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, which measured the exact block-wise projective capacity $\Sigma_n^{(c)} = \Delta r_n^{(c)} / |S_n|$ at $q \in \{29,53,61\}$ and derived its intrinsic asymptotic exponent exactly, $\delta_{\mathrm{exact}} = 3$, as a closed-form consequence of the observable's exact dependence on a projective pair $(\alpha,\beta)$.

O13 extends the exact computation to $q \in \{101,151,211\}$, with the $q=211$ run treated as a robustness extension under reduced BFS coverage, and analyses both convergence behaviour and variance structure of the finite-$q$ fitted slopes.

The exponent sequence $\hat\delta_{\mathrm{exact}} = 4.42, 4.80, 4.51, 4.27, 3.59$ at $q \in \{29,61,101,151,211\}$ exhibits a strict monotone decrease from $q=61$ onward, while measurement quality improves systematically: the fitting window lengthens 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. No fitting strategy across this range produces $\delta_\infty > 5.0$, consistent with O12's exact closed-form value $\delta_\infty = \delta_{\mathrm{exact}} = 3$.

Scope statement. This page provides a structured overview. The complete technical analysis — the extended exact runs, the fit-window diagnostics, the variance law $V_n^{\max}(q)$, and the connection to O12's closed-form asymptotic exponent — 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 $\hat\delta_{\mathrm{exact}} = 4.42, 4.80, 4.51, 4.27, 3.59$ shows a strict decrease from $q = 61$ onward, consistent with the exact asymptotic value derived in O12.
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.
Consistency with the exact asymptotic value: no fitting strategy across the tested range produces $\delta_\infty > 5.0$, matching O12's independently derived closed-form result $\delta_{\mathrm{exact}} = 3$.

Interpretation

O13 shows that the finite-$q$ fitted slopes identified in O12 behave exactly as crossover statistics should: as the fitting windows lengthen and the variance falls, the fitted exponent does not stabilise at its largest observed value. It decreases, tracking the exact asymptotic value derived independently in O12.

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, measured finite-$q$ slopes $\hat\delta_{\mathrm{exact}} \approx 4.3$–$4.8$, and derived the intrinsic asymptotic exponent exactly, $\delta_{\mathrm{exact}}=3$.
O13 tests whether the finite-$q$ slopes drift toward a larger value at bigger primes and finds instead that they decrease, consistent with O12's exact value.

The central coordinate $\gamma$, already shown in O12 to be algebraically inert for this observable, plays no role in this trend: the decrease is a direct signature of the finite-$q$ crossover statistics converging to the exact closed-form asymptote.

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 extends O12's finite-$q$ measurement and confirms, over a wider prime range and with improving measurement quality, the asymptotic value that O12 derives independently in closed form. The separate question of how the block-level capacity exponent relates to the cascade exponent $\beta^*$ is addressed downstream, beginning with O15.

Current outcome

The exact exponent sequence measured across $q \in \{29,61,101,151,211\}$ decreases monotonically from $q=61$ onward and does not exceed $\delta_\infty > 5.0$ under any tested fitting strategy — directly consistent with O12's independently derived closed-form asymptotic value $\delta_{\mathrm{exact}} = 3$.

The separate question of how this block-level capacity exponent connects to the cascade exponent $\beta^*$ is not addressed by extending the prime range further; it is taken up structurally starting with O15.

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.