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 $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, under the note's closure hypothesis (a supplied purity of the joint state, which neither the note nor E1 derives, with the diagonal Schmidt pairing, which E1 proves for matched blocks given that purity), 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. For independently sampled blocks, E2 adds a residual indiscernibility hypothesis and an O17 transfer gap.
Core results
- Closed conjugate pair (a K4 crossing). The conjugate pair $\{c, q-c\}$ is treated as a single infra-projectable object in a pure joint state — a supplied composition structure, not a consequence of admissibility. Given that supplied purity, Born–Infeld parity via $V_{q-c} \simeq \overline{V_c}$ does derive which Schmidt pairing is selected: the diagonal one, $e_j \leftrightarrow \overline{e_j}$.
- 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 every block by E1 (Theorem 4: every $k=3$ Weil fingerprint is a discrete Fourier mode, and distinct Fourier modes are orthogonal), and on the closure hypothesis (a supplied purity of the joint state, which E1 does not derive, with the diagonal pairing). The monotonicity of $S_{\mathrm{ent}}$ is structural (Gram–Schmidt). The capacity-level one-way activation $\Delta I(n) \geq 0$ is a conjecture of the projective temporal ordering analysis, supported numerically only within a tolerance ($\Delta I(n) \geq -10^{-6} I_{\max}$ after burn-in); exact non-negativity and its analytic proof remain open. The purity of the conjugate pair is a supplied composition structure (a K4 crossing), which E1 does not derive. For matched blocks, E1 proves the diagonal pairing given that purity and exhibits the maximally entangled state as the natural choice on the identified residual subspaces, rather than deriving purity from a more primitive input; the general (or mismatched-block) case used for the independently sampled canonical O25 pair observable remains open, and the literal Schmidt-partner pairing there is not fixed.
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, under a supplied purity of the joint state, which E1 does not derive. 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 $S_{\mathrm{ent}}^{\mathrm{adm}} = \log 2$ (conditional on two explicit hypotheses, (H-carrier) — E2's own atomic Fourier-triple carrier hypothesis, with zero-mode removal part of the sector definition — and Q3's (H-inv), for general canonical blocks) from the full residual-rank entropy $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ (proved for matched blocks by E1 under a supplied purity of the joint state, which E1 does not derive; conditional on that purity, $[\mathrm{H}_{\mathrm{res}}]$ and the O17 toy-model transfer gap 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 candidate projective temporal ordering, whose proxy is defined along a trajectory; an order on states would require the acyclicity hypothesis [H-acyc], which is not established. 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