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. O16–O24 established the canonical pair observable and its rank stability under vertical non-injectivity. The formerly appended chain \[ c_\chi \to \delta_{\mathrm{pair}} \to \beta^* \] is not closed: the reciprocal map to \(\beta^*\) imports a changing-degree LPS growth law into the fixed-degree Heisenberg cascade and has no native carrier there. 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 test the pair statistic across every conjugate pair and a substantially enlarged prime range, separating fixed-\(q\) concentration from unresolved cross-prime asymptotics.

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 a candidate finite-window control 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 every tested prime through \(q=211\). The extensions use 24 representative pairs at \(q=307\) and 12 at each of \(q=401,601\).
Inter-pair concentration: in the exhaustive campaign, the standard deviation decreases from \(0.54\) at \(q=29\) to \(0.109\) at \(q=211\), 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.
Finite-window diagnostic: the candidate control behaves as \[ \frac{\log q}{\log n_1(q)} \approx 1, \] over the accessible windows. This does not identify a unique asymptotic law.
Normalization correction: applying the O14 correction \[ \delta_{\mathrm{corr}}(q)= \delta_{\mathrm{pair}}(q)-\eta\frac{\log q}{\log n_1(q)} \] with \(\eta=1/2\) was useful at small \(q\), but the large-prime extension shows that it eventually overcorrects below the admissible window. The raw exponent reaches \(7.61\) at \(q=601\) without this correction.
Candidate asymptotic variable: O25 isolates \[ n_1(q)/q \] as a quantity to track explicitly; it does not prove a unique asymptotic limit.

Interpretation

O25 separates two effects that earlier stages conflated.

At fixed \(q\): the statistic concentrates across conjugate pairs.
Across primes: the raw exponent enters the admissible window, but the available data do not identify a unique asymptote.

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 (pair covariance, fibre identification open), 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, fixed-\(q\) pair concentration is established. The cross-prime asymptote, a native Heisenberg capacity-to-rate transfer, and the role of \(n_1(q)/q\) remain open.

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.
O14 correction: explain why the legacy \(\eta=1/2\) correction overcorrects at large \(q\).
Large-\(q\) campaign: extend the systematic computation beyond \(q=601\).
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.