This repository contains the source of the Entanglement Presentation Note Cosmochrony paper
Residual-Rank Entanglement of Conjugate Weil Pairs in the Spectral Admissibility Cascade.

This note computes the entanglement entropy of a closed conjugate Weil pair ${c,q-c}$ along the spectral admissibility cascade of the Cosmochrony programme. Entanglement is treated as the infra-projectable invariant of a non-decomposable joint fibre of the non-injective projection $\Pi$, not as an added postulate.

Central Question

Given the conjugate Weil structure that organises the spectral admissibility cascade on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$:

How much entanglement does a closed conjugate pair ${c,q-c}$ carry, and how does it evolve as the cascade resolves?

Core Result

The bipartite state lives on the residual fibre-level support of the Gram--Schmidt span of Weil fingerprints on the Heisenberg Cayley graph. With the diagonal Schmidt pairing $e_j\leftrightarrow\overline{e_j}$ selected by the Born--Infeld parity involution $V_{q-c}\simeq\overline{V_c}$:

• the normalized admissibility residual of each newly admitted independent direction is exactly $w_j=1$ in the tested finite-$q$ runs;
• admitted Weil fingerprint directions are therefore mutually orthogonal, so the reduced Schmidt spectrum on the residual support is flat;
• the reduced entropy is $$\Sent(n)=\log r_{\mathrm{pair}}(n),$$ with no surviving spectral weighting (the admissibility-weight deficit $\varepsilon_{\mathrm{adm}}(n)$ is identically zero).

Here $r_{\mathrm{pair}}(n)=R_\infty-R(n)$ is the residual fibre-level Gram--Schmidt rank: the number of independent admissible directions not yet activated at BFS depth $n$.

Monotonicity

Because the cascade only adjoins independent directions, $R(n)$ is non-decreasing, hence $r_{\mathrm{pair}}(n)$ is non-increasing. The entanglement profile is therefore monotone non-increasing in $n$, maximal at the earliest admissible stage and decreasing as independent fingerprint directions are stabilized into the projected span. This monotonicity is structural — it follows from the Gram--Schmidt construction alone, with no numerical input.

The decrease is not a loss of correlation to an environment: it is the conversion of residual correlated capacity into projected stable structure.

Status of Claims

Claim	Status
Closure of the conjugate pair (purity, diagonal pairing)	Structural (Born--Infeld parity)
Orthogonality of admitted Weil fingerprints ($w_j=1$)	Numerical, verified for $q\in{29,61,101,151}$
$\Sent(n)=\log r_{\mathrm{pair}}(n)$	Derived (conditional on orthogonality)
Monotone non-increase of $\Sent(n)$	Proved (Gram--Schmidt structural)
Capacity-level one-way activation $\Delta I(n)\ge 0$	Numerical (PTO analysis); analytic proof open

Numerical Deliverable

For $q\in{61,151,211,307}$ the residual rank trajectory $r_{\mathrm{pair}}(n)$ is reconstructed from the cumulative capacity profiles of the systematic pair campaign, via $$r_{\mathrm{pair}}(n)=R_\infty-\sum_{m\le n}\sigma_c(m),|S_m|,$$ and $\Delta I(n)=\sigma^{\mathrm{can}}{\mathrm{pair}}(n)-\sigma^{\mathrm{can}}{\mathrm{pair}}(n+1)$ follows directly from the stored pair-capacity profile.

The task is to measure the contraction law of $r_{\mathrm{pair}}(n)$, its scaling exponent, and its relation to the pair-capacity exponent $\delta_{\mathrm{pair}}$:

• if the contraction exponent agrees with $\delta_{\mathrm{pair}}$ within the expected finite-$q$ window, the entanglement trajectory and the capacity law share the same spectral engine;
• if it differs, the gap marks a genuine separation between residual rank contraction and shell-wise capacity decay.

Position in the Programme

• The non-injective projection $\Pi$ is the primitive shared with the white-paper and foundation paper.
• The Bell-correlation derivation of the bell-paper is the canonical empirical anchor for the same fibre mechanism; locality at the projected level is preserved here in the same way the Tsirelson bound is reached without superluminal signalling.
• The conjugate Weil pair, Born--Infeld parity, and the pair-capacity observable $\sigma^{\mathrm{can}}_{\mathrm{pair}}$ are imported from the O-series spectral admissibility sub-programme.
• The projected mass hierarchy is the downstream trace of the cascade stabilization recorded by $\delta_{\mathrm{pair}}$, not an independent input.

Build

bash compile.sh

This runs pdflatex → bibtex → pdflatex → pdflatex on tex/EntanglementNote.tex and produces out/EntanglementNote.pdf.