Asymptotic Calibration of the BFS Window and Effective Dimension of the Admissible Trajectory

The O25 programme identified the ratio n₁ (q) /q as the central open asymptotic variable controlling the finite-size drift of ₏₀₈ₑ (q), and O26–O27 identified the effective dimension r₄₅₅ = d_² of the per-pair covariance operator in End (V_) as the key falsifiability criterion for the representation-theoretic identification. The present paper reports two measurements from the Q5a-O5 checkpoints for q \29, 61, 101, 151\. First, we extract n₁ (q) from the auto-calibrated fitting windows and fit n₁ (q) = \, q + by ordinary least squares, obtaining 0. 053 with R² 0. 879 over q 211. Extending the window-depth measurement out of sample to q \307, 401, 503, 601\ excludes this linear law (it overpredicts n₁ by up to +88\%) and establishes n₁ (q) /q 0; the asymptotic law of n₁ otherwise remains open. The O14 correction yields ₂₎ₑₑ (q) 7. 4, 10. 6 for all q \29, 61, 101, 151, 211\, confirming the structural conclusion of O25. Second, using the per-block Weil vector projections c (v) stored in the Q5a-O5 checkpoints, we perform the formal effective-dimension computation of O26 Criterion 5. 4 in End (H₄₅₅), where H₄₅₅ = C³ is the admissible projection space (HEFF\DIM = 3, consistent with c (n₃) = 3 from O23). The covariance Cc has rank exactly r₄₅₅ = 3 for every conjugate pair and every prime q \61, 101, 151\ (100\ structure ₁: ₂: ₃ = 1: 12: 12 invariant across all pairs and primes. This rank-3 result fills H₄₅₅ completely. The gap from the spin-12 prediction d_² = 4 reflects that H₄₅₅ V_. The structural resolution of this gap is established in O29: the anti-linear Born–Infeld parity makes the target d_² = 4 inaccessible from conjugate-pair data, and r₄₅₅ = 3 identifies the irreducible adjoint carrier H₄₅₅ Sym² (V_) (spin-1) ; the spin-12 space V_ (d_ = 2) is recovered as the Veronese square-root structure, not as the rank measured by Test 4.