Measured inspection-memory admission and information boundaries

\section{Measured inspection-memory admission and information boundaries}
\label{sec:inspection-memory-20260914}

\paragraph{Capability and scope.}
The motivating capability is to retain compact physical evidence after
acquisition and answer a subsequently requested inspection question without
retraining a central model. A September 14, 2026 admission screen tests a
narrower prerequisite: held-view prediction from three simulated acquisition
groups on one real walnut CT slice. It does not establish defect detection,
new-object generalization, autonomous hardware, or an industrial inspection
system. Source measurements and the official finite-domain fan-beam matrix
come from H\"am\"al\"ainen et al.,
\emph{Tomographic X-ray data of a walnut},
\url{https://arxiv.org/abs/1502.04064}; payload provenance is
\url{https://zenodo.org/records/1254206}.

\paragraph{Frozen acquisition and controls.}
The $82\times82$ representation uses the supplied $9840\times6724$ matrix.
Of 120 views, zero-based indices $j\bmod4\ne3$ give 90 acquisition views;
$j\bmod8=3$ gives 15 development-validation views. Fifteen views with
$j\bmod8=7$ remain unscored. Three donors each receive 30 acquisition views.
Cardinal cubic and cosine spaces have equal dimensions 324, 676 and 1156;
exact one-dimensional pixel averages form the separable field map $B$.
This is not an exact continuous ray integral: the official pixel operator
acts on those pixel averages. Quadratic regularization uses the respective
continuous gradient-energy Gram, without an empirical circulant assumption.
Raw float32, 16-bit and 8-bit messages use the same acquisition data and
receive classical quadratic and nonnegative total-variation (TV) controls.
All tested regularization choices are reported; validation selects parameters.
The protocol was pushed before scoring, at checkpoint \texttt{29b940dc1}.

\begin{center}
\begin{tabular}{lrrr}
\hline
Representation & Dimension & Message bytes & Relative RMSE \\
\hline
Cardinal cubic & 324 & 4116 & 0.108249 \\
Cosine & 324 & 4116 & 0.092104 \\
Cardinal cubic & 676 & 8340 & 0.078360 \\
Cosine & 676 & 8340 & 0.069532 \\
Cardinal cubic & 1156 & 14100 & 0.063146 \\
Cosine & 1156 & 14100 & 0.055151 \\
Raw 8-bit + TV & --- & 7608 & 0.042262 \\
Raw 16-bit + TV & --- & 14988 & 0.042184 \\
Raw float32 + TV & --- & 29748 & 0.042184 \\
\hline
\end{tabular}
\end{center}
Errors concern measured development views, not ground-truth image error.
Bytes include three donors' numeric metadata, not file-container overhead
or shared operators. Every cardinal point is dominated by its matched cosine
point. The 8-bit raw control is smaller and more accurate than the two larger
cardinal messages. This rejects the tested compression route, not all splines.

\paragraph{Fixed-space memory allows changing a prior.}
Let $H_i=A_iB$ include any fixed known weighting/scaling and let
$J_i=H_i^\top H_i$, $b_i=H_i^\top y_i$. For any subsequent coefficient prior
or extended-valued constraint $R(c)$,
\[
 \frac12\sum_i\|H_ic-y_i\|^2+R(c)
 =\frac12c^\top\Big(\sum_iJ_i\Big)c-
 \Big(\sum_i b_i\Big)^\top c+R(c)+\frac12\sum_i\|y_i\|^2.
\]
Thus $(J,b)$ suffice for minimization over this fixed space even when $R$
is nonlinear. Absolute objective values additionally need the constant term;
unknown noise models or changing observation weights require further care.
This is a least-squares identity, not a validation of a Gaussian physical
noise model for this dataset.

The recipient implementation consequently accepts only $J$, $b$, $B$ and
the prior weight, not raw measurements or the ray operator. For
$R(c)=\lambda\operatorname{TV}(Bc)+\iota_{Bc\ge0}$, ordinary ADMM splits
$z=Bc$ and $v=DBc$. A cached factor solves
\[
 [J+\rho B^\top B+\rho B^\top D^\top DB]c
 =b+\rho B^\top(z-u)+\rho B^\top D^\top(v-t).
\]
Pixel projection and isotropic shrinkage provide the other updates.
For a separable field, the additional normal matrices are Kronecker sums of
one-dimensional product matrices. This exploits compiled inner products but
uses discrete pixel TV, not exact continuous TV of the underlying functions.
ADMM itself is established optimization; see Boyd et al. (2011),
\url{https://web.stanford.edu/~boyd/papers/admm_distr_stats.html}.

\paragraph{Prior-matched diagnostic, not confirmation.}
A separately frozen diagnostic (checkpoint \texttt{d1d917da9}) keeps the
six existing spaces and uses the raw control's five TV weights. At its stated
budget the best cardinal errors are $0.116842$, $0.084057$ and $0.061998$;
cosine errors are $0.101005$, $0.074667$ and $0.055811$. Fixed tighter checks
give respectively $0.117391$, $0.084271$, $0.062097$ and $0.101446$,
$0.074864$, $0.055911$. All six tighter checks miss their strict primal
feasibility tolerance, with small negative pixels remaining. These are
bounded-computation comparisons, not certificates of constrained optimality.
Extending the already selected raw TV solves to 7200 iterations without
reselecting penalties gives $0.04224944$ (float32) and $0.04230795$ (8-bit).
The tested nonlinear prior does not reverse the ranking.

Float32 retained-statistic and full-statistic recipient reconstructions
differ by at most $2.13\cdot10^{-6}$ relative image norm for cardinal spaces
and $3.30\cdot10^{-8}$ for cosine spaces under matched updates. This supports
the implementation's fixed-space reuse contract, not a unique spline benefit.

\paragraph{Information-equivalent fitted models.}
If $J_i+\lambda Q$ is invertible and the donor transmits
$c_i=(J_i+\lambda Q)^{-1}b_i$, then the known matrices recover
$b_i=(J_i+\lambda Q)c_i$. Model coefficients and right-hand sides therefore
need not differ in retained information. The largest cardinal coefficient
message gives held-view error $0.06314598$, versus $0.06314595$ from its
right-hand-side message. A naive average of independently fitted predictions
is not an adequate control for a claim of uniquely reusable evidence.

\paragraph{Refining functions does not recover missing evidence.}
Suppose a prospective new measurement feature $h$ is not in the column
space of the old feature matrix $H_0$. Let
$r=(I-P_{H_0})h\ne0$, with $P_{H_0}$ the orthogonal projector. The observation
vectors $y$ and $y+r$ have identical $H_0^\top y$, while their new statistic
differs by $h^\top r=\|r\|^2>0$. No deterministic decoding of the old
right-hand side and known geometry can recover the correct new statistic
for both observations. This is a linear-algebra counterexample, not a claim
that the constructed perturbation is a physically realizable new specimen.
If instead $H_1=H_0T$, both $b_1=T^\top b_0$ and $J_1=T^\top J_0T$ are
available. General enrichment needs retained extra statistics, sketches with
appropriate guarantees, or charged additional measurements.

\paragraph{Resource accounting and decision.}
The official operator uses 11,671,028 numeric bytes. Donor normal matrices
and quadratic prior add 3,359,232--42,762,752 bytes; the nonlinear diagnostic
adds 1,679,616--21,381,376 bytes for its constraint matrix and factor.
The complete admission and diagnostic take 30.24 and 40.00 seconds and peak
at 442.69 and 372.58 MiB RSS, respectively, on one CPU numerical worker.
These are local run receipts, not cross-hardware performance claims.
Eight primitive and information-contract tests pass. Full trial records,
source/data hashes, solver residuals and the corrected serialization incident
are retained in \texttt{inspection\_memory\_20260914/}.

The tested fixed-space compression mechanism is closed without opening the
reserved views. A future inspection proposal must earn its role through an
expensive physical decision, such as fewer additional measurements for a
new local defect question under controlled false confidence, and compare
against compressed-raw reconstruction with classical targeted acquisition.
No such capability, labelled-defect result or new acquisition algorithm is
established here. The broad vision remains a goal, not a conclusion inferred
from these negative admission experiments.