Systematic Pair-Level Campaign for \(\delta_{\mathrm{pair}}\): Convergence, Inter-Pair Concentration, and Normalization Structure

Overview

This paper continues the spectral admissibility programme after O24. O24 established the structural closure of the chain \[ c_\chi \to \delta_{\mathrm{pair}} \to \beta^* \] with respect to the fibre structure of the non-injective projection \(\Pi\). What remained open was the full numerical status of \(\delta_{\mathrm{pair}}\): was the exponent stable across all conjugate pairs, and what controlled its residual variation with \(q\)?

The central aim of O25 is: to show that \(\delta_{\mathrm{pair}}\) is a structural invariant of the Weil representation, and that its apparent drift across primes is caused by the normalization structure of the observable, not by a breakdown of the admissible mechanism.

The key observation is that direct extrapolation in \(q\) is structurally misleading. The dominant finite-size correction is not controlled by \(q\) alone, but by the BFS fitting-window depth \[ n_1(q), \] whose ratio to \(q\) has not yet stabilized over the tested range. As a consequence, multiple empirical asymptotic laws fit the same data equally well while predicting incompatible limits.

O25 therefore performs a full pair-level campaign, measures the inter-pair concentration of \(\delta_{\mathrm{pair}}\), and identifies \[ n_1(q)/q \] as the correct asymptotic variable.

Scope statement. This page summarizes the numerical and structural content of O25: systematic pair-level computation, convergence of \(\delta_{\mathrm{pair}}\), degeneracy of naive extrapolation, O14 normalization correction, and identification of \(n_1(q)/q\) as the central open direction.

Main contributions

First full pair-level campaign: O25 computes \(\delta_{\mathrm{pair}}\) across all \((q-1)/2\) conjugate pairs \((c,q-c)\) for each tested prime.
Inter-pair concentration: the standard deviation decreases from \(0.54\) at \(q=29\) to \(0.14\) at \(q=151\), showing that \(\delta_{\mathrm{pair}}\) is not a block-level fluctuation but a structural invariant of the representation.
Extrapolation degeneracy: fits of the form \[ \delta_\infty+\frac{a}{\log q},\qquad \delta_\infty+\frac{a}{(\log q)^2},\qquad \delta_\infty+\frac{a}{q^\alpha} \] all describe the accessible data well, while yielding incompatible asymptotic limits.
Structural explanation: the leading correction behaves as \[ \frac{\log q}{\log n_1(q)} \approx 1, \] because \(n_1(q)=\Theta(q)\) at the level of scaling behaviour. The obstruction to extrapolation is therefore structural, not statistical.
Normalization correction: applying the O14 correction \[ \delta_{\mathrm{corr}}(q)= \delta_{\mathrm{pair}}(q)-\eta\frac{\log q}{\log n_1(q)} \] with \(\eta=1/2\) brings the corrected values into the admissible window \([7.4,10.6]\) for all tested primes \(q \in \{61,101,151\}\).
Correct asymptotic variable: O25 identifies \[ n_1(q)/q \] rather than \(q\) itself as the quantity controlling convergence.

Interpretation

O25 changes the interpretation of the numerical drift observed in earlier stages of the programme.

Before O25: \(\delta_{\mathrm{pair}}\) seemed to drift with \(q\)
After O25: the drift is explained as a finite-size normalization effect

The crucial point is that the measured observable depends on the BFS shell geometry through the fitting window \([n_0,n_1]\). Since the ratio \(n_1(q)/q\) has not yet stabilized over the accessible range, the observable remains pre-asymptotic even when the numerical fits themselves look extremely good.

In other words, the paper shifts the asymptotic question:

from direct extrapolation in \(q\)
to analytical control of the internal window geometry
from a naive fit variable
to the structurally correct variable \(n_1(q)/q\)

Relation to the Cosmochrony programme

O25 occupies the numerical counterpart of O24 in the O-series. After the fibre-level construction of the observable (O16–O19), the persistence and intrinsic saturation framework (O20–O21), shell locking (O22), threshold derivation (O23), and rank stability under non-injectivity (O24), O25 shows that the measured exponent behaves exactly as expected once the normalization structure is taken into account.

The sequence now reads: O16 (pair observable), O17 (pair dynamics), O18 (minimal fibre structure), O19 (canonical normalization), O20 (persistence criterion), O21 (intrinsic saturation rank), O22 (projection locking), O23 (threshold dimension), O24 (rank stability), O25 (full pair-level campaign and normalization structure).

After O25, the open problem is no longer whether \(\delta_{\mathrm{pair}}\) is stable, but how the asymptotic ratio \(n_1(q)/q\) is determined analytically.

Current result and open directions

O25 establishes that \(\delta_{\mathrm{pair}}\) is numerically robust and strongly concentrated across conjugate pairs, and that naive extrapolation in \(q\) is not a meaningful way to infer \(\delta_\infty\).

The following directions remain open:

Asymptotic ratio \(n_1(q)/q\): determine whether the ratio stabilizes, and if so, to which constant \(\alpha\).
Next-to-leading corrections: derive the subleading dependence of \(\bar{\delta}_{\mathrm{pair}}(q)\) once \(n_1(q)=\alpha q+O(q^\beta)\) is known.
Validation of \(\eta=1/2\): test within the exact O25 pipeline the O14 hypothesis used in the normalization correction.
Large-\(q\) campaign: extend the systematic computation to \(q=211\) and beyond.
Analytical completion: determine \(\delta_\infty\) from the structure of \(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})\), rather than by numerical extrapolation.

Reference

Jérôme Beau. Systematic Pair-Level Campaign for \(\delta_{\mathrm{pair}}\): Convergence, Inter-Pair Concentration, and Normalization Structure.