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\begin{document}
\title{Lorentz / Preferred-Frame Consistency Candidate\\\large Physicist Reproduction and Audit Note}
\author{Frostyverse Laboratory}
\date{Website Laboratory v1.0.48}
\maketitle

\statusbox{\textbf{Scientific status.} This is a preserved historical software candidate for testing whether a preferred discrete update substrate can produce Lorentz-compatible local clock, ruler, synchronization, and light observables. The candidate calculation is generated and SHA-256 sealed before the Special Relativity (SR) comparison is opened. A PASS is a software/model result, not evidence that nature contains a FrostGrid.}

\begin{abstract}
This note states the mathematics actually exercised by the public Lorentz / Preferred-Frame Python Test. The candidate starts only from a unit signal-propagation limit, moving-anchor intercept geometry, a shared saturated-FULL checkpoint, and signed underfill/over-completion relaxation. From those rules it computes a transverse moving-clock slowdown, a longitudinal equilibrium scale, a synchronization offset, and local one-way/two-way light speeds. The public reference is $\beta=0.80$ with an off-axis boost direction. Only after the FV result is complete and fingerprinted are $\gamma_{\rm SR}$ and $1/\gamma_{\rm SR}$ evaluated. The immutable historical Reference Simulator is bundled for independent inspection and full-suite reruns.
\end{abstract}

\section{Audit question and claim boundary}
Let the preferred grid frame use units with $c=1$. A structure moves with
\[
\boldsymbol\beta=\beta\hat{\mathbf a},\qquad 0\leq\beta<1,
\]
where $\hat{\mathbf a}$ is any normalized 3-D boost direction.

The audit question is narrow:
\begin{quote}
Do the declared moving-target and saturation rules, without receiving a Lorentz factor as an input, generate the same clock slowdown, longitudinal equilibrium scale, synchronization, and local light observables that SR predicts for the tested inertial cases?
\end{quote}

The result does \emph{not} establish microscopic Lorentz symmetry of the grid, nor does it prove all relativistic phenomena follow from these historical candidate mechanics.

\section{FV mechanics supplied to the calculation}
The focused public wrapper imports the unchanged engine from
\begin{quote}
\path{historical/Frostyverse_Reference_Simulator_v1.0.zip}.
\end{quote}
The relevant candidate ingredients are:
\begin{enumerate}
\item a support influence propagates at the same normalized causal limit $c=1$ used by the simulator;
\item both endpoints of a bound relation co-move with velocity $\boldsymbol\beta$;
\item a saturated 3-D structure does not declare its local transfer complete until the required support cycle reaches the shared FULL checkpoint;
\item underfill at that checkpoint produces an inward signed relaxation pressure, while early completion reverses the sign;
\item ambient vacuum pressure may alter geometry but does not add packet energy;
\item the SR comparison routine is not called by the FV relaxation solver.
\end{enumerate}

\section{Moving-anchor intercept equation}
Let $\mathbf r$ be the target endpoint's separation from the emitting endpoint at the instant of launch. A unit-speed support influence must satisfy
\[
\left\lVert \mathbf r+\boldsymbol\beta t\right\rVert=t.
\]
Squaring gives the quadratic actually solved by the historical engine,
\[
(\beta^2-1)t^2
+2(\mathbf r\cdot\boldsymbol\beta)t
+\lVert\mathbf r\rVert^2=0.
\]
The code chooses the positive physical root. A complete support cycle is
\[
T(\mathbf r,\boldsymbol\beta)
=t(\mathbf r,\boldsymbol\beta)
+t(-\mathbf r,\boldsymbol\beta).
\]
This moving-target geometry is the input to both the transverse clock calculation and the longitudinal saturation relaxation.

\section{FV transverse moving-clock slowdown}
Choose a rest separation $L$ perpendicular to the boost direction. Then
\[
\mathbf r\cdot\boldsymbol\beta=0.
\]
The positive intercept root is
\[
t_\perp=\frac{L}{\sqrt{1-\beta^2}},
\]
so the moving transverse round-trip checkpoint is
\[
T_\perp=\frac{2L}{\sqrt{1-\beta^2}}.
\]
The rest cycle is $T_0=2L$. The candidate therefore defines its moving-clock slowdown directly from the support-cycle geometry as
\[
\boxed{
G_{\rm FV}=\frac{T_\perp}{T_0}
=\frac{1}{\sqrt{1-\beta^2}}
}.
\]
The Python engine obtains $T_\perp$ by solving the moving-intercept problem; it does not call the SR gamma routine to construct this result.

\section{FV longitudinal saturation relaxation}
Let $k$ be the longitudinal scale relative to rest length $L$, so the moving longitudinal separation is $kL\hat{\mathbf a}$. The forward and backward intercept times are
\[
t_+=\frac{kL}{1-\beta},
\qquad
 t_-=\frac{kL}{1+\beta},
\]
and therefore
\[
T_\parallel(k)
=t_++t_-
=\frac{2kL}{1-\beta^2}.
\]

The candidate compares this transfer period with the previously generated transverse FULL checkpoint. Its normalized fill ratio is
\[
f(k)=\frac{T_\perp}{T_\parallel(k)},
\]
and its signed snowplow pressure is
\[
P(k)=1-f(k).
\]
Thus:
\begin{align*}
f<1 &\Rightarrow P>0 &&\text{underfilled at FULL checkpoint: relax inward},\\
f>1 &\Rightarrow P<0 &&\text{completed early: rejected-pressure sign relaxes outward},\\
f=1 &\Rightarrow P=0 &&\text{equilibrium}.
\end{align*}

The implementation does \emph{not} directly solve for a desired Lorentz scale. Starting from $k=1$, it repeatedly applies
\[
\boxed{
k\leftarrow k\exp[-gP(k)]
}
\]
with the historical default relaxation gain $g=0.35$ until $|f-1|$ is below tolerance.

For audit purposes, the zero-pressure consequence of those same FV equations can be written analytically. Setting $T_\parallel=T_\perp$ gives
\[
\frac{2kL}{1-\beta^2}
=\frac{2L}{\sqrt{1-\beta^2}},
\]
so
\[
\boxed{
k_{\rm FV}=\sqrt{1-\beta^2}}.
\]
The numerical relaxation is therefore independently inspectable: the expected equilibrium follows from the declared moving-target and FULL-checkpoint rules, not from an SR target value passed to the iteration.

\section{Arbitrary 3-D bound orientations}
For an arbitrary rest-space relation $\mathbf r$, decompose it relative to the boost axis,
\[
\mathbf r=\mathbf r_\perp+\mathbf r_\parallel.
\]
The candidate uses the derived longitudinal scale to construct
\[
\boxed{
\mathbf r_{\rm moving}
=\mathbf r_\perp+k_{\rm FV}\mathbf r_\parallel
}.
\]
The public focused test samples 96 approximately uniform orientations on the sphere. Each strained relation is sent through the same moving-roundtrip and saturation-fill calculation. The maximum mismatch from the common FULL checkpoint is reported; no orientation receives its own fitted scale.

\section{Derived synchronization offset}
For a longitudinal rest separation $L$, the candidate derives the front/rear synchronization offset from the unequal forward and backward intercepts rather than inserting an SR relativity-of-simultaneity formula.

Let
\[
\Delta_{\rm sync}
=\frac{1}{2}\frac{t_++t_-}{G_{\rm FV}}
-\frac{t_+}{G_{\rm FV}}.
\]
At the FV equilibrium $k=\sqrt{1-\beta^2}$ and $G_{\rm FV}=1/\sqrt{1-\beta^2}$, the declared geometry gives
\[
\boxed{
\frac{\Delta_{\rm sync}}{L}=-\beta
}.
\]
This value is derived only after the FV clock and longitudinal scale have been generated.

\section{Emergent local event coordinates and light speed}
For a grid-frame event displacement $(\Delta t,\Delta\mathbf x)$, subtract the motion of the local structure,
\[
\Delta\mathbf x_{\rm rel}
=\Delta\mathbf x-\boldsymbol\beta\Delta t.
\]
Split this into parallel and perpendicular parts. The historical engine converts the derived moving geometry back to local ruler coordinates as
\[
\Delta\mathbf x'_{\rm local}
=\Delta\mathbf x_\perp
+\frac{\Delta\mathbf x_\parallel-\boldsymbol\beta\Delta t}{k_{\rm FV}},
\]
and uses the derived synchronization gradient for local elapsed time,
\[
\Delta t'
=\frac{\Delta t}{G_{\rm FV}}
+\left(\frac{\Delta_{\rm sync}}{L}\right)
\Delta x'_\parallel.
\]
Substituting the FV-derived values $k_{\rm FV}=\sqrt{1-\beta^2}$, $G_{\rm FV}=1/\sqrt{1-\beta^2}$, and $\Delta_{\rm sync}/L=-\beta$ gives the familiar algebraic form
\[
\Delta x'_\parallel
=\gamma_{\rm geom}(\Delta x_\parallel-\beta\Delta t),
\]
\[
\Delta t'
=\gamma_{\rm geom}(\Delta t-\beta\Delta x_\parallel),
\qquad
\gamma_{\rm geom}=\frac{1}{\sqrt{1-\beta^2}},
\]
but here this is shown as a consequence of the candidate's moving-target, saturation, ruler, and synchronization rules. The focused test samples 256 light-ray directions and checks that both local one-way and two-way speeds remain unity to numerical tolerance.

\section{Candidate-first comparison firewall}
The public wrapper deliberately separates generation from comparison:
\begin{enumerate}
\item normalize the user-supplied boost axis;
\item generate $G_{\rm FV}$ from the transverse moving-support cycle;
\item relax $k_{\rm FV}$ using only saturation fill/pressure;
\item audit arbitrary bound orientations, synchronization, and local light speeds;
\item serialize the complete FV candidate record and create a SHA-256 seal;
\item \emph{only now} call the historical comparison routine for SR.
\end{enumerate}

The external SR ruler is
\[
\boxed{
\gamma_{\rm SR}=\frac{1}{\sqrt{1-\beta^2}},
\qquad
k_{\rm SR}=\frac{1}{\gamma_{\rm SR}}
}.
\]
The equality of the closed-form expressions does not remove the firewall question: a reviewer can inspect whether the numerical FV solver is independently generated or merely receives the comparison answer. In this package, the SR routine is called only after the candidate dictionary has been sealed.

\section{Public reference case: $\beta=0.80$}
The current Python Test uses one focused reproduction case rather than requiring the reviewer to begin with the full historical seven-speed suite.

\begin{center}
\begin{tabular}{ll}
\toprule
Input & Reference value \\
\midrule
$\beta$ & $0.80$ \\
Raw boost axis & $(0.73,-0.41,0.55)$ \\
Normalized boost axis & $(0.728725844,-0.409284378,0.549040019)$ \\
Orientation rays & 96 \\
Local-light rays & 256 \\
Saturation threshold & 1.0 \\
Relaxation gain & 0.35 \\
\bottomrule
\end{tabular}
\end{center}

Expected focused output, allowing normal floating-point variation:
\begin{center}
\begin{tabular}{lr}
\toprule
Quantity & Value \\
\midrule
FV moving-clock slowdown & $1.66666666666667$ \\
FV longitudinal scale & $0.600000000000485$ \\
Relaxation steps & 65 \\
Maximum orientation mismatch & $\sim7.98\times10^{-13}$ \\
Post-seal $\gamma_{\rm SR}$ & $1.66666666666667$ \\
Post-seal $1/\gamma_{\rm SR}$ & $0.600000000000000$ \\
FV slowdown $-\gamma_{\rm SR}$ & approximately $0$ \\
FV scale $-1/\gamma_{\rm SR}$ & $\sim4.85\times10^{-13}$ \\
\bottomrule
\end{tabular}
\end{center}

Reference command:
\begin{verbatim}
python lab_test.py --reference
\end{verbatim}
The reference mode additionally invokes the unchanged historical \code{evaluate\_snowplow\_strain} calculation and reports the absolute differences between the public wrapper and the historical engine.

\section{Fresh custom-input test}
A reviewer can choose any subluminal speed and any nonzero 3-D boost direction. The direction vector is normalized automatically. For example:
\begin{verbatim}
python lab_test.py --custom --beta 0.73 \
  --axis 0.40 0.81 0.17 --label fresh_073
\end{verbatim}

This is the preferred attack after reproducing the reference row. Useful challenges include speeds not shown in the historical examples and strongly off-axis directions that do not align with any cubic-grid axis or face diagonal.

Focused runs save human-readable TXT and machine-readable JSON records under \path{results/}. The JSON record includes the FV candidate seal and a boolean stating that candidate generation occurred before SR comparison.

\section{Historical full-suite reproduction}
The immutable historical package can be run unchanged from the wrapper:
\begin{verbatim}
python lab_test.py --full-reference
\end{verbatim}
The preserved suite includes the broader isotropy/rotation, boundary-continuity, saturated matter, acceleration/deceleration, rigidity, and adversarial checks. A successful historical rerun ends with
\begin{verbatim}
RESULT: ALL REFERENCE VERIFICATION TESTS PASSED.
\end{verbatim}
The historical seven-speed ladder remains useful provenance, but it is not the required first step for the public focused audit.

\section{State-machine conservation and negative controls}
The preserved historical suite also audits
\[
\text{FULL source}\rightarrow\text{TRANSITION}\rightarrow\text{FULL receiver},
\]
with source-plus-destination occupancy conserved and relaunch blocked until the destination reaches the saturation threshold.

Additional useful falsification checks are:
\begin{itemize}
\item search the source for any call to \code{gamma\_reference} inside the FV relaxation path;
\item change the boost axis to arbitrary off-axis directions;
\item vary threshold and relaxation gain and verify the equilibrium scale is stable;
\item increase the orientation/light-ray sampling density;
\item reverse the historical pressure sign as a negative control and confirm the system moves away from equilibrium;
\item test observables not present in this finite suite rather than treating these checks as a complete derivation of relativity.
\end{itemize}

\section{Package contents and provenance}
The public Python Test contains
\begin{itemize}
\item \path{lab_test.py} --- focused reference/custom-input wrapper;
\item \path{README_FIRST.md} --- run instructions and scientific boundary;
\item \path{EXPECTED_RESULTS.txt} --- focused reference fingerprint;
\item \path{requirements.txt} --- no third-party dependencies;
\item \path{historical/Frostyverse_Reference_Simulator_v1.0.zip} --- immutable historical engine and test suite;
\item \path{SHA256SUMS.txt} --- package integrity manifest;
\item Windows and macOS/Linux launchers.
\end{itemize}

Historical archive SHA-256:
\begin{quote}\footnotesize
\path{2c6bd32527c4fa71dbe1b55b4b00843dfd6d50b91e9227626e0b1b7991cdb807}
\end{quote}
The complete public Python Test ZIP used for this Laboratory has SHA-256
\begin{quote}\footnotesize
\path{884f41fe1284071f1d6f1862d5684a46d58747af77fe0e2680fa52c6d6b1702b}
\end{quote}
Verify the preserved historical archive directly with
\begin{verbatim}
python lab_test.py --verify
\end{verbatim}
The historical artifact is kept unchanged because it is the software candidate that earned the original tested result. Later Frostyverse development changed parts of the physical interpretation, especially photon/FQ transport; those later mechanics are not retroactively inserted into this package.

\section{What a PASS does and does not mean}
A successful reproduction establishes only that:
\begin{itemize}
\item the stated historical FV equations are represented by the supplied code;
\item the focused calculation reproduces the documented $\beta=0.80$ result;
\item fresh subluminal speeds and boost orientations can be tested without changing source;
\item the candidate output is generated before the SR comparison routine is opened;
\item the tested local clock/ruler/light observables are Lorentz-compatible to the stated numerical tolerance.
\end{itemize}

It does \emph{not} establish that the FrostGrid exists, that exact microscopic Lorentz symmetry is fundamental, that every relativistic observable has been derived from these rules, or that the historical saturation/pressure interpretation is physically correct. Those remain independent scientific questions.

\end{document}