% Paper 30: The Geometric Observer Network — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: current canon; the GON architecture is implemented (kernel/gon/gon.py). Two confirmed-load-bearing items remain
% open: the norm-drift bound inconsistency (testable against the implementation, not yet run) and the unverified J3(O)
% conservation claim.
% Registry: P030_1 (scope), P030_2 (confirmed-load-bearing), P030_3_c (confirmed-load-bearing) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P030.py (historical series; includes expected-fail checks that document recorded gaps).

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\title{The Geometric Observer Network \\[0.5em]
\large A Zero-Parameter Architecture for Relativistic\\
Phase Dynamics from $S^3$ Geometry}

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

\begin{document}
\maketitle
\tableofcontents
\newpage

\begin{abstract}
We specify a concrete computational architecture, the Geometric Observer
Network ($\GON$), whose forward pass exactly computes the relativistic
lapse, Hopf observables, and phase dynamics established in the geometric
Theory of Everything.  The architecture contains zero trainable parameters:
every operation (quaternion multiplication, Hopf projection, log map,
measure scaling) is hardcoded from proven identities.  Inputs are initial
conditions and generator schedules; outputs are time-dependent observables
including the Lorentz lapse $\sqrt{1-v^2}$.  We define a rigorous
benchmarking protocol comparing the $\GON$ against trained multilayer
perceptrons (MLPs) of matched or greater capacity, and formulate three
falsifiable predictions: (1)~the $\GON$ achieves zero error on the Hopf
identity by construction; (2)~a trained MLP cannot reach zero error and
converges to a residual floor determined by its capacity; (3)~promoting
$\GON$ constants to trainable parameters and applying gradient descent
\emph{increases} mean error.  These predictions operationalize the
training--exposure distinction of Paper~28 as a reproducible experiment.
\end{abstract}


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

Paper~27 \cite{Paper27} derived a set of exact identities governing
dual-observer dynamics on $S^3$, and Paper~28 \cite{Paper28} established
that computational graphs implementing such identities are degraded by
gradient descent.  This paper completes the program by specifying the
architecture explicitly, defining the evaluation protocol, and stating
falsifiable predictions.

The term ``zero-parameter'' deserves precision at the outset.  Every
architecture contains numbers; the $\GON$ contains five fixed
geometric constants (Proposition~\ref{prop:params}) and the
coefficients of quaternion arithmetic.  What it does not contain is
any number set by fitting.  A parameter, in the sense used throughout
this paper, is a degree of freedom that an optimization procedure is
permitted to move, and the $\GON$ has none: each constant is the
unique value forced by a derivation, and perturbing any of them
breaks the identity the architecture computes.  There is no nearby
configuration that is ``almost as good''; correctness here is not a
scalar to be maximized but a property that either holds or fails.

The corpus needs such an object for three reasons.  First, an
explicit architecture makes the identities of Paper~27
\emph{executable}: a coding error, a wrong sign, or a misremembered
constant becomes immediately visible as a nonzero residual.  Second,
the training--exposure distinction of Paper~28 needs a concrete exact
graph as its experimental target; the $\GON$ supplies it.  Third,
downstream systems need an engine whose outputs carry a definite
epistemic status: a fitted engine delivers estimates, the $\GON$
delivers evaluations of theorems.

The Geometric Observer Network is not a neural network in the conventional
sense.  It has no hidden layers to be trained, no activation functions to
be chosen, no hyperparameters to be tuned.  It is a \emph{computational
graph} that implements quaternion algebra and the Hopf map with fixed
operations.  Its relationship to standard ML architectures is analogous to
the relationship between a hand-derived formula and a polynomial
regression: both can produce outputs from inputs, but one is exact by
construction while the other approximates.

Section~\ref{sec:arch} specifies the architecture layer by layer;
Section~\ref{sec:notraining} states the no-training property;
Sections~\ref{sec:benchmark} and~\ref{sec:predictions} define the
protocol and the predictions; Sections~\ref{sec:variants}
through~\ref{sec:discussion} cover variants, implementation, and
discussion.  Open items confirmed as load-bearing are recorded in
place, in the registry remarks and their status notes, rather than
deferred to an appendix.


%==========================================================================
\section{Architecture Specification}\label{sec:arch}
%==========================================================================

The $\GON$ is organized as six layers, evaluated by a single forward
pass; the difference from a trained network lies entirely in what the
layers do.  Layers~1 through~4 form the core pipeline: integrate two
observer states on $S^3$, form their relational quaternion, project
through the Hopf map, and read off the lapse.  Layers~5 and~6 are
optional stages for diagnostics and density coupling.  Each layer is
stated with its operations and its parameter count, which is in every
case zero.

\subsection{Inputs}

The $\GON$ accepts the following inputs at each evaluation:
\begin{enumerate}[label=(\roman*)]
\item Initial observer states:
  $q^{(+)}_0, q^{(-)}_0 \in S^3 \subset \R^4$
  (8 real numbers, subject to 2 unit-norm constraints).
\item Generator schedules:
  $\Omega^{(+)}(t), \Omega^{(-)}(t) \in \R^3$
  for $t \in [0, T]$, discretized at $N$ time steps
  (6$N$ real numbers).
\item Integration parameters: step size $h$, horizon $T$.
\end{enumerate}

\subsection{Layer 1: Quaternion Evolution}

For each observer $\sigma \in \{+, -\}$, the state is updated by the
Lie group integrator:
\begin{equation}\label{eq:evolution-step}
q^\sigma_{k+1} = \exp\!\Bigl(\frac{h}{2}\,\Omega^\sigma_k\Bigr)\,
q^\sigma_k,
\end{equation}
where the quaternion exponential of a pure imaginary quaternion
$\omega = \omega_x\mathbf{i} + \omega_y\mathbf{j} + \omega_z\mathbf{k}$
with $|\omega| = \phi$ is
\begin{equation}\label{eq:quat-exp}
\exp(\omega) = \cos\phi + \frac{\omega}{|\omega|}\sin\phi.
\end{equation}

\begin{remark}
Using the exponential map (rather than Euler integration) preserves the
unit-norm constraint $|q| = 1$ exactly at every step, eliminating drift.
This is a structural choice, not a numerical convenience.
\end{remark}

\textbf{Parameters:} None.  The factor $\frac{1}{2}$ is fixed by the
$\mathrm{SU}(2)$ Lie algebra structure.

Layer~1 is where the dynamics lives; everything downstream is
observation.  The integrator is a Lie group method in the sense of
geometric numerical integration \cite{HairerLubichWanner2006}:
instead of stepping in the ambient $\R^4$ and renormalizing, each
step multiplies the state by a group element, so the trajectory never
leaves $S^3$ in exact arithmetic.

\subsection{Layer 2: Relational Quaternion}

\begin{equation}\label{eq:q-rel-layer}
q_{\mathrm{rel},k} = q^{(-)}_k \cdot \overline{q^{(+)}_k}.
\end{equation}

\textbf{Operations:} Quaternion conjugation (negate imaginary part) and
quaternion multiplication (bilinear, 16 multiply-adds).

\textbf{Parameters:} None.

The relational quaternion is the object the identities of Paper~27
are stated on.  Everything downstream sees only $q_{\mathrm{rel}}$,
never the individual states: the architecture commits, structurally,
to the position that the physically meaningful quantity is the
relation between the observers rather than either observer alone.

\subsection{Layer 3: Hopf Projection}

From $q_{\mathrm{rel},k} = (a, b, c, d)$:
\begin{equation}\label{eq:hopf-layer}
v_k = (a^2 + b^2) - (c^2 + d^2).
\end{equation}

\textbf{Operations:} Four squarings, one addition, one subtraction.

\textbf{Parameters:} The coefficients $(+1, +1, -1, -1)$ are the Hopf
structure, not trainable weights.

The quantity $v_k$ is the Hopf latitude of the relational state:
writing $a^2 + b^2 = \cos^2\beta$ and $c^2 + d^2 = \sin^2\beta$ for
the Hopf angle $\beta$, equation~\eqref{eq:hopf-layer} reads
$v_k = \cos(2\beta)$.  For a unit quaternion, $v_k \in [-1, 1]$
automatically.  This is the quantity Paper~27 identifies with the
velocity observable, computed by six arithmetic operations with
hardwired signs.

\subsection{Layer 4: Lapse Computation}

\begin{equation}\label{eq:lapse-layer}
m_k = \sqrt{1 - v_k^2}.
\end{equation}

\textbf{Operations:} One squaring, one subtraction, one square root.

\textbf{Parameters:} None.

The lapse $m_k = \sqrt{1 - v_k^2}$ is the central output of the
network.  Because $v_k \in [-1, 1]$ for unit-norm input, the square
root is well defined wherever Layer~3 is.  Note what is absent: no
series expansion, no small-velocity approximation, no fitted
correction term.  The relativistic form is computed exactly because
it is an identity of the geometry, not a model of it.

\subsection{Layer 5: Log Map and Axis-Resolved Observable (Optional)}

For diagnostics and extended observables:
\begin{align}
r_k &= \arccos(a_k), \label{eq:logmap-r} \\
\hat{n}_k &= \frac{(b_k, c_k, d_k)}{|(b_k, c_k, d_k)|},
  \label{eq:logmap-n} \\
\eta_k &= 2r_k, \label{eq:rapidity} \\
n_{\parallel,k} &= \hat{n}_k \cdot \hat{e}_{\mathrm{Hopf}}.
  \label{eq:n-parallel}
\end{align}

The axis-resolved identity then provides a redundant check:
\begin{equation}\label{eq:axis-check}
v_k \stackrel{!}{=} n_{\parallel,k}^2 +
(1 - n_{\parallel,k}^2)\cos\eta_k.
\end{equation}

The redundancy of \eqref{eq:axis-check} is the point of the layer:
the same observable $v_k$ is computed along two independent
arithmetic routes, and the routes must agree to floating-point
precision at every step.  A zero-parameter architecture cannot be
validated by held-out data, since nothing was fitted; it is validated
by internal identities of exactly this kind.

\begin{remark}[TBS]\label{rmk:tbs:P030_1}
The Riemannian logarithmic map on $S^3$ is singular at the antipodal point of any given base point and is not globally defined on all of $S^3$; all results depending on the global log map must be restricted to the cut locus complement or reformulated using an alternative atlas.
\end{remark}

\paragraph{Status.} Recorded in the registry as P030\_1 (scope), confirmed by A295. The restriction to the cut locus complement is a domain hypothesis on the log-map observables, not a defect in the dynamics; corpus use is unaffected. Ledger: Paper 40.

\textbf{Parameters:} The Hopf axis $\hat{e}_{\mathrm{Hopf}} = (1, 0, 0)$
is determined by the complex-splitting convention, not by fitting.

\subsection{Layer 6: $\Theta$-Cycle and Monad Map (Optional)}

For coupling to the density structure:
\begin{align}
\dot{\Theta}_k &= \Omega_0\,\|\Omega_{\mathrm{rel},k}\|,
  \label{eq:theta-rate} \\
\Theta_k &= \Theta_{k-1} + h\,\dot{\Theta}_k,
  \label{eq:theta-accum}
\end{align}
with $\Omega_0 = \pi^3/4$ and the monad closure at
$\Omega_{\mathrm{monad}} = 4\pi^3 + \pi^2 + \pi$.

\textbf{Parameters:} $\Omega_0$ and $\Omega_{\mathrm{monad}}$ are fixed
constants derived in Paper~27 \cite{Paper27}.

The $\Theta$-cycle accumulates phase at a rate proportional to the
magnitude of the relational generator, with the proportionality fixed
by $\Omega_0$.  The closure constant
$\Omega_{\mathrm{monad}} = 4\pi^3 + \pi^2 + \pi$ is the expression
derived from $(B^4, S^3)$ geometry at the start of the corpus
\cite{Paper1}.  Both enter the layer as derived, to full precision.

\subsection{Complete Forward Pass}

\begin{equation}\label{eq:forward-pass}
\GON\bigl(q^{(\pm)}_0,\; \Omega^{(\pm)}_{0:N}\bigr)
\;=\; \bigl\{v_k,\; m_k,\; \eta_k,\; n_{\parallel,k},\;
\Theta_k\bigr\}_{k=0}^{N}.
\end{equation}

\begin{proposition}[Total Parameter Count]\label{prop:params}
The $\GON$ contains exactly zero trainable parameters and five fixed
geometric constants:
$\frac{1}{2}$ (Lie algebra factor),
$(+1,+1,-1,-1)$ (Hopf structure),
$\hat{e}_{\mathrm{Hopf}} = (1,0,0)$ (splitting convention),
$\Omega_0 = \pi^3/4$ (measure constant),
$\Omega_{\mathrm{monad}} = 4\pi^3 + \pi^2 + \pi$ (monad closure).
All are derived, none are fitted.
\end{proposition}

The proposition is checkable by inspection: the layer specifications
above are the complete architecture.  It is stated as a proposition
rather than a remark because it is the load-bearing structural fact
of the paper; every prediction in Section~\ref{sec:predictions} rests
on the network having exactly this inventory.


%==========================================================================
\section{The No-Training Property}\label{sec:notraining}
%==========================================================================

The $\GON$ is not merely untrained; it is an architecture for which
training is the wrong operation.  This section connects that
statement to the result of Paper~28 \cite{Paper28}.

Paper~28 established the training--exposure distinction: a
computational graph that implements an exact identity sits at an
isolated global minimum of the loss landscape, and gradient descent
applied to such a graph generically degrades it.  The intuition is
direct.  Optimization searches a neighborhood of configurations for
one with lower loss.  When the current configuration is exact, the
loss is already zero on clean data and every neighboring
configuration is strictly worse; any displacement, whether from
gradient noise, regularization pressure, or a finite learning rate,
moves the system off the identity.  For a fitted model, nearby
configurations are nearly as good, and training navigates among them
productively.  For an exact graph there is no ``nearly.''

The operational consequence is that the $\GON$ runs in exposure mode.
Inputs vary; the architecture does not.  A new generator schedule
updates nothing inside the network; it selects a different trajectory
through the same fixed graph, as evaluating a proven formula at a new
argument selects a different value without amending the proof.
Adaptation, where needed, belongs outside the geometric core
(Section~\ref{sec:variants}).

Two boundaries should be drawn honestly.  Zero trainable parameters
is a statement about the core architecture, not the full protocol,
which still chooses a step size, a horizon, and comparator
configurations.  And the property does not by itself prove that
training would degrade a parameterized copy of the $\GON$; that is
Conjecture~\ref{conj:degrade}, stated as a falsifiable prediction so
that it is settled by experiment rather than by appeal to Paper~28.


%==========================================================================
\section{Benchmarking Protocol}\label{sec:benchmark}
%==========================================================================

The protocol is not a horse race between architectures.  Its purpose
is to operationalize the training--exposure distinction as a
reproducible experiment: a task on which the $\GON$ is exact by
construction, a trained comparator of standard design, and metrics
under which the difference between exactness and approximation is
visible rather than averaged away.

\subsection{Task Definition}

\begin{definition}[Lapse Prediction Task]\label{def:task}
Given a generator schedule $\{\Omega^{(\pm)}_k\}_{k=0}^N$ and initial
conditions $q^{(\pm)}_0$, predict the time series
$\{m_k\}_{k=0}^N$ where $m_k = \sqrt{1 - v(q_{\mathrm{rel},k})^2}$.
\end{definition}

\subsection{Evaluation Scenarios}

The following scenarios span the kinematic range, matching those used in
the numerical experiments of Paper~27:

\begin{center}
\begin{tabular}{llc}
\toprule
\textbf{Scenario} & \textbf{Description} & \textbf{Regime} \\
\midrule
S1: Rest & $\Omega^{(+)} = -\Omega^{(-)}$, small & $v \approx 0$ \\
S2: Constant $v$ & Fixed generator ratio & $v = 0.6$ \\
S3: Acceleration & Smooth ramp + oscillation & $v \in [0, 0.85]$ \\
S4: Near-light & Large generator asymmetry & $v \to 0.999$ \\
\bottomrule
\end{tabular}
\end{center}

For each scenario, generate $M = 1000$ trajectories with random initial
conditions $q^{(\pm)}_0 \sim \mathrm{Uniform}(S^3)$ and $N = 2500$
time steps.

The scenarios are graded by difficulty for an approximator.  S4 is
the discriminating case: as $v \to 0.999$ the lapse becomes steep,
small errors in $v$ amplify into large errors in $m$, and an
approximator must spend capacity on a region the exact computation
handles with the same operations it uses everywhere else.

\subsection{Comparator: Trained MLP}

\begin{definition}[Baseline MLP]\label{def:mlp}
A multilayer perceptron with:
\begin{enumerate}[label=(\roman*)]
\item Input: $(\Omega^{(+)}_k, \Omega^{(-)}_k, q^{(+)}_k, q^{(-)}_k)
  \in \R^{14}$.
\item Hidden layers: $L$ layers of width $W$ with ReLU activation.
\item Output: $\hat{m}_k \in \R$.
\item Training: MSE loss, Adam optimizer, $10^5$ gradient steps.
\end{enumerate}
\end{definition}

Test configurations: $(L, W) \in \{(2, 64), (4, 128), (8, 256)\}$ to
span capacity from small to large.

The comparator is deliberately conventional and trained generously,
and it receives at each step the same per-step information the $\GON$
layers operate on.  The contrast being tested is therefore not one of
information but of structure: one system carries the geometry in its
architecture, the other must recover it from samples.

\subsection{Metrics}

\begin{enumerate}[label=(\roman*)]
\item \textbf{RMSE}: $\sqrt{\frac{1}{NM}\sum_{i,k}(m_{i,k} -
  \hat{m}_{i,k})^2}$.
\item \textbf{Max absolute error}: $\max_{i,k}|m_{i,k} -
  \hat{m}_{i,k}|$.
\item \textbf{Identity residual}: for the $\GON$, compute
  $|v_k - [n_\parallel^2 + (1-n_\parallel^2)\cos\eta_k]|$
  as an internal consistency check.
\item \textbf{Norm drift}: $\max_k \bigl||q_{\mathrm{rel},k}| - 1\bigr|$
  (should be $\sim 10^{-15}$ for exponential integrator).
\end{enumerate}


%==========================================================================
\section{Falsifiable Predictions}\label{sec:predictions}
%==========================================================================

Three predictions follow, stated as conjectures because each is
settled by running the protocol, not by argument.  The first concerns
the $\GON$ itself and would be falsified by any implementation error
or wrong constant; the second concerns trained approximators; the
third concerns what happens when the geometric constants are handed
to an optimizer, and is the direct experimental form of the Paper~28
result.

\begin{conjecture}[GON Exactness]\label{conj:exact}
The $\GON$ achieves:
\begin{enumerate}[label=(\alph*)]
\item RMSE $= 0$ (up to floating-point arithmetic, i.e.,
  $< 10^{-14}$) on the lapse prediction task across all scenarios
  S1--S4.
\item Identity residual $< 10^{-13}$ at every time step.
\item Norm drift $< 10^{-15}$.
\end{enumerate}
\end{conjecture}

\begin{remark}[TBS]\label{rmk:tbs:P030_2}
The claimed norm drift bound $<10^{-15}$ is inconsistent with the $O(\varepsilon\cdot N)$ estimate derived from the internal error propagation analysis; one of these estimates must be in error, and a coherent bound with proof must be supplied.
\end{remark}

\paragraph{Status.} Recorded in the registry as P030\_2 (confirmed-load-bearing), confirmed by A295. The inconsistency between the claimed $<10^{-15}$ bound and the $O(\varepsilon\cdot N)$ estimate stands. It is directly testable against the $\GON$ implementation; that test has not yet been run, and the item remains open until it is. Ledger: Paper 40.

This is ``falsifiable'' in the sense that a coding error or incorrect
constant would immediately produce nonzero error.

\begin{conjecture}[MLP Residual Floor]\label{conj:mlp-floor}
For any MLP configuration $(L, W)$ with finite capacity, there exists
a scenario-dependent residual floor $\epsilon(L, W) > 0$ such that:
\begin{enumerate}[label=(\alph*)]
\item $\mathrm{RMSE}_{\mathrm{MLP}} \geq \epsilon(L, W) > 0$.
\item $\epsilon$ decreases with capacity but does not reach zero.
\item The MLP generalizes poorly to scenarios not in its training set
  (e.g., trained on S1--S3, tested on S4).
\end{enumerate}
\end{conjecture}

\begin{conjecture}[Training Degradation]\label{conj:degrade}
If the five geometric constants of the $\GON$ are promoted to trainable
parameters and gradient descent is applied:
\begin{enumerate}[label=(\alph*)]
\item With clean data and zero-noise gradients: parameters remain at
  geometric values (zero gradient at minimum).
\item With stochastic mini-batch noise ($B < NM$): parameters drift
  from geometric values and RMSE increases monotonically with training
  steps.
\item With $L_2$ regularization ($\lambda > 0$): the regularizer
  penalizes the geometric constants (which are $O(\pi^3)$, not small),
  pulling them toward zero and increasing error.
\end{enumerate}
\end{conjecture}

%==========================================================================
\section{Architecture Variants}\label{sec:variants}
%==========================================================================

Three variants extend the base architecture, in decreasing order of
maturity: a design principle ready for use, a structured proposal,
and a sketch whose central claim is recorded below as open.

\subsection{Hybrid Architecture}

In practical applications, the generator schedule $\Omega^{(\pm)}(t)$
may not be known a priori but must be inferred from raw sensor data.
The hybrid architecture separates the system into:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Encoder} (trainable): maps raw input $x_{\mathrm{raw}}
  \in \R^d$ to $(\Omega^{(\pm)}, q^{(\pm)}_0)$.  This is a standard
  neural network, trained by gradient descent.
\item \textbf{Geometric core} (fixed): the $\GON$ layers 1--6, with
  all constants hardcoded.
\end{enumerate}

The design principle from Paper~28 applies: \emph{train the unknown,
freeze the proven}.  The boundary between the two components is the
boundary between what has been derived and what has not.  The encoder
faces a genuine unknown, the mapping from raw sensor data to geometric
inputs, and training is the right tool for it.  The core faces no
unknown at all, and the hybrid design keeps the optimizer strictly on
the encoder side of the interface.

\subsection{$Z_3$ Extension}

To incorporate the three-family structure from earlier papers
\cite{Paper6,Paper18}, the generator space is decomposed into three
sectors with a cyclic coupling:
\begin{equation}\label{eq:z3-extension}
\Omega^{(\pm)}_k = \sum_{j=0}^{2} \omega^j\,\Omega^{(\pm)}_{k,j},
\qquad \omega = e^{2\pi i/3}.
\end{equation}

This Hermitian cyclic structure was shown in the $Z_3$-phase experiments
to produce triplet splitting in the eigenvalue spectrum.  In the $\GON$
framework, it adds a discrete symmetry layer between the generator input
and the evolution layer, still with no trainable parameters.

\subsection{$J_3(\OO)$ Extension}

The upstream Jordan algebra embedding from Paper~27 suggests a
higher-dimensional variant where the state is a $3 \times 3$ Hermitian
matrix over an associative subalgebra of $\OO$, and the dual observers
are extracted from off-diagonal entries.  This extends the generator space
from $\R^6$ to $\R^{27}$ and introduces the cubic invariant $\det_J(X)$
as an additional conserved quantity alongside the quadratic $\Tr(X^2)$.

The dimension cascade
\begin{equation}\label{eq:cascade}
J_3(\OO)\;(27) \;\to\;
\R^6_\Omega \;\to\;
S^3 \;\to\;
S^2 \;\to\;
v \;\to\;
\sqrt{1-v^2}
\end{equation}
maps directly onto a layered computational graph with progressively
reducing dimensionality.

\begin{remark}[TBS]\label{rmk:tbs:P030_3_c}
The $J_3(\OO)$ extension claims that the cubic Jordan invariant $\det_J(X)$ is ``an additional conserved quantity alongside $\mathrm{Tr}(X^2)$'' in the $\GON$ evolution. This claim is not proved. Conserved quantities of a dynamical system are established by showing they commute with the flow (or satisfy a Noether argument); neither is provided here. The $J_3(\OO)$ extension modifies the generator space from $\R^6$ to $\R^{27}$ and involves the non-associative octonion algebra $\OO$; the $\GON$ evolution equations (Layers~1--6, all defined for associative quaternion arithmetic) do not obviously extend to $J_3(\OO)$, and the extension is sketched rather than specified. The dimension cascade diagram above is an informal illustration, not a derived result.
\end{remark}

\paragraph{Status.} Recorded in the registry as P030\_3\_c (confirmed-load-bearing), confirmed by A295. The $J_3(\OO)$ conservation claim is structural and unverified: no flow-commutation or Noether argument has been supplied here or in any later paper. The extension should be read as a sketch, not a result. Ledger: Paper 40.


%==========================================================================
\section{Implementation Notes}\label{sec:implementation}
%==========================================================================

\subsection{Numerical Precision}

The exponential-map integrator \eqref{eq:evolution-step} preserves
$|q| = 1$ exactly in exact arithmetic.  In floating-point, drift
accumulates at rate $O(\epsilon_{\mathrm{mach}} \cdot N)$.  For
$N = 2500$ steps in double precision ($\epsilon_{\mathrm{mach}} \approx
10^{-16}$), expected drift is $\sim 10^{-13}$, consistent with the
Hopf metric identity residuals observed in Paper~27.

This estimate is in tension with the $< 10^{-15}$ drift bound stated
in Conjecture~\ref{conj:exact}: the two cannot both be right as
written.  The inconsistency is recorded as P030\_2
(Remark~\ref{rmk:tbs:P030_2}) and is directly testable against the
implementation; the item remains open until that test is run.
Neither number is quietly adjusted here, because which estimate is in
error is precisely what the test will determine.

\subsection{Computational Cost}

Per time step, the $\GON$ requires:
\begin{center}
\begin{tabular}{lr}
\toprule
\textbf{Operation} & \textbf{FLOPs} \\
\midrule
Quaternion exponential ($\times 2$ observers) & $\sim 40$ \\
Quaternion multiplication ($\times 2$ evolutions) & $\sim 32$ \\
Conjugation + relational product & $\sim 20$ \\
Hopf projection & $\sim 8$ \\
Lapse computation & $\sim 4$ \\
\midrule
\textbf{Total per step} & $\sim 104$ \\
\bottomrule
\end{tabular}
\end{center}

For $N = 2500$ steps: $\sim 2.6 \times 10^5$ FLOPs total.  This is
orders of magnitude cheaper than a single forward pass through even a
small MLP on the same input sequence.

\subsection{Reproducibility}

The architecture is fully deterministic given inputs.  No random
initialization, no stochastic training, no hyperparameter search.
Two implementations of the $\GON$ that correctly implement quaternion
arithmetic will produce bitwise-identical outputs (in the same floating
point environment).


%==========================================================================
\section{Discussion}\label{sec:discussion}
%==========================================================================

\subsection{What the $\GON$ Is Not}

The $\GON$ is not a universal function approximator.  It computes a
specific set of geometric identities.  It cannot learn new functions,
adapt to distribution shift, or generalize beyond its mathematical
domain.  It is a \emph{calculator}, not a \emph{learner}.

This is precisely its strength.  Within its domain, it is exact.  Outside
its domain, it is silent: it produces no output rather than a wrong
output.  The failure mode of the $\GON$ is ``not applicable,'' never
``approximately wrong.''

Silence is the property that universal approximators structurally
lack.  A trained network always returns an answer; far from its
training distribution it extrapolates, and the answer degrades
without announcing that it has degraded.  The $\GON$'s contract is
the opposite: applicability is decided by whether the input lies in
the mathematical domain of the identities, before evaluation, not
graded afterward by how plausible the output looks.  Two caveats
attach.  The domain has genuine boundaries, of which the log-map
restriction P030\_1 (Remark~\ref{rmk:tbs:P030_1}) is the concrete
instance here.  And silence is a specification discipline, not a
mechanism the formulas enforce by themselves; an implementation must
validate its domain at the boundary, since the arithmetic will
evaluate whatever it is handed.

\subsection{What the $\GON$ Demonstrates}

The existence of a zero-parameter architecture that exactly computes
relativistic phase dynamics demonstrates that:
\begin{enumerate}[label=(\roman*)]
\item The identities are computationally real, not merely algebraic
  curiosities.
\item The ``unreasonable effectiveness of mathematics'' has a
  computational counterpart: proven identities can be hardcoded, and
  the resulting system outperforms any trained approximation within
  the identity's domain.
\item The training--exposure distinction from Paper~28 has concrete
  architectural consequences.
\end{enumerate}

The first point deserves emphasis: an identity wired into an
executable graph is checked every time the graph runs, along
redundant routes (equation~\eqref{eq:axis-check}).  The $\GON$ turns
the corpus's central kinematic results from claims that must be
trusted into computations that can be watched.

\subsection{Position in the Corpus}

The $\GON$ is the executable terminal of a three-paper arc.
Paper~27 \cite{Paper27} derived the identities: the relational
quaternion, the Hopf observable, the emergent Lorentz lapse, the
measure constants.  Paper~28 \cite{Paper28} established what those
identities imply for computation.  This paper turns both into an
artifact.  Its constants reach further back: the monad closure
$4\pi^3 + \pi^2 + \pi$ is the expression of \cite{Paper1}, and the
$Z_3$ and $J_3(\OO)$ variants connect to the three-family and
master-operator structure of \cite{Paper6,Paper18}.  The architecture
adds no new physics; its contribution is that the physics already
derived can now be run.

\subsection{Relationship to Geometric Deep Learning}

The $\GON$ shares the \emph{philosophy} of geometric deep learning
\cite{Bronstein2021}, encoding symmetries and invariances into
architecture rather than learning them from data, but takes it to its
logical endpoint.  In geometric deep learning, equivariance is a
\emph{constraint on trainable layers} (e.g., group-equivariant
convolutions \cite{CohenWelling2016}).  In the $\GON$, the geometry is
the \emph{entire computation}, and there is nothing left to train.
The progression is natural: as more of a problem's structure is known
exactly, the trainable fraction of the architecture shrinks.  The
$\GON$ is the limit point of that progression for a problem whose
structure is known completely, and the hybrid architecture of
Section~\ref{sec:variants} shows that the limit point composes with
conventional practice rather than replacing it.


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

We have specified the Geometric Observer Network: a zero-parameter
computational architecture whose forward pass is the physics of
dual-observer phase dynamics on $S^3$.  The architecture is:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Exact:} it computes proven identities, not approximations.
\item \textbf{Parameter-free:} every constant is derived from geometry.
\item \textbf{Efficient:} $\sim 100$ FLOPs per time step.
\item \textbf{Falsifiable:} three concrete predictions distinguish it
  from trained alternatives.
\item \textbf{Extensible:} hybrid, $Z_3$, and $J_3(\OO)$ variants
  are structurally defined.
\end{enumerate}

Three items remain open and are recorded in place.  P030\_1
(Remark~\ref{rmk:tbs:P030_1}) is a scope restriction: the log-map
observables carry a cut locus complement domain hypothesis.  P030\_2
(Remark~\ref{rmk:tbs:P030_2}) is a confirmed-load-bearing
inconsistency between the stated norm-drift bound and the
error-propagation estimate, testable and untested.  P030\_3\_c
(Remark~\ref{rmk:tbs:P030_3_c}) is the unverified conservation claim
of the $J_3(\OO)$ sketch.  None of the three touches the core
pipeline of Layers~1 through~4; all three bound what may be claimed
beyond it.

The $\GON$ is not a model of physics.  It is a direct computational
instantiation of the geometric identities proven in Papers~27--28.
Its outputs are not predictions; they are theorems, evaluated.


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

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L.~F.~Vlegels,
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\bibitem{Paper6}
L.~F.~Vlegels,
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\bibitem{Paper18}
L.~F.~Vlegels,
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\bibitem{Paper27}
L.~F.~Vlegels,
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\bibitem{Paper28}
L.~F.~Vlegels,
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\end{thebibliography}

\end{document}
