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\begin{document}
\title{FrostCell 4$\times$ Influence\\\large H2-092 48-to-12 Node-Local Response-Context Multiplicity Audit}
\author{Frostyverse Laboratory}
\date{Website Laboratory v1.0.48}
\maketitle

\statusbox{\textbf{Scientific status.} H2-092 is an FV-only structural audit inside the declared Frostyverse software mechanics. Its result is an exact fourfold \emph{node-local response-context multiplicity}. It does not create four independent energy reservoirs. H2-090/H2-091 conservation ownership remains one D-scale conservative FrostTransit handoff. No GR, observed precession value, or other external physics target is used to select the H2-092 result.}

\begin{abstract}
This note exposes the mathematics executed by the public FrostCell 4$\times$ Influence Python Test. A FrostCell has eight corner FrostNodes, twelve unique internal FrostSegment edges, and six signed H-021 support contexts at each node. H2-092 asks whether the apparent count ratio $48/12=4$ survives the inherited squared-transverse response weighting and the exact edge-to-node-context map. The focused wrapper permits arbitrary travel direction and a nonsingular $3\times3$ affine FrostCell deformation matrix. The unchanged historical package additionally tests six inherited cell families, all-eight-node participation, direct H-021 opening mechanics, and 822 cell--direction cases. Deliberately altered ownership/context definitions give exact controls of three and two.
\end{abstract}

\section{Question under attack}
A bare count
\[
\frac{8\ \text{nodes}\times6\ \text{local supports}}{12\ \text{internal edges}}=4
\]
is not, by itself, a mechanical result. The audit therefore asks a sharper question:

\begin{quote}
If the same H-021 transverse response law is applied to the twelve-edge FrostCell ledger and to all node-local support contexts, does the four survive direction changes and cell deformation, and does it disappear in the expected way when the bookkeeping definition is changed?
\end{quote}

The comparison is entirely internal to FV. The desired value four is not selected by a GR/precession ruler. The historical falsification order explicitly closes those external targets during H2-092.

\section{FrostCell geometry used by the public wrapper}
\subsection{Affine cell construction}
Let a logical FrostCell corner be indexed by
\[
\mathbf b=(b_x,b_y,b_z),\qquad b_i\in\{0,1\}.
\]
The centered logical coordinate is
\[
\mathbf q_{\mathbf b}=\mathbf b-\left(\frac12,\frac12,\frac12\right).
\]
For a user-supplied affine deformation matrix $A\in\mathbb R^{3\times3}$, the physical corner is
\[
\boxed{\mathbf r_{\mathbf b}=A\mathbf q_{\mathbf b}}.
\]
The wrapper rejects singular or nearly singular matrices,
\[
|\det A|<10^{-9},
\]
so the three transformed edge families remain linearly independent.

The public convenience presets are a flat cube, mild shear, anisotropic stretch, and a skewed cell. A reviewer may instead enter all nine matrix components directly.

\subsection{Twelve unique internal edges}
Two corners form an internal edge if their logical bit labels differ in exactly one coordinate. Counting every physical cell edge once gives
\[
\boxed{N_{\rm edge}=12}.
\]
For each edge $e$, with endpoint positions $\mathbf r_a$ and $\mathbf r_b$, define one oriented unit edge direction
\[
\hat{\mathbf s}_e
=\frac{\mathbf r_b-\mathbf r_a}
       {\|\mathbf r_b-\mathbf r_a\|}.
\]
The sign chosen for this twelve-edge ledger is immaterial to the response rule below.

\section{Inherited H-021 transverse response}
The user enters a nonzero FrostTransit travel direction $\mathbf u$, which is normalized internally:
\[
\hat{\mathbf u}=\frac{\mathbf u}{\|\mathbf u\|}.
\]
For one local support direction $\hat{\mathbf s}$, H2-092 uses the inherited squared transverse-opening weight
\[
\boxed{
w(\hat{\mathbf u},\hat{\mathbf s})
=1-(\hat{\mathbf u}\cdot\hat{\mathbf s})^2
}.
\]
Equivalently,
\[
w=\sin^2\theta,
\]
where $\theta$ is the angle between travel and support directions. Therefore
\[
\boxed{w(\hat{\mathbf u},-\hat{\mathbf s})
=w(\hat{\mathbf u},\hat{\mathbf s})}.
\]
This sign symmetry is central to the exact response-context identities tested by the code.

\section{The four response ledgers}
\subsection{Twelve-edge baseline}
The one-cell internal-edge response is
\[
\boxed{
S_{12}=\sum_{e=1}^{12}
w(\hat{\mathbf u},\hat{\mathbf s}_e)
}.
\]
Because a nonsingular affine cell has three independent edge families, $S_{12}$ cannot vanish for a nonzero travel direction. The wrapper nevertheless guards against a numerically degenerate denominator.

\subsection{Twenty-four internal endpoint contexts}
Each physical internal edge has two endpoint FrostNodes. Looking inward along the edge from the two endpoints produces the signed pair
\[
\{\hat{\mathbf s}_e,-\hat{\mathbf s}_e\}.
\]
Thus the 24 internal endpoint contexts have response
\[
\begin{aligned}
S_{24}
&=\sum_{e=1}^{12}
\left[
w(\hat{\mathbf u},\hat{\mathbf s}_e)
+w(\hat{\mathbf u},-\hat{\mathbf s}_e)
\right]\\
&=2S_{12}.
\end{aligned}
\]
Hence
\[
\boxed{M_{24}=\frac{S_{24}}{S_{12}}=2}.
\]
This is the internal-endpoint-only negative control.

\subsection{Twenty-four opposing external partner contexts}
H-021's six-support local geometry supplies an opposing partner for each of those 24 inward endpoint contexts. In the focused wrapper these partner directions are the negatives of the corresponding inward directions, so the squared-transverse weight is unchanged:
\[
S_{24}^{\rm ext}=S_{24}=2S_{12}.
\]

\subsection{All 48 node-local contexts}
The complete six-support node-local response contains both sets:
\[
S_{48}=S_{24}+S_{24}^{\rm ext}=4S_{12}.
\]
Therefore
\[
\boxed{
M_{48}=\frac{S_{48}}{S_{12}}=4
}.
\]

This is stronger than merely printing $48/12=4$: the same direction-dependent response weight is applied before the ratio is formed. At the same time, the algebra above makes clear \emph{why} the weighted result is exact once the H-021 context map is accepted: every internal edge response appears in four sign-related node-local contexts, and $w$ is even under support reversal.

\subsection{Thirty-six physical-segment proxy control}
If node-local response contexts are instead deduplicated toward physical-segment ownership, the historical mapping contains the 12 unique internal edges plus 24 external neighboring partners. The wrapper scores
\[
S_{36}=S_{12}+S_{24}^{\rm ext}=3S_{12},
\]
so
\[
\boxed{
M_{36}=\frac{S_{36}}{S_{12}}=3
}.
\]
The $4/3/2$ pattern therefore distinguishes the full node-local response inventory from two deliberately altered ledgers.

\section{Transparent flat-cell reference calculation}
The published focused reference uses
\[
A=I,
\qquad
\mathbf u=(2,-1,3),
\qquad
\hat{\mathbf u}
=\frac{(2,-1,3)}{\sqrt{14}}.
\]
For a flat cube there are four parallel internal edges in each Cartesian direction. Hence
\[
\begin{aligned}
S_{12}
&=4\left[(1-u_x^2)+(1-u_y^2)+(1-u_z^2)\right]\\
&=4\left[3-(u_x^2+u_y^2+u_z^2)\right]\\
&=4(3-1)=8.
\end{aligned}
\]
The remaining sums follow directly:
\[
\boxed{
S_{12}=8,
\quad
S_{24}=16,
\quad
S_{36}=24,
\quad
S_{48}=32
}.
\]
Thus the public reference must report
\[
\boxed{
M_{48}=4,
\qquad
M_{36}=3,
\qquad
M_{24}=2
}.
\]
The wrapper PASS tolerance is
\[
\max\bigl(|M_{48}-4|,|M_{36}-3|,|M_{24}-2|\bigr)<10^{-11}.
\]
A fresh reference run gives zero printed error in all three lanes.

\section{Fresh deformed-cell attack}
The focused wrapper is not limited to the flat cube. For example, its public skewed preset is
\[
A=
\begin{pmatrix}
1.00&0.22&-0.08\\
-0.12&1.00&0.14\\
0.09&-0.07&0.94
\end{pmatrix},
\qquad
\det A=0.983916.
\]
For a fresh direction $\mathbf u=(1.7,-0.4,2.3)$, the weighted sums are no longer the flat-cell values:
\[
\begin{aligned}
S_{12}&=8.144977490486914,\\
S_{24}&=16.289954980973828,\\
S_{36}&=24.434932471460741,\\
S_{48}&=32.579909961947656.
\end{aligned}
\]
Nevertheless the response multiplicities remain
\[
M_{48}=4,
\qquad
M_{36}=3,
\qquad
M_{24}=2
\]
to floating-point precision. A 64-direction Fibonacci-sphere stress of the same skewed cell gives maximum errors of $0$ for $M_{48}$, approximately $4.44\times10^{-16}$ for $M_{36}$, and $0$ for $M_{24}$ in the tested run.

\statusbox{\textbf{Interpretation of fresh-input tests.} For this particular wrapper, changing direction or applying a nonsingular affine deformation attacks the implementation and context mapping, but it does not change the exact algebraic identity once the context sets are constructed as declared. A reviewer attempting to break the claim should therefore inspect the physical legitimacy of the H-021 context map and response rule as well as numerical execution.}

\section{Historical H2-092 checks beyond the focused wrapper}
The unchanged archive at \path{original/Frostyverse-Test-H2-092.zip} contains the publication-era H2-092 program and inherited H-015/H-017/H-021 engine files. It adds checks that the lightweight wrapper intentionally does not reimplement.

\subsection{All-eight-node geometric participation}
H2-092 reconfirms the preceding H2-091 result with trilinear corner weights. For logical position $(x,y,z)$ inside a cell,
\[
w_{ijk}(x,y,z)=
[x^i(1-x)^{1-i}]
[y^j(1-y)^{1-j}]
[z^k(1-z)^{1-k}],
\]
with
\[
\sum_{i,j,k\in\{0,1\}}w_{ijk}=1.
\]
For ordinary interior paths, all eight weights remain nonzero. The historical full H2-092 run reports a maximum partition-of-unity error of approximately
\[
2.220\times10^{-16}.
\]

\subsection{Inherited cell families and direct H-021 control}
The historical test uses one flat plus five inherited deformed FrostCell families, named directions, and a Fibonacci-sphere direction set. It also evaluates the H-021 opening rule directly rather than relying only on the wrapper's compact edge/context reconstruction.

The validation profile uses 16 sphere directions and reports 150 cell--direction cases. The full profile uses 128 sphere directions and reports
\[
\boxed{822\ \text{cell--direction cases}}.
\]
The full-profile result is
\[
M_{48}=4,
\qquad
M_{36}=3,
\qquad
M_{24}=2
\]
throughout the tested set, with decision
\begin{center}
\code{A\_48\_TO\_12\_RESPONSE\_CONTEXT\_FOUR\_SURVIVES}.
\end{center}
Its sealed historical fingerprint is
\begin{quote}\footnotesize
\path{94d127bbf6ee6631dd2082a9bbbe49929d9d789ea837bd50268f9a3deedd9d78}
\end{quote}

\section{Why the result is not four copies of energy}
The H2 chronology fixes conservative work ownership \emph{before} H2-092. H2-090/H2-091 retain one D-scale conservative handoff ledger, schematically
\[
W_{\rm source}=0.5D,
\qquad
W_{\rm destination}=0.5D,
\qquad
W_{\rm handoff}=D.
\]
All-eight-node participation partitions that same handoff; H2-092 does not multiply its stored work.

The publication-safe statement is therefore
\[
\boxed{
\text{one D-scale conservative handoff}
\times
\text{fourfold node-local response-context multiplicity}
}.
\]
When imported into the frozen weak-field precession closure, this is used as an \emph{effective} $+4D$ response. It is not four independent D-energy channels.

\section{No external comparison ruler}
Unlike the Photon Fly-by, Shapiro, Redshift, or Precession Laboratories, H2-092 has no GR/SR comparison equation to open after the FV result. Its question is purely structural:
\[
\text{Does the declared H-021 node-local response bookkeeping preserve }4?
\]
Accordingly, no observed perihelion advance, GR coefficient, or target amplitude enters the decision. A later theory-level use of the multiplicity in precession is separate from this audit.

\section{How to reproduce the public Python test}
Python 3.10 or newer is recommended. The wrapper requires no third-party packages.

From the extracted \path{Frostyverse_FrostCell_4x_Influence_Python_Test} directory:

\textbf{Interactive menu}
\begin{verbatim}
python lab_test.py
\end{verbatim}

\textbf{Focused published reference}
\begin{verbatim}
python lab_test.py --reference
\end{verbatim}

\textbf{Custom direction and FrostCell deformation}
\begin{verbatim}
python lab_test.py --custom
\end{verbatim}

\textbf{Fresh distributed-direction stress}
\begin{verbatim}
python lab_test.py --stress
\end{verbatim}

\textbf{Unchanged historical validation / full profile}
\begin{verbatim}
python lab_test.py --validation
python lab_test.py --full
\end{verbatim}

\textbf{Verify the bundled historical archive}
\begin{verbatim}
python lab_test.py --verify
\end{verbatim}

\textbf{Show command-line help}
\begin{verbatim}
python lab_test.py --help
\end{verbatim}

Windows users may double-click \path{run_test.bat}; macOS/Linux users may run \path{run_test.sh}. Focused and historical runs save timestamped TXT/JSON records under \path{results/}.

\section{What a reviewer should try to break}
Useful attacks include:
\begin{itemize}
\item enter arbitrary non-axis-aligned travel directions;
\item use the supplied shear/stretch/skew presets;
\item enter a fresh nonsingular $3\times3$ affine deformation matrix;
\item stress one deformation over many distributed directions;
\item inspect whether the 12-edge, 24-endpoint, 24-partner, 36-physical-proxy, and 48-node-context sets really correspond to the declared H-015/H-021 mechanics;
\item verify that the $3$ and $2$ controls arise from their altered bookkeeping definitions rather than from hard-coded output values;
\item run the unchanged historical validation and full profiles; and
\item challenge the physical interpretation of node-local response contexts even if the numerical identities reproduce exactly.
\end{itemize}

A fresh-input failure is scientifically useful. If a nonsingular input breaks the declared mapping or a historical result cannot be reproduced, the corresponding claim should be revised rather than the test adjusted to force a PASS.

\section{Provenance and integrity}
The unchanged research artifact bundled with the public Python Test is
\begin{quote}\footnotesize
\path{original/Frostyverse-Test-H2-092.zip}\\
SHA-256: \path{f7352267aa6e6394d7dcc9b72bf6928dcbf1c4ccc655c1b8d072bbacb3afbcb2}
\end{quote}
The reviewer-facing wrapper itself is listed in \path{SHA256SUMS.txt}; in the current public package its \path{lab_test.py} SHA-256 is
\begin{quote}\footnotesize
\path{810f1e66e1955f016f7cbe176443c5b03f55c2458ca73c0a146f1f95b5ae64a6}.
\end{quote}
The distributed Python Test ZIP has SHA-256
\begin{quote}\footnotesize
\path{a12966eba94e20268117e231b0859ea1464a1d987db359d44b743b703f3d55b1}.
\end{quote}

\section{Claim boundary}
A PASS establishes only that the declared H2-092 response-context calculation reproduces the exact $4/3/2$ identities across the tested geometry/direction inputs and historical controls. It supports the statement that the inherited H-021 six-support node-local bookkeeping carries a fourfold response-context multiplicity relative to the twelve-edge FrostCell ledger.

It does \textbf{not} establish four independent stored-energy ledgers, does not by itself establish the complete Frostyverse precession model, does not prove that H-021 is the correct mechanics of nature, and is not empirical evidence that physical space is a FrostGrid.

\end{document}