\documentclass[11pt]{article} \usepackage[margin=0.82in]{geometry} \usepackage{amsmath,amssymb} \usepackage{booktabs,array} \usepackage{enumitem} \usepackage{xcolor} \usepackage{hyperref} \usepackage{url} \usepackage[T1]{fontenc} \hypersetup{colorlinks=true,linkcolor=blue!45!black,urlcolor=blue!45!black} \setlist{nosep,leftmargin=1.4em} \setlength{\parindent}{0pt} \setlength{\parskip}{0.45em} \setlength{\emergencystretch}{3em} \newcommand{\FV}{\textsc{Frostyverse}} \newcommand{\code}[1]{\texttt{#1}} \newcommand{\statusbox}[1]{\noindent\fbox{\begin{minipage}{0.94\linewidth}\small #1\end{minipage}}} \begin{document} \title{FrostTanglement / Bell--CHSH Rule Audit\\\large Declared Compatibility Mathematics, Monte Carlo Consequences, and Open 3-D FrostGrid Derivation} \author{Frostyverse Laboratory} \date{Website Laboratory v1.0.48} \maketitle \statusbox{\textbf{Scientific status.} This is a \emph{declared-rule consequence audit}. The two-component $\cos^2\delta$ compatibility rule is supplied as a Frostyverse postulate and is not independently derived here from raw FrostNode positions, FrostSpace distances, FrostSegment/FrostLine dynamics, FrostCell handoffs, detector boundary conditions, or a complete 3-D FrostGrid update. A software PASS means that the supplied rule has the stated mathematical/statistical consequences. It is not experimental confirmation of Frostyverse and is not a lower-level derivation of Bell correlations.} \begin{abstract} This note exposes the exact mathematics executed by the public FrostTanglement Python Test. Unlike the gravitational and kinematic Frostyverse Labs, the central Bell-compatible functional form is not presently the output of a frozen microscopic H-test. It is a declared compatibility postulate. Given that postulate, the audit constructs balanced joint probabilities, verifies exact $50/50$ local marginals, derives $E(\delta)=\cos(2\delta)$ in the ideal same-result convention, obtains the canonical CHSH value $S=2\sqrt2$, samples those probabilities by Monte Carlo, exposes a provisional coherence control $C$, and checks the stated parity-key consequence used in the entanglement-swapping narrative. The decisive unresolved problem is to obtain the compatibility rule, or an equivalent, independently from ordinary 3-D FrostGrid mechanics without inserting the Bell-compatible angular law by hand. \end{abstract} \section{Question under attack} The physical FrostTanglement proposal is that two daughter structures may remain part of one shared relational state carried through the connected FrostGrid/FrostLine/FrostCell network. As the daughters move, the relationship is proposed to be relayed through successive local grid handoffs rather than by stretching one original literal segment indefinitely. That narrative is not yet enough. The mathematical question is whether ordinary lower-level FrostGrid mechanics generate the required detector-angle dependence. The present public Lab therefore starts at a clearly marked boundary: \[ \boxed{\text{declared compatibility rule}} \quad\Longrightarrow\quad \boxed{\text{joint probabilities and CHSH consequences}}. \] The left-hand box is supplied. The right-hand box is what this Lab audits. A future complete detector simulation must instead establish a chain of the form \[ \begin{gathered} \boxed{\text{raw 3-D FrostGrid mechanics}}\\[2pt] \Downarrow\\[2pt] \boxed{\text{compatibility rule}} \quad\Longrightarrow\quad \boxed{\text{Bell/CHSH statistics}}. \end{gathered} \] Failure of the first arrow would require the proposed physical mechanism to be revised or rejected. \section{Declared two-component compatibility rule} Let detector settings about the common propagation axis be $a$ and $b$. The Python wrapper retains the signed relative angle \[ \delta=a-b, \] which is sufficient because the relevant functions are $180^\circ$ periodic. For a constituent relation $i$, let $(\hat f_{1,i},\hat f_{2,i})$ denote two orthogonal transverse directions associated with the shared FrostArc/FrostLine frame, and let $(\hat d_1,\hat d_2)$ denote the corresponding detector frame. The current proposal supplies the two-component product \[ \boxed{ w_i=(\hat f_{1,i}\cdot\hat d_1) (\hat f_{2,i}\cdot\hat d_2) }. \] In the ideal coherent symmetric case, both component misalignments are the same relative rotation. Each overlap contributes a factor $\cos\delta$, giving the declared aggregate weight \[ \boxed{W(\delta)=\cos^2\delta}. \] This multiplication is the key \emph{postulate}. The present Lab does not derive it from lower-level grid geometry. \section{Provisional coherence control} The public test also exposes a deliberately provisional scalar coherence parameter \[ 0\le C\le1. \] The declared degraded weight is \[ \boxed{ W_C(\delta)=C\,W(\delta)+(1-C)\frac12 }. \] Using $\cos^2\delta=(1+\cos2\delta)/2$, \[ W_C(\delta)=\frac12\left[1+C\cos(2\delta)\right]. \] Thus $C=1$ returns the ideal rule, while $C=0$ returns a random favored/complementary relation with probability $1/2$. \statusbox{\textbf{Important boundary.} No physical law currently maps separation distance, intervening material, field deformation, temperature, detector properties, or any other environmental variable to $C$. The coherence control is an exposed rule parameter for stress testing. It is not presently a prediction of an entanglement range or decoherence length.} \section{Exact routing algorithm used by the Python audit} The implementation separates the local outcome choice from the shared relation choice. For each trial it draws two independent uniform random coordinates: \[ u_{\rm local},u_{\rm route}\sim U[0,1). \] The first coordinate assigns the local A outcome without reference to detector B: \[ A= \begin{cases} +1,&u_{\rm local}<1/2,\\ -1,&u_{\rm local}\ge1/2. \end{cases} \] The second coordinate determines whether the favored joint relation occurs: \[ u_{\rm route}