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\begin{document}
\title{Photon Fly-by / Light-Deflection Audit\\\large Frostyverse Steering Mathematics, Reproduction Case, and Post-Seal GR Ruler}
\author{Frostyverse Laboratory}
\date{Website Laboratory v1.0.48}
\maketitle

\statusbox{\textbf{Scientific status.} This document reproduces the mathematics used by the published Frostyverse photon fly-by software candidate. The Frostyverse path is generated first from the frozen local-grid steering rule. A General Relativity (GR) null-geodesic ruler is opened only after the Frostyverse trajectory has been sealed. The residual is nonzero and is preserved. H-109 is a later exact-computation/storage audit and does not introduce new photon physics. A computational pass is not experimental proof that nature uses Frostyverse.}

\begin{abstract}
This note is intended to let a physicist see, without reverse-engineering the Python source, what the public Photon Fly-by Python Test actually computes. The Frostyverse candidate first constructs its normalized spherical deformation field and advances a continuous photon through successive deformed FrostCells using a frozen current-cell steering rate. The public reference case is then sealed and compared with an independently evaluated Schwarzschild null-geodesic path. The same program exposes fresh impact parameter, resolution, grid orientation, and path-length inputs while deliberately refusing to invent a Sun/Jupiter/planet source-mass mapping that the frozen historical photon test did not establish.
\end{abstract}

\section{What is being tested}
The audit question is deliberately narrow:
\begin{quote}
Given the already-frozen Frostyverse field and local photon-steering law, does the resulting continuous trajectory remain close to an independently generated GR light path for both the published reference case and reasonable fresh normalized geometries?
\end{quote}

The public test therefore separates three layers:
\begin{enumerate}
\item \textbf{Frostyverse generation:} build the FV field and photon trajectory with GR closed;
\item \textbf{FV seal:} hash the trajectory/state arrays and release the large FV field from memory;
\item \textbf{external scoring:} generate the Schwarzschild ruler and measure the difference.
\end{enumerate}

No GR bend angle, GR coordinate, Newtonian inverse-square target, future-cell look-ahead, or post-hoc target answer is supplied to the FV path generator.

\section{Normalized geometry and public inputs}
The historical H-099 photon family is expressed in normalized source-radius units. The public wrapper exposes exactly the geometric/numerical variables that the historical calculation itself can vary:
\begin{center}
\begin{tabular}{ll}
\toprule
Symbol / input & Meaning \\
\midrule
$R$ & FrostCells per normalized source radius \\
$b_{\rm req}$ & requested signed impact parameter, in source radii \\
$b$ & realized grid-legal impact parameter \\
$H$ & path half-length, so the path spans $[-H,+H]$ source radii \\
orientation & travel/impact plane: XY, XZ, YX, or ZX \\
$K$ & frozen steering coefficient \\
$p$ & inherited FrostSegment length exponent, frozen here at $0$ \\
\bottomrule
\end{tabular}
\end{center}

The legal impact parameter is snapped exactly as in the historical code,
\[
 b=\frac{\operatorname{round}(R b_{\rm req})}{R}.
\]
For example, $R=11$ and $b_{\rm req}=4.85$ gives
\[
 b=\frac{53}{11}=4.818181818181818\ldots .
\]

\statusbox{\textbf{Source-mapping boundary.} The public custom mode does \emph{not} convert an arbitrary physical Sun, Jupiter, or planet mass/radius into a new FV source. The original photon development kept that physical transfer/holdout question separate. The custom mode attacks fresh geometry of the same normalized frozen source model rather than silently adding a new source law.}

\section{Frostyverse calculation}
\subsection{Current-cell convergence vector}
For the occupied FrostCell, the inherited solver constructs a local convergence vector $\mathbf c$ from the deformed cell geometry. Let
\[
 C=\lVert\mathbf c\rVert,
 \qquad
 \widehat{\mathbf c}=\frac{\mathbf c}{C},
 \qquad
 \vhat=\frac{\mathbf v}{\lVert\mathbf v\rVert}.
\]
Only the component of the convergence direction transverse to the current photon direction contributes to steering:
\[
 \mathbf p_\perp
 =\left(I-\vhat\vhat^{\mathsf T}\right)\widehat{\mathbf c},
 \qquad
 s=\lVert\mathbf p_\perp\rVert.
\]
Geometrically, $s=\sin\theta$, where $\theta$ is the angle between the convergence direction and the photon direction. Define the transverse steering direction
\[
 \widehat{\mathbf d}_\perp=\frac{\mathbf p_\perp}{s}
\]
when $s>0$.

\subsection{Frozen local steering rate}
The exact frozen H-084/H-099 rate used by the public test is
\[
 \boxed{
 \mathbf r_{\rm FV}
 = K R^2\sqrt{C}\,s^3\,\widehat{\mathbf d}_\perp
 }
\]
with
\[
 \boxed{K=8.6816265415871847\times10^{-5}},
 \qquad p=0.
\]
Equivalently,
\[
 \lVert\mathbf r_{\rm FV}\rVert
 =K R^2\sqrt{C}\,\sin^3\theta.
\]
The $R^2$ factor is part of the frozen resolution normalization used by this software candidate. Numerical resolution itself is not claimed to be a physical constant.

If $C=0$ or the convergence vector is exactly parallel to the photon direction, the transverse rate is zero.

\subsection{H-089 WALL\_CONTINUOUS propagation}
The public focused test uses the frozen H-089 \code{WALL\_CONTINUOUS} lane. Inside the current occupied cell, the incoming photon is carried in a straight line to the \emph{first} lattice face it reaches. Let $\Delta s_n$ be that physical/model-coordinate distance. The position update is
\[
 \mathbf x_{n+1}=\mathbf x_n+\Delta s_n\,\vhat_n.
\]
The steering accumulated during that cell transit changes the outgoing direction according to
\[
 \boxed{
 \mathbf v_{n+1}
 =\operatorname{unit}\!\left(
 \mathbf v_n+\Delta s_n\,\mathbf r_{{\rm FV},n}
 \right)
 }.
\]
The next occupied FrostCell is then chosen from the new continuous position/direction and the process repeats. There is no projection of the physical photon direction onto an axis or node-to-node departure vector.

This distinction matters: the continuous FV path is the physical trajectory under audit; the discrete completed-FrostNode route is a separate representation layer.
\n\subsection{Reviewer finite-endpoint correction}\nThe preserved historical H-089 archive still completes whole wall-to-wall transits exactly as originally published.  The reviewer-facing custom/reference wrapper, however, accepts an arbitrary finite half-length $H$, so $+H$ need not coincide with a lattice wall.  Package v1.0.2 therefore leaves every full-cell transit unchanged but caps the \emph{final} FrostCell exposure at the exact requested endpoint.  If the next wall lies beyond $+H$, the wrapper uses only the remaining distance\n\[\n \Delta s_{\rm end}=\frac{H-x_{\rm travel}}{\widehat v_{\rm travel}}\n\]\nin the same frozen steering update and then stops at $x_{\rm travel}=H$.  The forward-wall sanity count is correspondingly the number of travel-axis wall planes actually crossed inside $[-H,+H]$, rather than the difference of rounded endpoint indices.\n\nThis is a finite-path bookkeeping correction, not a new photon law: $K$, the $R^2$ normalization, the $\sqrt C\sin^3\theta$ response, and $p=0$ are unchanged.  The post-seal GR wire is sampled at the same exact finite endpoints and intervening travel-axis wall planes so the comparison uses like-for-like path limits.\n
\section{FV seal before the answer sheet opens}
For a focused run, the wrapper hashes the inputs plus the generated FV position, direction, and cell-index arrays:
\[
 \mathcal H_{\rm FV}
 =\operatorname{SHA256}(\text{inputs}\,\Vert\,\mathbf x\,\Vert\,\mathbf v\,\Vert\,\text{cell telemetry}).
\]
The large FV field object is then released. Only after that step does the program call the GR reference functions. The exact seal bytes can vary with array/platform representation, so the numerical trajectory is the cross-platform reproduction target rather than one universal wrapper hash.

\section{Independent General Relativity ruler}
The public test uses a normalized synthetic compactness
\[
 \mu=\frac{GM}{R_{\rm source}c^2}=10^{-6}
\]
for the independent Schwarzschild comparator. This number belongs to the comparison source model; it is not passed into the FV field/steering calculation.

\subsection{Exact Schwarzschild deflection used by the reference machinery}
For impact parameter $b$ at infinity, the closest approach $r_0$ is obtained from
\[
 b=\frac{r_0}{\sqrt{1-2\mu/r_0}}.
\]
Writing $m=\mu/r_0$, the historical comparator evaluates the exact asymptotic Schwarzschild bending angle as
\[
 \boxed{
 \alpha_{\rm GR}
 =2\int_0^{\pi/2}
 \frac{\cos\vartheta\,d\vartheta}
 {\sqrt{1-2m-\sin^2\vartheta+2m\sin^3\vartheta}}
 -\pi
 }.
\]
In the weak-field limit this approaches the familiar result
\[
 \alpha_{\rm GR}\simeq\frac{4GM}{bc^2}
 =\frac{4\mu}{b},
\]
but the historical scoring code does not merely substitute that approximation.

\subsection{Finite-path GR wire used by the focused Python test}
To compare the complete path over the same finite $x$ interval as FV, the GR code solves the Schwarzschild null-orbit equation in reciprocal radius $u=1/r$,
\[
 \frac{d^2u}{d\chi^2}=3\mu u^2-u,
\]
with the closest-approach condition above, and samples the resulting position and heading at the same travel-axis ticks used by the FV wire. The focused wrapper then compares both finite-path heading change and the pointwise transverse trajectory.

This is why the reported reference bend over $[-3,+3]$ source radii is not identical to simply inserting $4\mu/b$ as an asymptotic angle.

\section{Comparison observables}
For matched FV and GR transverse positions, after subtracting each path's own starting point,
\[
 \Delta\mathbf x_{\perp,j}
 =\mathbf x^{\rm FV}_{\perp,j}-\mathbf x^{\rm GR}_{\perp,j}.
\]
The most direct H-099 physical-miss quantity is
\[
 \boxed{
 \Delta_{\max}=\max_j\lVert\Delta\mathbf x_{\perp,j}\rVert
 }.
\]
The nominal FrostCell width is
\[
 L_{\rm nominal}=\frac1R,
\]
so
\[
 \Delta_{\max,\rm nominal\ cells}
 =\frac{\Delta_{\max}}{1/R}=R\Delta_{\max}.
\]
H-099 also measures the actual local deformed impact-axis FrostCell width $L_{{\rm local},j}$ directly from the eight FV cell corners before the field is released, and reports
\[
 \Delta_{\max,\rm local\ cells}
 =\frac{\Delta_{\max}}{L_{{\rm local},j_{\max}}}.
\]

The inherited wire normalized RMS error is decomposed as
\[
 \mathrm{nRMSE}_{\rm wire}
 =
 \frac{
 \sqrt{\left\langle
 \lVert\Delta\mathbf x_\perp\rVert_{\rm components}^{2}
 \right\rangle}}
 {
 \sqrt{\left\langle
 \lVert\mathbf x^{\rm GR}_\perp\rVert_{\rm components}^{2}
 \right\rangle}}
 .
\]
For very weak bends the GR denominator can be tiny, so H-099 deliberately reports absolute path separation in FrostCell widths in addition to percentage-style residuals.

\section{Public reference reproduction}
The reviewer-facing Python Test reference is deliberately small enough to run on ordinary hardware while using the unchanged H-099 mechanics:
\begin{center}
\begin{tabular}{lr}
\toprule
Quantity & Reference value \\
\midrule
Resolution & $R=11$ cells/source radius \\
Requested $b$ & $4.85$ \\
Realized $b$ & $4.818181818181818$ \\
Orientation & XY (travel $+X$, impact $+Y$) \\
Path half-length & $H=3.0$ source radii \\
Wire nRMSE & $8.55205019718325\times10^{-3}$ \\
Maximum FV--GR path separation & $9.751939371748257\times10^{-9}$ \\
Maximum separation / nominal cell & $1.0727133308923083\times10^{-7}$ \\
FV finite-path signed bend & $-0.12488312900311113$ arcsec \\
Post-seal GR finite-path signed bend & $-0.12312153841364273$ arcsec \\
Signed bend difference & $-0.001761590589468398$ arcsec \\
Relative signed-bend difference & $+1.430773699\%$ \\
\bottomrule
\end{tabular}
\end{center}

The last percentage is \emph{not} the H-099 freeze-safety criterion. The path miss is only about $1.07\times10^{-7}$ nominal FrostCells in this reference case. The nonzero residual is intentionally retained.

\section{Custom-input attack}
A reviewer can change the same normalized geometry without changing the FV law. For example:
\begin{verbatim}
python lab_test.py --custom --resolution 11 --impact 3.17 \
    --orientation XZ --half 3.0 --workers 1
\end{verbatim}
Allowed focused inputs are:
\begin{itemize}
\item $6\le R\le24$ (R9--R11 recommended for a quick check);
\item $1.1\le |b_{\rm req}|\le8.0$ source radii;
\item orientation XY, XZ, YX, or ZX;
\item $1.5\le H\le8.0$ source radii.
\end{itemize}
The requested $b$ is snapped to the legal grid value and both are printed. TXT and JSON result records are written under \code{results/}.

\section{Predeclared H-099 freeze-safety gates}
The historical full H-099 audit uses thresholds external to FV propagation:
\begin{align*}
\Delta_{\max}/L_{\rm nominal} &\le 0.05,\\
\Delta_{\max}/L_{\rm local} &\le 0.05,\\
\text{growth above first audited }R &\le 0.01\text{ highest-}R\text{ cell},\\
\text{orientation max-miss spread} &\le 0.01\text{ nominal cell}.
\end{align*}
The preserved full grid is
\[
R\in\{52,55,58,60\},
\qquad
b\in\{2.35,4.85,6.25\}.
\]
Its peak physical miss / nominal FrostCell values are:
\begin{center}
\begin{tabular}{lr}
\toprule
Impact parameter & Peak miss / nominal cell \\
\midrule
$b=2.35$ & $2.430601057\times10^{-6}$ \\
$b=4.85$ & $4.031342048\times10^{-6}$ \\
$b=6.25$ & $4.591972922\times10^{-6}$ \\
\bottomrule
\end{tabular}
\end{center}
These values are nonzero. The defensible statement is that the frozen candidate stayed far inside the declared physical-miss ceiling on the tested grid, not that FV and GR are pointwise identical.

\section{Run commands for the current Python Test ZIP}
Interactive menu:
\begin{verbatim}
python lab_test.py
\end{verbatim}
Focused reference:
\begin{verbatim}
python lab_test.py --reference
\end{verbatim}
Focused fresh geometry:
\begin{verbatim}
python lab_test.py --custom --resolution 11 --impact 3.17 \
    --orientation XZ --half 3.0
\end{verbatim}
Unchanged historical validation profiles:
\begin{verbatim}
python lab_test.py --h099-validation
python lab_test.py --h109-validation
\end{verbatim}
Unchanged heavy historical audits:
\begin{verbatim}
python lab_test.py --h099-full
python lab_test.py --h109-full
\end{verbatim}
Verify frozen archive hashes:
\begin{verbatim}
python lab_test.py --verify
\end{verbatim}
The public package requires Python 3.10+ (recommended), NumPy, and SciPy.
The exact dependency specifications are listed in \path{requirements.txt}.

\section{H-109 exact-provider architecture audit}
H-109 changes the storage/I/O architecture while keeping the photon physics unchanged. Its exact direct-read symmetry-wedge provider uses a packed disk wedge and a fixed direct-mapped scalar cache. The preserved R80 result is:
\begin{center}
\begin{tabular}{lr}
\toprule
Quantity & Preserved result \\
\midrule
Full exact reference field & 30.460 GiB \\
Exact symmetry wedge on disk & 1.910 GiB \\
Explicit active exact-field cache & 64 KiB \\
Sampled field relative difference & $2.123\times10^{-16}$ \\
Maximum position/vector path miss & $0/0$ \\
FrostCell route / wall hits identical & true / true \\
\bottomrule
\end{tabular}
\end{center}
The 64 KiB number is only the explicit active exact-field cache during photon propagation. It is not the wedge storage size and is not the source-to-wedge generation-memory requirement.

\section{Recommended independent attacks}
A reviewer does not need to accept the Frostyverse ontology. Useful attacks include:
\begin{itemize}
\item inspect the source/dataflow and verify that the GR functions are not called while the FV field/path is being generated;
\item reproduce the R11/$b=4.85$ public reference case;
\item choose a fresh $b$ and/or orientation and look for a large residual or lattice-direction dependence;
\item inspect both finite-path bend and absolute path separation rather than relying on one percentage metric;
\item rerun the historical H-099 validation or full grid if hardware permits;
\item verify that changing the comparison code cannot steer an already-sealed FV trajectory;
\item do not retune $K$, $p$, or the $\sin^3\theta$ law after seeing a poor fresh result---a failure is a valid audit outcome;
\item independently challenge the still-open physical source-transfer question rather than assuming the normalized source automatically represents every real gravitating body.
\end{itemize}

\section{Provenance and integrity}
Primary H-099 historical package:
\begin{quote}\footnotesize
\path{original/Frostyverse-Test-H-099.zip}\\
SHA-256: \path{b26e38ce66dbfa7d18897d59b7c7f744dade66c26a011da36cfa0b4f53ff9d20}
\end{quote}
H-109 exact-provider package:
\begin{quote}\footnotesize
\path{original/Frostyverse-Test-H-109.zip}\\
SHA-256: \path{7bf655837903d7a6d77b56b03c076870c1594a902ce3be1df0f4249a9b2e8463}
\end{quote}
The complete public Python Test ZIP used for this Laboratory has SHA-256
\begin{quote}\footnotesize
\path{b2c739ed2d32f41fc16d088b55a398383f16f9a40f1a5b358e0828b551db2d42}
\end{quote}
Use \code{SHA256SUMS.txt} in the Python Test ZIP to verify the Laboratory-facing files.

\section{Claim boundary}
The strongest supported claim from this Laboratory item is computational and conditional: the frozen FV steering candidate, when applied to the tested normalized spherical source and geometries, produces trajectories that remained within the declared H-099 physical-miss gates and can be reproduced on fresh normalized inputs. H-109 additionally shows that one exact symmetry-reduced field-access architecture preserves that photon path while reducing active exact-field RAM.

This does \textbf{not} establish that nature uses Frostyverse, does not establish a universal physical mass-to-FV-source mapping, does not claim zero FV--GR residual, and does not permit the GR ruler to be reinterpreted as an input to FV generation.

\end{document}