The O-series establishes the transfer chain c_ ₀₈ₑ ^* unconditionally with respect to the fibre structure of the non-injective projection. The value ₀₈ₑ 7. 44 extracted in O16~Beau2026a20 was based on a single conjugate pair per prime and a limited prime range. The present paper reports the first systematic campaign computing ₀₈ₑ across all (q-1) /2 conjugate pairs (c, q-c) for q \29, 61, 101, 151, 211\, with M = 50 independent block samples per pair. Three results are established. First, ₀₈ₑ (q) converges robustly: the inter-pair standard deviation decreases from 0. 54 at q = 29 to 0. 11 at q = 211, confirming that ₀₈ₑ is a structural invariant of the Weil representation rather than a block-level fluctuation. Second, empirical fits of the form _ + a/\! q describe the data excellently over the tested range, but carry no asymptotic meaning: the leading finite-size correction is governed by the BFS window depth n₁ (q), which satisfies n₁ (q) = (q) by the Window-Depth Theorem, making the correction approximately constant in q and rendering direct extrapolation structurally degenerate. Third, when the O14 normalization correction \, q / n₁ (q) ~Beau2026a18 is applied, the corrected values ₂₎ₑₑ (q) 7. 7, 9. 0 fall within the admissible window 7. 4, 10. 6 consistently across q \61, 101, 151, 211\. This supports the interpretation that the apparent decrease of raw ₀₈ₑ (q) is a finite-size normalization effect rather than a signal of asymptotic drift below the admissible window. The identification of the asymptotic value _ requires analytical control of n₁ (q) /q, identified here as the central open direction.
Jérôme Beau (Wed,) studied this question.