Asymptotic Stability of Exact Weil-Block Capacity on Heisenberg Graphs: Extended Prime Range, Variance Reduction, and Requalification of the δ–β* Tension

Overview

This article continues the spectral admissibility programme after O12. While O12 established that the exact Weil projection yields a larger decay exponent than the proxy observable of O11, it still left open the most natural interpretation of the mismatch with the phenomenological target for $\beta^*$: a finite-size effect at the accessible primes.

O13 tests that hypothesis directly by extending the exact computation to $q \in \{101,151,211\}$, with the $q=211$ run treated as a robustness extension under reduced BFS coverage. The observable remains the exact block-wise incremental projective capacity $\Sigma_n^{(c)} = \Delta r_n^{(c)} / |S_n|$ measured inside generic irreducible Weil blocks of dimension $q$.

The outcome is negative in a precise and scientifically useful sense: the exact exponent does not drift upward toward the target range. Instead, the sequence $\hat{\delta}_{\mathrm{exact}} = 4.42, 4.80, 4.52, 4.27, 3.59$ at $q \in \{29,61,101,151,211\}$ decreases monotonically from $q=61$ onward, while the measurement quality improves at the same time.

Scope statement. This page provides a structured overview. The complete technical analysis, including the extended exact runs, the fit-window diagnostics, the variance law $V_n^{\max}(q)$, and the requalification of the δ–β* tension, is presented in the preprint linked above.

Core contributions

Extended exact computation: O13 extends the exact Weil-block analysis of O12 to $q = 101$ and $151$, with $q = 211$ added as a robustness extension under reduced BFS coverage.
Absence of upward drift: the exact exponent sequence $\hat{\delta}_{\mathrm{exact}} = 4.42, 4.80, 4.52, 4.27, 3.59$ shows a strict decrease from $q = 61$ onward rather than any drift toward the phenomenological range $[7.4,10.6]$.
Improving measurement quality: the fitting window grows from 4 to 11 points, the inter-block variance drops from $V_n^{\max}=5.20$ to $0.30$, and $R^2 > 0.993$ throughout.
Block-by-block universality: condition (E2) is satisfied at $q \geq 151$, so the extracted exponent becomes a property of individual exact blocks, not merely of their average.
Variance scaling law: the peak inter-block variance follows the empirical law $V_n^{\max}(q) \approx 754\,q^{-1.41}$, showing that block heterogeneity decays rapidly as the prime grows.
Finite-size hypothesis rejected: the strong version of the O12 interpretation is ruled out: larger primes do not move the exact observable toward the phenomenological target.
Structural mismatch isolated: the remaining problem is identified as the exact-block relation $\delta \mapsto \beta^*$, not as a numerical or finite-size issue.

Interpretation

O13 shows that the tension identified in O12 survives precisely in the regime where the measurement becomes cleaner. This is the decisive point: as the fitting windows lengthen, the variance falls, and the exact exponent does not rise toward the phenomenological target. On the contrary, it decreases.

O11 established the proxy-level Weil-block observable and found a decay exponent around $\hat{\delta}_{\mathrm{cap}} \approx 3.39$.
O12 replaced the proxy by the exact Weil projection and revealed a regime shift to $\hat{\delta}_{\mathrm{exact}} \approx 4.3$--$4.8$.
O13 tests whether that shift disappears at larger primes and shows that it does not. The mismatch is therefore structural rather than numerical.

In this picture, the central coordinate $\gamma$ is not a small correction to the proxy dynamics. It remains dynamically active, affects the exact observable class, and forces a revision of the proxy-level map from decay exponent to cascade exponent.

Relation to the Cosmochrony program

O13 follows directly from O12. Spectral admissibility, capacity, rigidity, and stratigraphy define the spectral backbone of the theory. O1 restores ordering through projective dynamics, O3 amplifies the hierarchy via valence growth, O4 constrains the cascade exponent from bounded relational flux, O5 localises the failure of vertex-level mechanisms, O6 proves the no-go for fixed finite-dimensional fingerprints, O7 reformulates the observable in terms of projective capacity, O8 identifies geometric compression on LPS graphs, O9 removes that compression by moving to polynomial-growth Heisenberg geometry, O10 isolates the dense-sketch bottleneck, O11 restores observability at the proxy representation level, and O12 implements the exact Weil-block observable.

The present paper is the first in the series to falsify an explicit rescue hypothesis of the exact-block regime. It does not merely extend the computation: it shows that the remaining discrepancy with phenomenology is intrinsic to the exact observable and must now be addressed at the structural level.

Current outcome and open problem

Propagated through the O7 relation $\beta^* = 1 / (\delta + \tfrac{1}{2})$, the present data imply values around $\beta^* \approx 0.19$ at $q=151$, $\beta^* \approx 0.24$ at $q=211$, and $\beta^* \approx 0.24$--$0.26$ under log-linear extrapolation, all well above the target window $\beta^* \in (0.09, 0.13)$.

O13 therefore closes the strong finite-size explanation and leaves a more precise open problem: how must the structural relation $\delta \mapsto \beta^*$ be reformulated in the exact-block setting, where the observable is normalised by $|S_n|$ and retains the central phase through $\gamma$?

The next step of the programme is therefore not simply larger-prime numerics, but a theoretical rederivation of the exact-block map from projective-capacity decay to the cascade exponent. Larger primes remain useful as robustness checks, but they are no longer the primary conceptual bottleneck.

References

Jérôme Beau. Asymptotic Stability of Exact Weil-Block Capacity on Heisenberg Graphs: Extended Prime Range, Variance Reduction, and Requalification of the δ–β* Tension. Preprint.