Systematic Pair-Level Campaign for δpair: Convergence, Inter-Pair Concentration, and Normalization Structure

Abstract

The O-series defines a canonical pair-level capacity observable and establishes that vertical non-injectivity does not change its observable rank. The further prescription ${\beta^{*}}=1/({\delta_{\mathrm{pair}}}+\tfrac12)$, however, imports the changing-degree LPS growth equation into a fixed-degree Heisenberg cascade and has no native carrier there. The value ${\delta_{\mathrm{pair}}} \approx 7.44$ extracted in O16 was based on a single conjugate pair per prime and a limited prime range. The present paper reports a systematic campaign computing ${\delta_{\mathrm{pair}}}$ across the $(q-1)/2$ conjugate pairs $(c, q-c)$, originally for $q \in \{29, 61, 101, 151, 211\}$ with $M = 50$ block samples per pair, and here extended to $q \in \{307, 401, 601\}$ by a capped breadth-first construction that reproduces the pair observable exactly at a fraction of the cost (with reduced sampling: $M = 16$ at $q = 307$, $M = 8$ at $q = 401, 601$). Three results are established. First, ${\delta_{\mathrm{pair}}}(q)$ is extracted reproducibly and concentrates across pairs, confirming that it is a stable fixed-$q$ pair statistic rather than a block-level fluctuation. Second, and centrally, the extended campaign shows that the raw exponent $\delta_{\mathrm{global}}(q)$ descends monotonically into the admissible window $[7.4, 10.6]$ on its own, reaching $7.61$ at $q = 601$, without any finite-size correction; the O14 normalization correction, needed to bring the small-$q$ values into the window, becomes progressively unnecessary at large $q$ and eventually overcorrects, sending the corrected quantity below the lower edge $7.4$. The admissible-window agreement is therefore carried by the raw observable, not by the corrected one. Third, the asymptotic value $\delta_\infty$ remains insufficiently constrained by the accessible range: competing convergence laws — notably $1/q$ and $1/\sqrt{q}$ — remain statistically viable, so no single extrapolated $\delta_\infty$ is claimed. For comparison only, applying the legacy reciprocal map ${\beta^{*}} = 1/({\delta_{\mathrm{pair}}} + \tfrac12)$ produces the narrow interval $0.108$–$0.123$ across $q \in \{211, \dots, 601\}$. This is a phenomenological numerical coincidence, not a Heisenberg capacity-to-rate inference. Keywords. Cosmochrony; spectral admissibility; pair-level observable; Weil representation; Heisenberg graphs; capacity exponent; convergence; inter-pair concentration; normalization correction; window depth; BFS; asymptotic analysis; large-prime extension