Overview
In the Cosmochrony framework, quantum states arise as equivalence classes of $\chi$-configurations under the non-injective projection $\Pi$. Composite systems inherit a fibre structure in which sub-system configurations are not independently labelled at the $\chi$-level. Entanglement is then not an added postulate: it is the infra-projectable invariant of a non-decomposable joint fibre of $\Pi$.
The presentation note locates entanglement inside the spectral admissibility cascade. For a closed conjugate Weil pair $\{c, q-c\}$ on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, the reduced entropy on the residual fibre-level support of the Gram–Schmidt span of Weil fingerprints is exactly $$ S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n), $$ the logarithm of the residual Gram–Schmidt rank, with no surviving spectral weighting.
Core results
- Closed conjugate pair. Born–Infeld parity binds $\{c, q-c\}$ into a single infra-projectable object via $V_{q-c} \simeq \overline{V_c}$, with diagonal Schmidt pairing $e_j \leftrightarrow \overline{e_j}$.
- Flat Schmidt spectrum. Admitted Weil fingerprint directions are mutually orthogonal ($w_j = 1$, verified for $q \in \{29, 61, 101, 151\}$), so the reduced spectrum on the residual support is uniform.
- Residual-rank entropy. $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ with $r_{\mathrm{pair}}(n) = R_\infty - R(n)$; the admissibility-weight deficit is identically zero.
- Structural monotonicity. $S_{\mathrm{ent}}$ is non-increasing in $n$: residual entanglement is maximal at the earliest admissible stage and decreases as independent fingerprint directions stabilise into the projected span.
- Mass hierarchy as downstream trace. The projected mass hierarchy follows the pair-capacity exponent $\delta_{\mathrm{pair}}$, not an independent dynamical input.
Status of claims
The reduction $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ is derived, conditional on the orthogonality of admitted Weil fingerprints — now proved analytically for matched single-character blocks by E1 (Theorem 1: every $k=3$ Weil fingerprint is a discrete Fourier mode, and distinct Fourier modes are orthogonal). The monotonicity of $S_{\mathrm{ent}}$ is structural (Gram–Schmidt). The capacity-level one-way activation $\Delta I(n) \geq 0$ is numerically established in the projective temporal ordering analysis; a general analytic proof remains open. The purity of the conjugate pair is proved by E1 for matched blocks (Theorem 2); the literal Schmidt-partner pairing for independently sampled canonical O25 blocks remains open.
Constituent papers
- E1 — Weil Fingerprint Orthogonality and the Flat Schmidt Spectrum of Conjugate Pairs. Proves $w_j = 1$ as an exact algebraic identity and derives $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ analytically for matched single-character blocks. DOI 10.5281/zenodo.20499309.
- E2 — Two Entanglement Entropies of the Conjugate Weil Pair: the Admissible Pair Sector and the Full Residual Rank. Separates the rank-2 admissible pair entropy $S_{\mathrm{ent}}^{\mathrm{adm}} = \log 2$ (the Bell–Tsirelson singlet sector, unconditional for general canonical blocks via O21 atomicity + mean-zero + Q3 singlet) from the full residual-rank entropy $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ (unconditional for matched blocks by E1, conditional on $[\mathrm{H}_{\mathrm{res}}]$ for general blocks; metaplectic dilation excluded as a bridge). DOI 10.5281/zenodo.20499440.
Position within Cosmochrony
The sub-programme sits alongside the foundational papers: it relies on the projective formalism stated in the white-paper and the Foundation paper; it inherits the Bell-correlation derivation as its canonical empirical anchor; and it imports the conjugate Weil pair, Born–Infeld parity, and the pair-capacity observable $\sigma^{\mathrm{can}}_{\mathrm{pair}}$ from the O-series spectral admissibility sub-programme.
Locality at the projected level is preserved in the same way the Tsirelson bound is reached without superluminal signalling: a fully infra-projectable invariant has no position in the projected temporal ordering. Entanglement is, in this sense, the not-yet-time-ordered sector of the cascade.
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. The task is to measure the contraction law of $r_{\mathrm{pair}}(n)$ and compare its scaling exponent with the pair-capacity exponent $\delta_{\mathrm{pair}}$. Agreement would mean the entanglement trajectory and the capacity law share the same spectral engine; a gap would mark a genuine separation between residual rank contraction and shell-wise capacity decay.
References
Beau, J. Residual-Rank Entanglement of Conjugate Weil Pairs in the Spectral Admissibility Cascade. Working paper, 2026. https://doi.org/10.5281/zenodo.20498835