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\begin{document}
\title{Static Gravitational Redshift\\\large H-017 Clock Exponent, Three-Axis FrostCell Coupling, and Post-Seal Schwarzschild Ruler}
\author{Frostyverse Laboratory}
\date{Website Laboratory v1.0.48}
\maketitle

\statusbox{\textbf{Scientific status.} This Laboratory reproduces the provisionally frozen H-117/H-118 static gravitational-clock candidate. The flat-grid FrostTransit clock exponent is re-derived from H-017; the H-116 three-axis FrostCell coupling and radial field construction are then used without retuning. The FV clock-rate ratio is generated and SHA-256 sealed while the General Relativity (GR) redshift ruler is closed. The Schwarzschild stationary-clock comparison is opened only afterward. A software PASS is an internal computational result, not experimental evidence that nature contains FrostCells.}

\begin{abstract}
This note exposes the mathematics executed by the public Gravitational Redshift Python Test so a physicist can audit the candidate without reverse-engineering the historical source. The calculation is intentionally independent of the Shapiro path-timing observable: two completed-state clocks are held at different normalized radii and the same gravitational clock mechanics must predict their relative rate without a new fitted coefficient. H-117 first measures how the inherited flat-grid H-017 clock responds to local FQ--FrostSegment coupling, then applies that measured exponent to the frozen H-116 three-axis gravitational coupling profile. The FV result is sealed before a Schwarzschild stationary-clock ruler is evaluated. H-118 carries the unchanged mechanism to higher numerical resolution.
\end{abstract}

\section{Question under attack}
The Shapiro Laboratory integrates timing along a moving photon trajectory. This Laboratory asks a different question: if two stationary completed-state clocks occupy different gravitational depths, does the same FV clock machinery predict the relative rate of the inner and outer clocks?

No new redshift coefficient is fitted. The public wrapper calls the unchanged H-117 machinery and preserves the following sequence:
\begin{enumerate}
\item derive a flat-grid clock exponent from H-017 itself;
\item solve the frozen symmetry-clean $p=0$ FrostField;
\item construct the inherited H-116 three-axis clock coupling;
\item integrate the corresponding logarithmic coupling deficit radially;
\item predict the stationary FV clock-rate ratio and seal it; and
\item only then evaluate an external Schwarzschild clock comparison.
\end{enumerate}

This is the static gravitational-clock candidate. It is not the separate cosmological redshift/tired-light proposal discussed elsewhere in the Frostyverse manuscript.

\section{Normalized public inputs}
The focused public test exposes only quantities already meaningful inside H-117:
\begin{center}
\begin{tabular}{ll}
\toprule
Input & Meaning \\
\midrule
$R$ & numerical FrostField resolution \\
$r_i$ & normalized radius of the inner/deeper stationary clock \\
$r_o$ & normalized radius of the outer/reference stationary clock \\
workers & numerical worker count passed to the historical solver \\
\bottomrule
\end{tabular}
\end{center}

The focused domain is
\[
12\le R\le80,
\qquad
1.10\le r_i<r_o\le9.90.
\]
The outer radial zero shell is frozen at
\[
\boxed{r_0=9.90}.
\]

\statusbox{\textbf{Source-mapping boundary.} The source normalization is deliberately not exposed as a free physical mass knob. H-117 was defined for its frozen normalized FV source, while its external Schwarzschild ruler uses a synthetic normalized mass parameter $M=10^{-6}$. This Laboratory does not invent a general physical mass-to-FrostField-source map that the historical test did not establish.}

\section{Step 1: derive the flat-FV clock exponent from H-017}
H-117 does not insert the number $1/2$ into the gravitational redshift calculation. It first measures the clock response of the inherited flat-grid FrostTransit mechanism.

The local FQ--FrostSegment coupling is sampled at
\[
k\in\{0.25,\ 0.50,\ 1,\ 2,\ 4\}
\]
for three direction classes:
\[
\text{axis}=(1,0,0),\qquad
\text{face}=(1,1,0),\qquad
\text{body}=(1,1,1).
\]
For each run the historical H-017 solver returns a turning time $T(k)$, and H-117 defines the clock rate
\[
\Gamma(k)=\frac{1}{T(k)}.
\]
The exponent for each direction family is the log--log slope
\[
\boxed{
\alpha_{\rm FV}
=\frac{d\ln\Gamma}{d\ln k}
}.
\]
Equivalently, the regression being tested is
\[
\ln\Gamma=\alpha_{\rm FV}\ln k+b.
\]
The public reference freshly re-derives
\[
\alpha_{\rm axis}
=\alpha_{\rm face}
=\alpha_{\rm body}
=0.500000000000
\]
to numerical precision, with zero direction-family spread at the printed precision and a maximum log-fit residual of approximately $7.38\times10^{-15}$.

Thus the candidate flat-grid scaling is measured as
\[
\Gamma\propto k^{\alpha_{\rm FV}},
\qquad
\alpha_{\rm FV}\simeq\frac12,
\]
but the numeric $1/2$ is an \emph{output} of this pre-gravity calculation, not an input supplied to the redshift fit.

\section{Step 2: frozen three-axis FrostCell clock coupling}
\subsection{Principal-strain scalar}
At each sampled FrostCell, the inherited H-116 code fits the local affine deformation of the eight cell corners. If $F$ is the local deformation matrix and
\[
\sigma_1,\sigma_2,\sigma_3>0
\]
are its singular values, the principal logarithmic strains are
\[
\boxed{\epsilon_j=\ln\sigma_j}.
\]
Let
\[
e_j=|\epsilon_j|,
\qquad
e_{\max}=\max_j e_j.
\]
The frozen H-116 completed-state clock coupling is
\[
\boxed{
C_{3D}
=\frac{\frac13(e_1+e_2+e_3)}{e_{\max}}
=\frac{\operatorname{mean}_j|\epsilon_j|}
       {\max_j|\epsilon_j|}
}.
\]
For an effectively undeformed numerical cell, the implementation uses the neutral upper-bound value $C_{3D}=1$ instead of dividing by floating-point noise. The scalar is built from all three principal deformation directions; no GR clock factor enters its construction.

\subsection{Raw and clock-weighted radial envelopes}
At radial samples $r_n=n/R$ in the H-117 profile, the inherited field machinery supplies the current-cell convergence magnitude $C_{\rm cell}(r_n)$. Using the previously frozen constant
\[
K=8.681626541587185\times10^{-5},
\]
H-117 forms the raw envelope
\[
\boxed{
A_{\rm raw}(r)
=K R^2\sqrt{C_{\rm cell}(r)}
}.
\]
The gravitational-clock candidate weights that envelope by the local three-axis coupling:
\[
\boxed{
A_{\rm clock}(r)
=C_{3D}(r)A_{\rm raw}(r)
=C_{3D}(r)K R^2\sqrt{C_{\rm cell}(r)}
}.
\]
The unweighted $A_{\rm raw}$ lane is retained as a negative control.

\section{Step 3: logarithmic coupling deficit}
H-117's new static-clock hypothesis is that $A_{\rm clock}$ is the radial gradient of an accumulated logarithmic local-coupling deficit. With the finite outer shell $r_0$ set to zero,
\[
\boxed{
\Phi_{\rm clock}(r)
=\int_r^{r_0}A_{\rm clock}(s)\,ds,
\qquad
\Phi_{\rm clock}(r_0)=0
}.
\]
The program evaluates this inward on the discrete radial table using the trapezoidal update
\[
\Phi_j
=\Phi_{j+1}
+\frac12(A_j+A_{j+1})(r_{j+1}-r_j).
\]
The associated effective local coupling ratio is
\[
\boxed{
\frac{k_{\rm eff}(r)}{k_{\rm eff}(r_0)}
=e^{-\Phi_{\rm clock}(r)}
}.
\]
For two clocks, define
\[
\Delta\Phi_{\rm clock}
=\Phi_{\rm clock}(r_i)-\Phi_{\rm clock}(r_o).
\]
Because the inner clock is deeper in the tested field, the candidate has $\Delta\Phi_{\rm clock}>0$.

\section{Step 4: FV stationary-clock prediction}
The independently measured H-017 exponent converts the local-coupling difference into a completed-state clock-rate ratio:
\[
\frac{\Gamma_i}{\Gamma_o}
=\left(\frac{k_{{\rm eff},i}}{k_{{\rm eff},o}}\right)^{\alpha_{\rm FV}}.
\]
Substituting the exponential coupling profile gives the exact H-117 candidate used by the public Python Test:
\[
\boxed{
\frac{\Gamma_i}{\Gamma_o}
=\exp\!\left[-\alpha_{\rm FV}\,\Delta\Phi_{\rm clock}\right]
}.
\]
The historical software defines its positive weak-field rate deficit as
\[
\boxed{
z_{\rm FV}
=1-\frac{\Gamma_i}{\Gamma_o}
}.
\]
This is the quantity scored by H-117. For the tiny shifts in this normalized test it agrees at first order with the corresponding reciprocal redshift convention, but reviewers should reproduce the historical definition above rather than silently substitute another convention.

Three controls are generated before the external ruler is opened:
\begin{align*}
\left(\frac{\Gamma_i}{\Gamma_o}\right)_{\alpha=1}
&=e^{-\Delta\Phi_{\rm clock}},\\
\left(\frac{\Gamma_i}{\Gamma_o}\right)_{\rm no\ C3D}
&=e^{-\alpha_{\rm FV}\Delta\Phi_{\rm raw}},\\
\left(\frac{\Gamma_i}{\Gamma_o}\right)_{\rm reversed}
&=e^{+\alpha_{\rm FV}\Delta\Phi_{\rm clock}}.
\end{align*}
The reversed-sign lane should make the deeper clock faster and is therefore deliberately the wrong-direction control.

\section{FV-first firewall and candidate seal}
The focused wrapper executes the following order:
\begin{enumerate}
\item re-derive $\alpha_{\rm FV}$ from H-017 while gravity/GR scoring is closed;
\item solve the unchanged H-117 symmetry-clean $p=0$ field;
\item construct $C_{3D}(r)$, $\Phi_{\rm raw}(r)$, and $\Phi_{\rm clock}(r)$;
\item compute the FV clock ratio, $z_{\rm FV}$, controls, and X/Y/Z covariance diagnostics;
\item serialize the FV-only scientific payload and SHA-256 seal it; and
\item only then call the historical Schwarzschild stationary-clock function.
\end{enumerate}
Runtime is excluded from the scientific seal because it is machine dependent. The wrapper prints
\begin{verbatim}
GR consulted before seal: NO
\end{verbatim}
so the sequencing is visible during reproduction.

\section{External Schwarzschild stationary-clock ruler}
For a stationary clock outside a spherical mass, the standard Schwarzschild coordinate-time relation is
\[
\frac{d\tau}{dt}
=\sqrt{1-\frac{2GM}{rc^2}}.
\]
Therefore, for stationary clocks at $r_i$ and $r_o$, the external rate-ratio ruler is
\[
\boxed{
\left(\frac{\Gamma_i}{\Gamma_o}\right)_{\rm GR}
=
\sqrt{
\frac{1-2GM/(r_i c^2)}
     {1-2GM/(r_o c^2)}
}
}.
\]
The historical H-117 comparator uses normalized units $G=c=1$ and a synthetic comparison mass
\[
M=10^{-6},
\]
so its implemented ruler is
\[
\boxed{
\left(\frac{\Gamma_i}{\Gamma_o}\right)_{\rm GR}
=
\sqrt{
\frac{1-2M/r_i}
     {1-2M/r_o}
}
}.
\]
It then uses the same historical rate-deficit convention,
\[
\boxed{
z_{\rm GR}=1-\left(\frac{\Gamma_i}{\Gamma_o}\right)_{\rm GR}}.
\]
Only after the FV seal are the score quantities calculated:
\[
\mathcal R_z=\frac{z_{\rm FV}}{z_{\rm GR}},
\qquad
\epsilon_z=|\mathcal R_z-1|.
\]

\statusbox{\textbf{Comparison firewall.} The Schwarzschild expression is an external ruler. Its result does not choose $\alpha_{\rm FV}$, $C_{3D}$, the sign of the log-coupling deficit, $K$, the radial profile, or any correction coefficient. The normalized comparison mass $M=10^{-6}$ is not an FV source knob exposed to the reviewer.}

\section{Public focused reference}
The lightweight public reference is the preserved H-117 R20 clock pair
\[
R=20,
\qquad
r_i=1.60,
\qquad
r_o=2.70.
\]
A fresh run of the public package gives:
\begin{center}
\begin{tabular}{lr}
\toprule
Quantity & Reference result \\
\midrule
Derived $\alpha_{\rm FV}$ & $0.500000000000$ \\
FV inner/outer clock ratio & $0.999999750696187$ \\
FV $z$ & $2.493038133711\times10^{-7}$ \\
FV SHA-256 seal & \code{e985b940...b0fd76} \\
Post-seal GR clock ratio & $0.999999745370149$ \\
Post-seal GR $z$ & $2.546298506489\times10^{-7}$ \\
FV / GR & $0.979083217210$ \\
Relative difference & $2.091678\%$ \\
\bottomrule
\end{tabular}
\end{center}
The same run seals the negative/control lanes before GR opens:
\[
\Gamma_i/\Gamma_o|_{\alpha=1}=0.999999501392435,
\]
\[
\Gamma_i/\Gamma_o|_{\rm no\ C3D}=0.999999627688988,
\]
\[
\Gamma_i/\Gamma_o|_{\rm reversed}=1.000000249303876.
\]
The last value is greater than one, visibly confirming that reversing the predeclared sign makes the deeper clock faster.

\section{Fresh/custom-input attack}
A reviewer can choose a new legal stationary clock pair while leaving all FV mechanics frozen. For example,
\begin{verbatim}
python lab_test.py --custom --R 20 --inner 2.35 --outer 6.70
\end{verbatim}
A fresh run of that example produced:
\begin{center}
\begin{tabular}{lr}
\toprule
Quantity & Fresh result \\
\midrule
FV clock ratio & $0.999999726990419$ \\
FV $z$ & $2.730095809422\times10^{-7}$ \\
Post-seal GR $z$ & $2.762783042343\times10^{-7}$ \\
FV / GR & $0.988168729712$ \\
Relative difference & $1.183127\%$ \\
\bottomrule
\end{tabular}
\end{center}
The source normalization remains frozen. A failure on a fresh legal pair is a useful audit result and should be reported rather than tuned away.

\section{Historical holdouts and resolution behavior}
The complete H-117 profile uses five clock pairs,
\[
(1.60,2.70),\ (2.20,4.10),\ (3.75,6.25),\ (5.00,8.20),\ (1.85,8.50),
\]
with the main ladder $R=20,40,80$ and $R=12$ as validation.

The predeclared highest-resolution interpretation gates are
\begin{align*}
\max|z_{\rm FV}/z_{\rm GR}-1| &\le 1.0\%,\\
\text{pair-to-pair ratio spread} &\le 1.0\%,
\end{align*}
together with decreasing worst error over $R20\to R40\to R80$ and the inherited X/Y/Z coupling-covariance check. These are decision criteria, not fitted parameters.

The preserved full H-117 summary is:
\begin{center}
\begin{tabular}{lrrr}
\toprule
Resolution & Mean FV/GR & Worst error & Pair-ratio spread \\
\midrule
R20 & 0.987119 & 2.0917\% & 1.4421\% \\
R40 & 0.993491 & 1.0428\% & 0.7026\% \\
R80 & 0.996654 & 0.5274\% & 0.3457\% \\
\bottomrule
\end{tabular}
\end{center}
A linear $1/R$ diagnostic gives
\[
\left(z_{\rm FV}/z_{\rm GR}\right)_{1/R\to0}=0.999839941.
\]
That extrapolation is diagnostic only and is never fed back into the candidate.

H-118 keeps the same mechanism and extends the preserved worst redshift residual to approximately
\[
0.3380\%\quad\text{at R120},
\qquad
0.2723\%\quad\text{at R160}.
\]

\section{Run the public Python audit}
Install the public dependencies with
\begin{verbatim}
python -m pip install -r requirements.txt
\end{verbatim}
and open the interactive reviewer menu with
\begin{verbatim}
python lab_test.py
\end{verbatim}
The focused reference can be run directly:
\begin{verbatim}
python lab_test.py --reference
\end{verbatim}
A custom static clock pair is, for example,
\begin{verbatim}
python lab_test.py --custom --R 20 --inner 2.35 --outer 6.70
\end{verbatim}
The unchanged historical programs remain available:
\begin{verbatim}
python lab_test.py --historical h117-validation
python lab_test.py --historical h117-full
python lab_test.py --historical h118-validation
python lab_test.py --historical h118-full
\end{verbatim}
Verify the bundled historical archives with
\begin{verbatim}
python lab_test.py --verify
\end{verbatim}
The wrapper accepts \code{--workers N} for numerical worker control.

\subsection{Resource warning}
The historical high-resolution calculations are intentionally preserved and can be large. Reported main-array sizes are approximately
\[
3.810\ \text{GiB at R80},\qquad
12.864\ \text{GiB at R120},\qquad
30.499\ \text{GiB at R160},
\]
before Python/process overhead. H-118 also performs disk-backed slabbed exact-generation work. The public focused mode defaults to R20.

\section{Package map and provenance}
The reviewer-facing ZIP contains
\begin{itemize}
\item \code{lab\_test.py}: focused reference/custom wrapper and historical launcher;
\item \code{README\_FIRST.md}: quick-start instructions and scope;
\item \code{EXPECTED\_RESULTS.txt}: preserved result fingerprints;
\item \code{requirements.txt}: NumPy/SciPy requirements;
\item \code{run\_test.bat} and \code{run\_test.sh}: convenience launchers;
\item \code{results/}: timestamped focused TXT/JSON outputs; and
\item \code{original/}: unchanged H-117 and H-118 historical archives.
\end{itemize}

Historical H-117 archive:
\begin{quote}\footnotesize
\path{original/Frostyverse-Test-H-117.zip}\\
SHA-256: \path{814e6884f42c960c72ba4eb06795f171c7c4bd419cc5732f5235bda3e8cdf8ae}
\end{quote}
Historical H-118 archive:
\begin{quote}\footnotesize
\path{original/Frostyverse-Test-H-118.zip}\\
SHA-256: \path{21ecd71ea57392a42ca01b2cc35b294a69f3bcce5b11005c96477a95295588c0}
\end{quote}
Public Python Test ZIP:
\begin{quote}\footnotesize
\path{Frostyverse_Redshift_Python_Test.zip}\\
SHA-256: \path{a4be6cfbe4c1deca473949aba8f59a2e376c0ecf2f1fcc98d00623555ca2da41}
\end{quote}
The package-level \code{SHA256SUMS.txt} can be used to verify its individual files.

\section{Browser Laboratory scope}
The browser Laboratory is an orientation/replay surface for the preserved redshift calculation. It is not the full FrostField/H-017 solver. A physicist who wants to regenerate the FV field, re-derive $\alpha_{\rm FV}$, change a clock pair, inspect the seal, or run the historical resolution ladder should use the Python package.

\section{Claim boundary}
H-117/H-118 support a provisionally frozen \FV{} gravitational-clock candidate under the specified software tests. The calculation has a clear internal chain,
\[
\text{H-017 clock response}
\longrightarrow
\alpha_{\rm FV}
\longrightarrow
C_{3D}A_{\rm raw}
\longrightarrow
\Phi_{\rm clock}
\longrightarrow
\Gamma_i/\Gamma_o
\longrightarrow
z_{\rm FV},
\]
and the external Schwarzschild value is opened only after the FV result is sealed.

This does not establish that nature contains FrostNodes/FrostCells, does not prove GR from a grid, and does not establish a physical mass calibration for the normalized source. A fresh-input failure is scientifically useful evidence against the candidate and should not be repaired by fitting to the comparison ruler.

\end{document}