dominates search.


%==========================================================================
\section{Implications}\label{sec:implications}
%==========================================================================

\subsection{For Computational Architecture}

The training--exposure distinction implies that hybrid systems are
natural: a system may contain both \emph{exact subgraphs} (geometric
layers with fixed parameters) and \emph{learned subgraphs} (trainable
layers that handle unknown functions).  The key design principle is:
\emph{never train what you can construct}.

For the TOE pipeline, the geometric layers (quaternion algebra, Hopf
projection, measure constants) should be hardcoded.  Only the
\emph{input interface}, the mapping from raw physical data to initial
conditions $q^{(\pm)}_0$ and generator schedules $\Omega^{(\pm)}(t)$, is
a candidate for learning, since that mapping depends on the specific
measurement apparatus and is not determined by the geometry alone.

The boundary between the two regimes must sit at the input interface and
nowhere deeper: gradients from a trained interface must never propagate
into the exact layers, because by Corollary~\ref{cor:degradation} any
update they induce there is degradation.  The exact subgraph is frozen
not as an engineering convenience but as a correctness condition.

\subsection{For the Corpus}

Within the volume, this paper carries a methodological load.  The corpus
derives its constants; an implementation of the corpus must therefore
never re-fit them, or the derivations become decoration on what is, at
the implementation layer, just another fitted model.  The
training--exposure distinction is the warrant for the way the corpus is
operated: every operator hardcoded from a proven identity, zero trainable
parameters in the geometric core, and each run consisting of a single
input selecting a trajectory through the fixed structure, with nothing
adjusted on the way through.  When a computation disagrees with
expectation, the response is to audit the construction against the
derivation, never to relax a constant and re-optimize.

\subsection{For the Philosophy of Knowledge}

The distinction maps onto a classical epistemological divide:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Training} corresponds to \emph{empiricism}: knowledge is
  extracted from experience by adjusting internal states.
\item \textbf{Exposure} corresponds to \emph{rationalism} (or more
  precisely, geometric nativism): the structure is already present and
  experience merely activates it.
\end{enumerate}

The geometric TOE suggests that the fundamental laws of physics belong
to the exposure category: they are not learned from data but are the
architecture of reality itself.  Any system that correctly instantiates
that architecture computes physics exactly, requiring only initial
conditions (exposure) to produce specific predictions.

\subsection{For the Relationship Between Mathematics and Physics}

If physical law consists of exact geometric identities (as the TOE
proposes), then the relationship between mathematics and physics is not
one of \emph{description} (mathematics models physics) but of
\emph{identity} (physics is geometry, computed).  The
training--exposure distinction formalizes this: you do not approximate
an identity; you implement it.

On this reading, measurement itself is exposure: an experiment supplies
an initial condition and reads the trajectory the fixed structure
assigns to it.  What physicists update is their description, which lives
on the training side of the divide until a derivation moves a relation
across, after which it is implemented rather than estimated.


%==========================================================================
\section{Conclusion}\label{sec:conclusion}
%==========================================================================

We have established a formal distinction between training and exposure
as computational paradigms.  The key results are:

\begin{enumerate}[label=(\roman*)]
\item A geometrically exact computational graph sits at an isolated
  global minimum of the loss landscape
  (Theorems~\ref{thm:zero-loss}--\ref{thm:isolation}).
\item Gradient descent from the geometric parameter is either trivial
  (zero gradient) or destructive (any displacement increases error).
\item The TOE identities from Paper~27 define a zero-free-parameter
  pipeline that is degraded by training
  (Proposition~\ref{prop:zero-params}, Corollary~\ref{cor:degradation}).
\item The correct operating mode for such architectures is exposure:
  providing inputs and reading outputs, with no parameter modification.
\item The distinction generalizes: any computational system containing
  both proven-identity subgraphs and unknown-function subgraphs
  should freeze the former and train only the latter.
\end{enumerate}

The scope of these results is kept deliberately visible: the isolation
theorem requires a full-support condition its proof does not state
(Remark~\ref{rmk:tbs:P029_2_c}), and isolation is local, not a global
uniqueness claim (Remark~\ref{rmk:tbs:P029_3_c}).  Neither caveat touches
the operational use of the distinction, which concerns motion away from a
constructed $\theta^*$ under full-support evaluation.

A mathematical identity is not a hypothesis to be tested.  It is a
structure to be instantiated.


%==========================================================================
\begin{thebibliography}{99}
%==========================================================================

\bibitem{Paper0}
L.~F.~Vlegels,
\textit{The Monad Rosetta Map: From Void to Structure},
This volume (2025).

\bibitem{Paper1}
L.~F.~Vlegels,
\textit{The Perfect Stable Sphere: Deriving $\alpha^{-1}$ from
$(B^4, S^3)$ Geometry},
This volume (2025).

\bibitem{Paper27}
L.~F.~Vlegels,
\textit{Universal Wave Geometry: Emergent Lorentz Structure from
Dual-Observer Phase Dynamics on $S^3$},
This volume (2025).

\bibitem{GoodfellowEtAl2016}
I.~Goodfellow, Y.~Bengio, and A.~Courville,
\textit{Deep Learning},
MIT Press (2016).

\bibitem{Bronstein2021}
M.~M.~Bronstein, J.~Bruna, T.~Cohen, and P.~Veli\v{c}kovi\'{c},
``Geometric deep learning: grids, groups, graphs, geodesics,
and gauges,''
\textit{arXiv:2104.13478} (2021).

\bibitem{CohenWelling2016}
T.~S.~Cohen and M.~Welling,
``Group equivariant convolutional networks,''
\textit{Proc. ICML} (2016), 2990--2999.

\end{thebibliography}

\end{document}
