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\begin{document}
\title{FrostField Gravity: Exact Spherical Deformation / Symmetry Wedge\\\large Physicist Reproduction and Audit Note}
\author{Frostyverse Laboratory}
\date{Website Laboratory v1.0.48}
\maketitle

\statusbox{\textbf{Scientific status.} This Laboratory tests a numerical symmetry and storage claim inside the Frostyverse gravity-field machinery. For a centered symmetry-clean spherical source, the declared six-neighbor discrete field equation can be solved on a canonical component wedge and reconstructed over the full cube without a measurable field change beyond floating-point/numerical-solver error. The full-cube calculation is an \emph{internal Frostyverse numerical ruler for the same equation}, not General Relativity (GR) or Newtonian gravity. A PASS does not establish that the FrostField equation is physically correct or that arbitrary nonspherical sources admit the same reduction.}

\begin{abstract}
This note exposes the calculation performed by the public FrostField Gravity Python Test so a reviewer need not reverse-engineer the source. The compact reference builds a normalized fractional spherical source, forms the $x$-component right-hand side of the inherited six-neighbor FrostField equation, solves only the canonical symmetry wedge with red-black successive over-relaxation (SOR), seals the candidate summary, and only then independently solves the corresponding full cube. The wedge and full-cube vector fields are compared at deterministic interior samples. The preserved H-103 research audit independently establishes the underlying 48-fold cubic symmetry using the historical direct solver; H-108/H-109 then exploit that symmetry for exact reduced storage and bounded current-cell access.
\end{abstract}

\section{Audit question}
The narrow computational claim is:
\begin{quote}
For the declared centered spherical-source problem, can the same discrete FrostField solution be represented by the canonical one-component wedge
\[
x\geq0,\qquad y\geq z\geq0,
\]
then reconstructed everywhere by the cube's own sign/permutation symmetries, without introducing an approximate radial surrogate?
\end{quote}

This Lab does \emph{not} ask whether the FrostField equation equals GR or Newtonian gravity. It asks whether the stated symmetry reduction is faithful to the same Frostyverse discrete equation.

\section{Compact public Frostyverse calculation}
\subsection{Normalized fractional spherical source}
The compact public test uses normalized FrostGrid lattice units. Let the requested source radius be $a_s$. Each lattice cell in a bounding region is subdivided into $m^3$ subcells, with
\[
m=3,
\qquad
\delta_q=\frac{q+1/2}{m}-\frac12,
\qquad q=0,1,2.
\]
For lattice coordinate $(i,j,k)$, count the subcell centers satisfying
\[
(i+\delta_p)^2+(j+\delta_q)^2+(k+\delta_r)^2\leq a_s^2.
\]
If that count is $c_{ijk}$, the normalized occupancy is
\[
\rho_{ijk}=\frac{c_{ijk}}{\sum_{abc}c_{abc}},
\qquad
\sum_{ijk}\rho_{ijk}=1.
\]
The compact wrapper fixes the normalized source amplitude to
\[
Q=1.
\]
The $x$-component source term is then the centered occupancy difference
\[
\boxed{
b_{ijk}=Q\left(\rho_{i+1,j,k}-\rho_{i-1,j,k}\right).
}
\]
The corresponding $y$ and $z$ components are obtained by coordinate permutation when the vector field is reconstructed.

\subsection{Six-neighbor FrostField equation}
For the scalar $x$ component $u_x$ at an interior coordinate $(i,j,k)$, the declared discrete equation is
\[
\boxed{
6u_{i,j,k}
-u_{i-1,j,k}-u_{i+1,j,k}
-u_{i,j-1,k}-u_{i,j+1,k}
-u_{i,j,k-1}-u_{i,j,k+1}
=b_{i,j,k}.
}
\]
Equivalently, if $\Sigma_{ijk}$ denotes the sum of the six neighboring scalar values,
\[
u_{ijk}=\frac{\Sigma_{ijk}+b_{ijk}}{6}
\]
at a converged interior point.

The compact Python/browser reproducer solves this equation iteratively. For SOR relaxation parameter $\omega$, one update is
\[
\boxed{
u^{\rm new}_{ijk}
=u^{\rm old}_{ijk}
+\omega\left[
\frac{\Sigma_{ijk}+b_{ijk}}{6}-u^{\rm old}_{ijk}
\right].
}
\]
Updates are performed in red-black parity order. The published wrapper fixes
\[
\omega=1.82,
\qquad
\varepsilon_{\rm SOR}=10^{-10},
\]
and stops when the maximum update in a sweep is at most $\varepsilon_{\rm SOR}$, or when the requested maximum sweep count is reached.

\subsection{Compact boundary closure}
For the small public/browser domain, boundary values and the initial interior guess use the same declared continuum closure. Define
\[
r^2=x^2+y^2+z^2,
\qquad
r_{+}=\sqrt{r^2+2x+1},
\qquad
r_{-}=\sqrt{r^2-2x+1}.
\]
Then
\[
\boxed{
u_x^{\rm bdry}(x,y,z)
=
\frac{Q}{4\pi}
\frac{-4x}{r_{+}r_{-}(r_{+}+r_{-})}
}
\]
when the denominator is nonzero, and zero otherwise.

\statusbox{\textbf{Boundary-closure interpretation.} This expression is part of the compact declared numerical problem and is used identically by both the wedge and full-cube lanes. It is \emph{not} an independently opened GR/Newton comparison value. Consequently, wedge/full agreement tests the symmetry reduction of this numerical problem; it does not validate the physical correctness of the boundary closure itself.}

\section{Exact cubic symmetry representation}
For the centered spherical problem, the scalar $x$ component has the expected parity/permutation structure
\[
u_x(-x,y,z)=-u_x(x,y,z),
\]
\[
u_x(x,-y,z)=u_x(x,y,z),
\qquad
u_x(x,y,-z)=u_x(x,y,z),
\]
\[
u_x(x,z,y)=u_x(x,y,z).
\]
Therefore every interior scalar query can be mapped to the canonical wedge. Let
\[
s=\operatorname{sgn}(x),
\qquad
a=|x|,
\qquad
b=\max(|y|,|z|),
\qquad
c=\min(|y|,|z|).
\]
For $x\neq0$,
\[
\boxed{
u_x(x,y,z)=s\,U(a,b,c),
\qquad a>0,\quad b\geq c\geq0,
}
\]
while $u_x(0,y,z)=0$ by odd parity.

The full vector deformation-state field is reconstructed by component permutation,
\[
\boxed{
\mathbf u(x,y,z)
=\bigl(
 u_x(x,y,z),
 u_x(y,x,z),
 u_x(z,y,x)
\bigr).
}
\]
More generally, the historical H-103 audit checks covariance under all 48 signed permutations $S$ of the cubic group,
\[
\boxed{
\mathbf u(S\mathbf x)=S\,\mathbf u(\mathbf x).
}
\]

\section{Storage count and why the reduction is testable}
For a compact cube with half-width $h$, the full solved interior contains
\[
\boxed{N_{\rm full}=(2h-1)^3}
\]
scalar $u_x$ unknowns. The canonical wedge stores $x=1,\ldots,h-1$ and $0\leq z\leq y<h$. Its triangular $yz$ count is
\[
T_h=\frac{h(h+1)}{2},
\]
so
\[
\boxed{N_{\rm wedge}=(h-1)\frac{h(h+1)}{2}.}
\]
The raw scalar unknown-count reduction is therefore
\[
\boxed{
\mathcal R_{\rm store}
=\frac{(2h-1)^3}{(h-1)h(h+1)/2}.
}
\]
For the published $h=9$ case,
\[
N_{\rm wedge}=360,
\qquad
N_{\rm full}=4913,
\qquad
\mathcal R_{\rm store}=13.647222\ldots .
\]

\section{Residual and candidate seal}
After the wedge solve, the discrete equation is recomputed rather than inferred from the SOR stopping criterion. Define the point residual
\[
R_{ijk}=6u_{ijk}-\Sigma_{ijk}-b_{ijk}.
\]
The wrapper reports the normalized worst residual
\[
\boxed{
R_{\rm rel}
=
\frac{\max|R_{ijk}|}
{\max\left(|6u_{ijk}|,|\Sigma_{ijk}|,|b_{ijk}|\right)}
}
\]
where the denominator maximum is taken over the audited wedge points.

Before any full-cube comparison is opened, the wrapper hashes the wedge case summary---geometry, achieved sweep count, residual, and storage counts---and prints a SHA-256 seal. This is a software firewall: the candidate wedge result exists before the internal full-cube ruler is evaluated.

For the fresh published reference run used for this note, the seal was
\begin{quote}\footnotesize
\path{f407628d0a461102fd8cdb1cd843295f753f7c0c8938d8c6cd0776f3ecea5f59}.
\end{quote}

\section{Post-seal full-cube numerical ruler}
Only after the wedge has been sealed does the public test solve the full cube. The full-cube lane uses the \emph{same}
\begin{itemize}
\item fractional source occupancy $\rho$;
\item source term $b$;
\item six-neighbor equation;
\item boundary closure;
\item $\omega=1.82$ SOR relaxation;
\item $10^{-10}$ stopping tolerance and requested sweep budget.
\end{itemize}
It differs only in domain representation: every full interior scalar value is stored and updated directly.

At deterministic sampled interior coordinates, the wrapper reconstructs the three-component wedge field and compares it with the independently stored full-cube vector field. If $\mathbf u_W$ and $\mathbf u_F$ are the two values,
\[
\Delta_{\rm abs}
=\max_{q,k}|u_{W,k}(\mathbf x_q)-u_{F,k}(\mathbf x_q)|,
\]
\[
\Delta_{\rm rel}
=\frac{\Delta_{\rm abs}}
{\max_{q,k}(|u_{W,k}|,|u_{F,k}|)}.
\]
A material mismatch here would falsify the compact symmetry reconstruction even if both solvers individually reached small equation residuals.

\statusbox{\textbf{Comparison firewall.} The full cube is not GR, Newton, or an observational target. It is an internal numerical ruler for the same Frostyverse equation. No external gravity value is supplied to the wedge solver or used to tune its source, SOR update, boundary closure, or reconstruction rule.}

\section{Published compact Python reference}
Install the public dependencies first:
\begin{verbatim}
python -m pip install -r requirements.txt
\end{verbatim}
Then run the interactive package with
\begin{verbatim}
python lab_test.py
\end{verbatim}
or use \code{run\_test.bat} on Windows / \code{run\_test.sh} on macOS or Linux.

The focused published case is also available directly:
\begin{verbatim}
python lab_test.py --reference
\end{verbatim}
with
\begin{center}
\begin{tabular}{ll}
\toprule
Input & Published value \\
\midrule
Cube half-width $h$ & 9 \\
Normalized spherical-source radius $a_s$ & 1.75 \\
Maximum SOR sweeps & 450 \\
SOR relaxation $\omega$ & 1.82 \\
SOR stopping tolerance & $10^{-10}$ \\
Wedge/full sample count & 320 \\
\bottomrule
\end{tabular}
\end{center}

A fresh run of the finished public Python Test produced:
\begin{center}\small
\begin{tabular}{lr}
\toprule
Observable & Result \\
\midrule
Wedge convergence sweep & 93 \\
Wedge equation residual, relative & $1.555753314230\times10^{-9}$ \\
Wedge scalar unknowns & 360 \\
Full interior scalar unknowns & 4913 \\
Raw unknown/storage reduction & $13.647222\times$ \\
Full-cube equation residual, relative & $1.555753355803\times10^{-9}$ \\
Maximum wedge/full absolute mismatch & $1.387778780781\times10^{-17}$ \\
Maximum wedge/full relative mismatch & $4.784067286065\times10^{-16}$ \\
Worst sampled coordinate & $(-1,-2,-1)$ \\
\bottomrule
\end{tabular}
\end{center}
The residuals are not claimed to be machine zero; they reflect the compact iterative stopping rule. The key symmetry result is that the independently represented wedge and full-cube solutions agree to floating-point scale in the sampled field values.

\section{Fresh custom-input attack mode}
A reviewer can change the compact geometry and numerical work budget without editing the Frostyverse equation. For example,
\begin{verbatim}
python lab_test.py --custom \
  --half-width 10 \
  --radius 2.10 \
  --sweeps 500 \
  --samples 400
\end{verbatim}
A fresh run of that example produced
\[
R_{W,\rm rel}=1.737487727452\times10^{-9},
\]
\[
R_{F,\rm rel}=1.737487551259\times10^{-9},
\]
\[
\Delta_{\rm abs}=1.387778780781\times10^{-17},
\qquad
\Delta_{\rm rel}=5.285765651838\times10^{-16}.
\]
The wrapper accepts
\[
5\leq h\leq18,
\qquad
0.5\leq a_s\leq h-2,
\]
\[
20\leq N_{\rm sweeps}\leq3000,
\qquad
20\leq N_{\rm samples}\leq5000.
\]
The public interface holds $\omega=1.82$ and the tolerance $10^{-10}$ fixed. A reviewer who wants to attack those numerical choices can inspect/edit the short public wrapper directly.

The internal full-cube ruler can also be suppressed:
\begin{verbatim}
python lab_test.py --custom --half-width 10 \
  --radius 2.10 --sweeps 500 --samples 400 --no-ruler
\end{verbatim}
This is useful for verifying that the wedge candidate is able to run and seal without opening the comparison lane.

Each focused run writes human-readable \code{.txt} and machine-readable \code{.json} records under \code{results/}.

\section{Historical H-103 direct-solver audit}
The public ZIP preserves
\begin{quote}
\path{original/Frostyverse-Test-H-109.zip}
\end{quote}
unchanged. H-103 inside that archive attacks the same cubic-symmetry premise with the historical direct field solver rather than the compact SOR demonstration.

For the inherited direct solver, one scalar component is transformed with a type-I discrete sine transform. The one-dimensional mode eigenvalues are
\[
\lambda_m=2-2\cos\left(\frac{\pi m}{M+1}\right),
\]
and the transformed 3-D coefficient is divided by
\[
\boxed{\lambda_i+\lambda_j+\lambda_k.}
\]
The six-neighbor equation residual is then recomputed in real space.

H-103 independently checks:
\begin{enumerate}
\item cubic symmetry of the fractional-sphere source itself;
\item canonical $u_x$ wedge reconstruction;
\item complete 48-fold vector covariance;
\item all 12 FrostSegment lengths and bends/tilts under those transformations;
\item the complete finite-difference RHS symmetry at manageable resolution;
\item raw octant and tight-wedge storage accounting.
\end{enumerate}

Run the unchanged validation profile through the public wrapper:
\begin{verbatim}
python lab_test.py --historical-validation
\end{verbatim}
Representative preserved validation results are
\begin{center}\small
\begin{tabular}{lr}
\toprule
Historical H-103 observable & Result \\
\midrule
R10/R11 source cubic-generator residual & $0/0$ \\
Worst $u_x$ wedge reconstruction relative error & $1.591249\times10^{-16}$ \\
Worst 48-fold vector covariance relative error & $1.591249\times10^{-16}$ \\
Complete RHS symmetry residual at R10 & $0$ \\
R11 full / octant / tight wedge & $0.078/0.010/0.005$ GiB \\
R11 tight-wedge raw-data reduction & $15.641\times$ \\
Validation verdict & PASS \\
\bottomrule
\end{tabular}
\end{center}
The much heavier historical profile is available with
\begin{verbatim}
python lab_test.py --historical-full
\end{verbatim}
and should be treated as a deeper audit rather than the first reproduction step.

\section{Source-symmetry control}
The wedge reduction is only legitimate if the declared source and discrete operator carry the required symmetries. H-103 therefore checks the fractional source under cubic generators before interpreting the field result. If a source sampler produces a finite symmetry residue at some resolution, that residue is classified as source discretization, not silently rebranded as FrostCell physics.

H-103 also contains a symmetry-projected source \emph{shadow} diagnostic. It is an investigative control only: it does not feed the inherited production solve. This distinction matters because forcing the source to be symmetric and then ``discovering'' a symmetric field would not independently test the original source construction.

\section{H-108/H-109 storage and current-cell architecture}
H-103 first established that the exact field contains exploitable cubic redundancy; it did not by itself prove a production wedge-only solver. H-108/H-109 subsequently attacked that engineering question while preserving exact field values.

In the preserved H-109 R80 result:
\begin{center}
\begin{tabular}{lr}
\toprule
Quantity & Preserved result \\
\midrule
Full field storage & 30.460 GiB \\
Exact packed disk wedge & 1.910 GiB \\
Explicit exact current-cell cache & 64 KiB \\
Maximum compact/reference path position difference & $0$ \\
Maximum compact/reference vector difference & $0$ \\
Route / wall-hit identity & true / true \\
\bottomrule
\end{tabular}
\end{center}
The H-109 provider does not memory-map the wedge. A cache miss seeks to the exact packed float64 value and reads 8 bytes. The direct-mapped cache contains 4096 uint64 keys and 4096 float64 values:
\[
4096(8\ \mathrm{bytes})+4096(8\ \mathrm{bytes})
=65536\ \mathrm{bytes}=64\ \mathrm{KiB}.
\]
Eviction changes disk-I/O frequency only; it does not alter the field value returned.

\statusbox{\textbf{64 KiB limitation.} The 64 KiB figure is the explicit exact-field cache used during the H-109 photon lane. It is \emph{not} total source-to-field generation RAM and it is not total storage. The exact packed wedge remains on disk.}

\section{Useful falsification and negative controls}
A physicist reviewing this Lab should try to break it rather than merely rerun the default case. Useful attacks include:
\begin{enumerate}
\item vary $h$ and source radius over unrelated legal compact cases;
\item increase the SOR sweep budget and inspect residual convergence rather than accepting a single stopping point;
\item increase deterministic comparison sampling, or replace sampling with an exhaustive small-domain comparison;
\item deliberately perturb the source construction so cubic symmetry is broken and confirm that the wedge premise fails or is flagged;
\item audit signed reflections and component permutations, not merely one positive octant;
\item compare the complete finite-difference RHS under symmetry operations, not only solved field values;
\item alter the boundary closure in both lanes and check whether wedge/full identity persists as a symmetry statement;
\item apply a nonspherical or off-center source as a scope-breaking control---no exact spherical-wedge claim should be transferred to it without a new derivation;
\item run the unchanged H-103 validation and, resources permitting, the historical full profile.
\end{enumerate}
A failure on a fresh legal case is a useful audit result and should be preserved rather than tuned away.

\section{Reproduction files and provenance}
The reviewer-facing Python Test ZIP contains:
\begin{itemize}
\item \code{lab\_test.py}: compact wedge-first solver, candidate seal, full-cube ruler, custom mode, and historical launcher;
\item \code{README\_FIRST.md}: run instructions and scientific scope;
\item \code{EXPECTED\_RESULTS.txt}: published numerical fingerprint;
\item \code{requirements.txt}: NumPy/SciPy requirements;
\item \code{SHA256SUMS.txt}: package-internal hashes;
\item \code{run\_test.bat} and \code{run\_test.sh}: convenience launchers;
\item \code{original/Frostyverse-Test-H-109.zip}: immutable historical research archive.
\end{itemize}
Python 3.10 or newer is recommended. The public dependencies are NumPy $\geq1.24$ and SciPy $\geq1.10$; third-party packages are not bundled.

The preserved H-109 archive SHA-256 is
\begin{quote}\footnotesize
\path{7bf655837903d7a6d77b56b03c076870c1594a902ce3be1df0f4249a9b2e8463}.
\end{quote}
Verify it with
\begin{verbatim}
python lab_test.py --verify
\end{verbatim}
The public FrostField Gravity Python Test ZIP used for this note has SHA-256
\begin{quote}\footnotesize
\path{8183e1ab85f57ed9d9ff1a48cef11e943aaa92dfa1d44fb1e491d2c1c55c7018}.
\end{quote}

\section{Claim boundary}
The supported statement is narrow: for the declared centered, symmetry-clean spherical-source/cubic-domain FrostField problem, the cubic sign/permutation relations support an exact symmetry representation, and the compact wedge reconstruction reproduces the corresponding full-cube numerical solution to floating-point scale in the tested cases. Historical H-103 independently verifies the symmetry structure, while H-108/H-109 demonstrate exact reduced storage/current-cell access in the preserved research architecture.

This does \emph{not} establish that the Frostyverse gravity equation describes nature, does not constitute an empirical test of gravity, does not establish equality with GR or Newtonian gravity, does not validate the compact continuum boundary closure as a physical law, and does not automatically generalize the wedge to nonspherical, off-center, time-dependent, or otherwise symmetry-breaking sources.

\end{document}