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\begin{document}
\title{Perihelion / Apsidal Precession\\\large Frostyverse Calculation, Blind-Icarus Reproduction, and Post-Seal GR Ruler}
\author{Frostyverse Laboratory}
\date{Website Laboratory v1.0.48}
\maketitle

\statusbox{\textbf{Scientific status.} The precession investigation is frozen for publication at H2-092. The public Python Test reproduces the historical Icarus calculation and exposes a fresh-orbit mode using the final H2-089 pair-total bookkeeping and H2-092 response multiplicity. The Frostyverse result is generated and SHA-256 sealed before the standard weak-field GR value is evaluated. A software PASS is not experimental proof that nature uses the Frostyverse.}

\begin{abstract}
This note exposes the actual numerical Frostyverse calculation used by the public Precession Python Test. A spherical FrostGrid profile supplies a dimensionless radial clock/deformation quantity $N(r)$. The selected pair mass scale is mapped into the finite numerical orbit, the inherited secular Hamiltonian-like stable and $D$ families are averaged around the orbit, and their eccentricity derivatives generate apsidal response. The final public prediction is
\[
\Delta\varpi_{\rm FV}=\Delta\varpi_{\rm stable}+4\,\Delta\varpi_D,
\]
where the factor four is the H2-092 response-context multiplicity acting on one conservative $D$-scale FrostTransit handoff, not four copies of stored energy. Only after the FV payload is sealed does the wrapper evaluate
\[
\Delta\varpi_{\rm GR}=\frac{6\pi G(M_1+M_2)}{a(1-e^2)c^2}
\]
as an external weak-field two-body comparison ruler. The preserved blind H2-087 Icarus R80 result is $10.062877461882765$ arcsec/century versus a post-seal GR ruler of $10.063089091251317$ arcsec/century.
\end{abstract}

\section{What this Laboratory actually tests}
The public test has two distinct scientific questions.
\begin{enumerate}
\item \textbf{FV calculation:} given source/body gravitational parameters, semi-major axis, eccentricity, and numerical resolution, does the frozen Frostyverse calculation generate a reproducible apsidal response without receiving the GR answer?
\item \textbf{External comparison:} after that result is sealed, how close is it to the standard isolated weak-field two-body GR perihelion contribution for the same physical inputs?
\end{enumerate}

The historical H2 chain also asks a deeper internal-mechanics question: why does the final weak-field response behave like stable $\sim2D$ plus an effective $\sim4D$ response? H2-090 through H2-092 are retained because they prevent the structural factor four from being misrepresented as duplicated energy.

\section{Physical inputs and final two-body bookkeeping}
For a fresh orbit the reviewer supplies
\begin{itemize}
\item source gravitational parameter $GM_1$ in km$^3$/s$^2$;
\item secondary/body gravitational parameter $GM_2$ in km$^3$/s$^2$ (zero is allowed for a test particle);
\item physical semi-major axis $a_{\rm phys}$ in km (the interactive wrapper also accepts AU);
\item eccentricity $e$;
\item numerical resolution $R\in\{20,40,80\}$.
\end{itemize}

The final H2-089 working bookkeeping is
\[
GM_{\rm pair}=GM_1+GM_2.
\]
In the public custom mode the same pair-total scale is used for both the effective FV field scale and the relative-orbit scale. This is an \emph{effective two-domain bookkeeping rule}; it is not a claim that one body literally exerts a Newtonian self-force on itself.

The historical blind H2-087 Icarus run predates this final bookkeeping selection and was test-particle-like ($GM_2=0$), so the later pair-total rule does not alter its sealed prediction.

\section{Numerical orbit normalization: geometry only}
The historical solver works inside a finite dimensionless radial profile. The physical orbit is mapped to a numerical semi-major axis $a_{\rm n}$ with
\[
r_{p,{\rm n}}=a_{\rm n}(1-e),\qquad
r_{a,{\rm n}}=a_{\rm n}(1+e).
\]
The public wrapper uses the fixed safe profile interval
\[
0.62\le r_{\rm n}\le9.70.
\]
Therefore
\[
\frac{0.62}{1-e}\le a_{\rm n}\le\frac{9.70}{1+e}.
\]
Unless the reviewer explicitly requests another valid value, the program chooses $a_{\rm n}$ from this geometry-only interval, centered around the inherited default where possible. No GR target enters this choice. A second valid $a_{\rm n}$ is also evaluated when possible and reported as an FV-only coordinate-sensitivity check.

Very low eccentricity is numerically delicate because perihelion angle becomes poorly conditioned. The public test warns for $e<0.02$ and limits this finite-profile implementation to approximately $e<0.88$.

\section{Mapping the physical pair into the dimensionless FV orbit}
The historical profile uses a baseline dimensionless mass parameter $\mu_0=10^{-6}$. The public wrapper forms the scale factor
\[
s_\mu
=\frac{GM_{\rm pair}\,a_{\rm n}}
{c^2 a_{\rm phys}\,\mu_0}.
\]
The orbit calculation then uses
\[
\mu=\mu_0 s_\mu
=\frac{GM_{\rm pair}\,a_{\rm n}}
{c^2a_{\rm phys}}.
\]
Thus $\mu_0$ is a historical numerical normalization; it cancels from the mapped physical dimensionless strength and is not a newly fitted physical constant.

\section{Frozen radial FrostGrid profile}
At the requested resolution, the public wrapper calls the historical H2-089 code to build the same compact spherical FrostGrid field and radial profile used by that package. The inherited profile construction uses the frozen current-cell convergence envelope, three-axis FrostCell clock coupling, and the H-017-derived flat-clock exponent $\alpha=1/2$ to form a radial factor $N(r)$.

For orientation, the relevant inherited construction has the schematic form
\[
A_{\rm raw}(r)=K R^2\sqrt{C_{\rm cell}(r)},
\]
\[
\Phi(r)=\int_r^{r_0} A_{\rm raw}(r')\,C_{3D}(r')\,dr',
\qquad
N(r)=\exp[-\alpha\Phi(r)],
\]
with the historical frozen $K$ and $p=0$ gravity field mechanics. The precession secular lane described below consumes $N(r)$ through $\ln N(r)$; it does not receive a GR precession angle.

This document does not re-derive the entire source-to-profile gravity program. That provenance remains in the frozen historical archives. The important audit point here is the dataflow from the frozen FV profile into the apsidal calculation.

\section{Exact Frostyverse secular calculation used by the public Python test}
The following equations mirror the H2-089 functions called by the public wrapper.

For eccentric anomaly $E$ sampled uniformly around the orbit,
\[
r(E)=a_{\rm n}(1-e\cos E).
\]
The radial FV quantity used by the secular families is
\[
n(r)=s_\mu\,\ln N(r).
\]
Define
\[
L=\sqrt{\mu a_{\rm n}(1-e^2)},
\]
\[
p_r(E)=\frac{\sqrt{\mu a_{\rm n}}\,e\sin E}{r(E)},
\qquad
v_t^2(E)=\frac{L^2}{r(E)^2},
\]
and
\[
q_0(E)=p_r(E)^2+v_t(E)^2.
\]

The historical stable family is
\[
h_{\rm stable}(E)
=\frac12n^2
+\frac12nq_0
-\frac18q_0^2,
\]
while the inherited $D$ family is
\[
h_D(E)=-\frac18q_0^2.
\]
For either family $f\in\{\mathrm{stable},D\}$ the orbit average used by the code is
\[
\overline H_f(e^2)
=\left\langle h_f(E)\,[1-e\cos E]\right\rangle_E.
\]
The program evaluates the derivative with respect to
\[
s=e^2
\]
using a centered fourth-order finite difference. With
\[
L_d=\sqrt{\mu a_{\rm n}},
\qquad
n_{\rm orb}=\sqrt{\frac{\mu}{a_{\rm n}^3}},
\]
the secular apsidal contribution returned by the historical function is
\[
\boxed{
\Delta\varpi_f
=-\frac{4\pi}{n_{\rm orb}}
\frac{\sqrt{1-e^2}}{L_d}
\frac{\partial\overline H_f}{\partial(e^2)}
}
\]
in radians per orbit (the program converts this to degrees).

The final public Frostyverse prediction is therefore
\[
\boxed{
\Delta\varpi_{\rm FV}
=\Delta\varpi_{\rm stable}
+4\,\Delta\varpi_D
}.
\]
This is the calculation a reviewer should compare against the source code. It is not a wrapper that simply evaluates the GR expression under a different name.

\section{Where the $D$ and the structural four come from}
\subsection{D-scale foundation}
H2-038 traces the lower motion-strain channel to inherited H-025/H-041 FrostSegment mechanics. At low completion strain $q$,
\[
W\sim\frac{q^2}{8},
\]
and in the natural strain coordinate the local elastic work curvature is
\[
W\propto\frac12\varepsilon^2.
\]
H2-044/H2-045 found the stable resolved weak-field orbital response near
\[
H_{\rm stable}\approx2D.
\]

\subsection{H2-090: one conservative handoff still owns one D}
H2-090 directly feeds the exact H-025 strained FrostSegment vector into the H-017 FrostTransit displacement vector. The shared-coordinate decode closes to floating precision and gives
\[
\frac{W_{\rm source}}{D}=0.5,
\qquad
\frac{W_{\rm destination}}{D}=0.5,
\qquad
\frac{W_{\rm full\ handoff}}{D}=1.
\]
Four transverse modes partition that one handoff,
\[
\sum_{m=1}^4 W_m=D,
\]
so ``four modes = four D-scale energy copies'' is explicitly rejected.

\subsection{H2-091: eight FrostNodes participate without cloning D}
A normal interior FrostTransit has nonzero trilinear participation from all eight FrostCell corners, but
\[
\sum_{n=1}^8W_n=D.
\]
The naive ownership pictures $8\times\frac12D$ and $4\times D$ are therefore double counting.

\subsection{H2-092: response-context multiplicity}
A FrostCell has eight corner FrostNodes, each with six H-021 support contexts, giving 48 node-local response contexts. Relative to the 12 internal FrostCell edges, the inherited transverse squared-response weighting
\[
w(\hat u,\hat s)=1-(\hat u\cdot\hat s)^2
\]
gives
\[
M_{48}=\frac{S_{48}}{S_{12}}=4
\]
to floating precision across the published deformation/direction audit. Deliberately changed controls give
\[
\frac{S_{36}}{S_{12}}=3,
\qquad
\frac{S_{24}}{S_{12}}=2.
\]
The final four is therefore a \emph{response-context multiplicity acting on the D-scale handoff}, not new stored energy.

\section{From per-orbit FV response to arcseconds per century}
For the physical pair,
\[
T=2\pi\sqrt{\frac{a_{\rm phys}^3}{GM_{\rm pair}}}.
\]
If $T_{\rm day}$ is the period in days and $\Delta\varpi_{\rm FV,deg}$ is the FV result in degrees/orbit, the reported century rate is
\[
\dot\varpi_{\rm FV}
=\Delta\varpi_{\rm FV,deg}\times3600
\times\frac{36525}{T_{\rm day}}
\]
in arcsec/century.

\section{FV seal and independent GR comparison}
Before evaluating GR, the wrapper serializes the complete FV input/result payload and computes a SHA-256 fingerprint. Only after that seal does it evaluate the standard isolated weak-field two-body ruler
\[
\boxed{
\Delta\varpi_{\rm GR}
=\frac{6\pi GM_{\rm pair}}
{a_{\rm phys}(1-e^2)c^2}
}
\]
in radians per orbit.

The same physical pair period converts the GR value to arcsec/century. The comparison printed by the Python test is
\[
\delta_{\rm rel}
=\frac{\Delta\varpi_{\rm FV}-\Delta\varpi_{\rm GR}}
{\Delta\varpi_{\rm GR}}.
\]

\statusbox{\textbf{Comparison firewall.} $\Delta\varpi_{\rm GR}$, the GR coordinates, and the target GR residual are not inputs to the FV profile, the stable-family calculation, the $D$ calculation, the H2-092 factor, or the geometry-only numerical normalization. The GR value is evaluated only after the FV payload has been sealed.}

The GR expression above is only the isolated relativistic two-body apsidal contribution. It does not include third-body perturbations, source oblateness, tides, or other classical contributions that can dominate the observed total apsidal motion of systems such as the Moon or Europa.

\section{Reference reproduction: Icarus 1566}
The public reference mode uses the historical H2-087 Icarus physical setup
\[
GM_1=1.32712440041279419\times10^{11}\ \mathrm{km^3/s^2},
\qquad GM_2=0,
\]
\[
a_{\rm phys}=1.078\ \mathrm{AU},
\qquad e=0.827,
\]
and the exact historical primary mapping
\[
a_{\rm n}=\frac{9.5}{1+e}.
\]

The preserved ladder is
\begin{center}
\begin{tabular}{lrr}
\toprule
Resolution & FV prediction (arcsec/century) & Comment \\
\midrule
R20 & 10.0605278916 & quick reference \\
R40 & 10.0623924511 & recommended workstation check \\
R80 & 10.062877461882765 & publication-resolution sealed row \\
\bottomrule
\end{tabular}
\end{center}

The post-seal ruler is
\[
\Delta\varpi_{\rm GR}=10.063089091251317\ \mathrm{arcsec/century},
\]
so the preserved R80 relative difference is approximately
\[
-0.002103\%.
\]
The R80 calculation is not replaced by a fitted low-memory surrogate. The public README warns that its main preserved field array is roughly 4.1 GB before NumPy/Python overhead; R40 is the recommended ordinary-workstation reproduction.

\section{Fresh-orbit mode}
The independent value of the public package is not limited to replaying Icarus. The reviewer can enter a new physically reasonable orbit. The program then repeats the same sequence:
\begin{enumerate}
\item build the frozen spherical FV profile at the selected $R$;
\item map the physical $GM_{\rm pair},a_{\rm phys},e$ into the dimensionless orbit using geometry-only normalization;
\item calculate $\Delta\varpi_{\rm stable}$ and $\Delta\varpi_D$ from the historical H2-089 secular functions;
\item form $\Delta\varpi_{\rm FV}=\Delta\varpi_{\rm stable}+4\Delta\varpi_D$;
\item optionally repeat at an alternate legal $a_{\rm n}$ as an FV-only coordinate check;
\item SHA-256 seal the FV payload;
\item only then evaluate the GR formula and print the difference.
\end{enumerate}
A fresh-input failure is scientifically useful and should be reported rather than repaired by changing the factor four or fitting a new orbit-specific coefficient.

\section{Run the public Python Test}
Requirements are Python 3.10+ and NumPy. From the extracted Python Test ZIP:

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

Recommended Icarus reference reproduction:
\begin{verbatim}
python lab_test.py --reference 40
\end{verbatim}

Publication-resolution Icarus reproduction (high RAM):
\begin{verbatim}
python lab_test.py --reference 80
\end{verbatim}

Example fresh orbit using explicit physical inputs:
\begin{verbatim}
python lab_test.py --custom \
  --source-name CUSTOM_SOURCE \
  --source-gm 132712440041.279419 \
  --body-gm 0 \
  --a-km 149597870.7 \
  --e 0.20 \
  --R 40
\end{verbatim}

Verify bundled historical hashes:
\begin{verbatim}
python lab_test.py --verify
\end{verbatim}

Run an unchanged historical validation profile:
\begin{verbatim}
python lab_test.py --historical H2-087 --profile validation
python lab_test.py --historical H2-089 --profile validation
python lab_test.py --historical H2-092 --profile validation
\end{verbatim}
Replace \code{validation} with \code{full} for the exact historical full profile when the machine has sufficient resources.

\section{What to inspect in the source}
A short independent source audit should verify the following concrete dataflow:
\begin{itemize}
\item \code{build\_compact\_field} and \code{build\_compact\_profile} generate the inherited FV radial profile;
\item \code{h\_average} implements the stable and $D$ families written above;
\item \code{secular\_deg} differentiates the orbit average with respect to $e^2$ and returns apsidal response;
\item public custom mode uses the H2-089 pair-total scale for both field and relative orbit;
\item the public wrapper forms \code{stable + 4*D};
\item the FV payload is fingerprinted before \code{\_gr\_after\_seal} is called;
\item changing a GR target cannot change the already generated FV payload.
\end{itemize}

\section{Named-case interpretation and low-e guardrail}
The historical chain contains Mercury, Venus, Icarus, Moon, and Europa cases, but they do not all carry the same evidentiary weight.
\begin{itemize}
\item Mercury was a development/control case.
\item Icarus is the clean high-e blind holdout.
\item Moon is a useful Earth--Moon cross-source test, but observed lunar apsidal motion contains large solar/three-body and Earth-figure contributions.
\item Venus and Europa have very low eccentricity and are conditioning-sensitive in perihelion-coordinate extraction. Their raw residuals must not automatically be interpreted as physical falsifications.
\end{itemize}

\section{Historical chronology that must remain separate}
\begin{itemize}
\item \textbf{H2-087:} blind cross-source prediction with structural coefficient frozen before GR/context opened; official Icarus R80 FV value $10.062877461882765$ arcsec/century.
\item \textbf{H2-089:} separate FV-only mass-ratio homogeneity stress selects \code{PAIR\_TOTAL\_EVERYWHERE}, $M_{\rm pair}=M_1+M_2$, as the simple universal two-domain bookkeeping rule.
\item \textbf{H2-090/H2-091:} conservative work ownership closes at one $D$ per complete handoff and rejects duplicated-energy interpretations.
\item \textbf{H2-092:} exact H-021 node-local response-context multiplicity four survives weighted geometry/deformation stress; 3x and 2x definitions remain negative controls.
\end{itemize}
The later H2-089 pair-total rule must not be retroactively described as an ingredient that produced the already sealed H2-087 Icarus result.

\section{Provenance and integrity}
Public reviewer package:
\begin{quote}\footnotesize
\path{Frostyverse_Precession_Python_Test.zip}\\
SHA-256: \path{45591303279370232e07e5438c598f5c0d2312b1e432fcd074cfc4be8bb5f5a5}
\end{quote}

Frozen historical archives inside \code{original/}:
\begin{quote}\footnotesize
\path{Frostyverse-Test-H2-087.zip}\\
SHA-256: \path{1a703ee7e47a0d96bd01dbb2eda160c265bed9f649969b8a34ef92528a892f22}\\[0.5em]
\path{Frostyverse-Test-H2-089.zip}\\
SHA-256: \path{3d7732996163158df58cd9f266573ec7f657b65d15611561c8e927149c18f260}\\[0.5em]
\path{Frostyverse-Test-H2-092.zip}\\
SHA-256: \path{f7352267aa6e6394d7dcc9b72bf6928dcbf1c4ccc655c1b8d072bbacb3afbcb2}
\end{quote}
The historical ZIPs are verified before use and are executed from temporary extracted copies; the public wrapper does not modify them in place.

\section{Recommended falsification attacks}
\begin{itemize}
\item reproduce Icarus at R20/R40/R80 and verify convergence without changing the FV equations;
\item choose a fresh moderate-e orbit and compare the generated FV result with the post-seal GR ruler;
\item inspect whether the GR function is called only after the FV fingerprint is created;
\item vary the legal numerical $a_{\rm n}$ and examine the reported coordinate sensitivity;
\item run the H2-089 $q$-homogeneity audit and verify why pair-total survives while the alternatives drift;
\item rerun H2-092 and confirm the 4/3/2 response-definition controls;
\item do not rescue a failed fresh case by retuning the response factor, source strength, eccentricity, or coordinate map.
\end{itemize}

\section{Claim boundary}
The following are mechanically/internal-software results within the declared Frostyverse program: the D-scale work family, stable weak-field response near $2D$, the H-025$\to$H-017 physical coordinate bridge, one-D conservative handoff ownership, all-eight-node geometric participation, H-021 response-context multiplicity four, and the H2-089 pair-total homogeneity result.

The close Icarus agreement with the post-seal GR ruler is an external/effective validation of this frozen calculation. It does not establish that the Frostyverse ontology is physically real, and it does not replace independent tests on fresh systems.

\end{document}