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 was based on a single conjugate pair per prime and a limited prime range. The present paper reports a systematic campaign computing ₀₈ₑ across the (q-1) /2 conjugate pairs (c, q-c), originally for q \29, 61, 101, 151, 211\ with M = 50 block samples per pair, and here extended to q \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, ₀₈ₑ (q) converges robustly and concentrates across pairs, confirming that it is a structural invariant of the Weil representation rather than a block-level fluctuation. Second, and centrally, the extended campaign shows that the raw exponent ₆₋₎₁₀₋ (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 _ remains insufficiently constrained by the accessible range: competing convergence laws — notably 1/q and 1/q — remain statistically viable, so no single extrapolated _ is claimed. As a secondary consequence, the inferred transfer parameter ^{*} = 1/ (₀₈ₑ + 12) stays in the narrow interval 0. 108–0. 123 across q \211, , 601\, consistent with the O24 transfer analysis. 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
Jérôme Beau (Fri,) studied this question.