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 δ̄ₚₐᵢᵣ(q), and O26–O27 identified the effective dimension reff = d_ρ² of the per-pair covariance operator in End(V_ρ) as the key falsifiability criterion for the representation-theoretic identification. The present paper reports two analyses built from the Q5a-O5 checkpoints and their documented prime-range extensions. First, we extract n₁(q) from the auto-calibrated fitting windows and fit n₁(q) = α̂\,q + β̂ by ordinary least squares, obtaining α̂ ≈ 0.053 with R² ≈ 0.879 over q ≤ 211. Extending the window-depth measurement out of sample to q ∈ \307, 401, 503, 601\ makes this calibrated linear extrapolation fail (it overpredicts n₁ by up to $+88%$). The exact law is established in the companion Critical Coverage note: n₁(q) → 22 in probability for five independently and uniformly sampled generic blocks, with critical coverage x₁(q) = |Bn₁|/q² q⁻²; the measured decrease of n₁(q)/q belongs to the transitional resonance regime of that law. The declared finite-size adjustment δcorr(q) = δ̄ₚₐᵢᵣ(q) - ηlog q/log n₁(q) lies in $[7.4, 10.6]$ for all q ∈ \29, 61, 101, 151, 211\; O14 establishes that this term does not correct the intra-q fitted exponent, so the stability is a diagnostic and not a corrected capacity exponent. Second, using the per-block Weil vector projections πc(v) stored in the Q5a-O5 checkpoints, we perform the formal effective-dimension computation of O26 Criterion 5.4 in End(Heff), where Heff = C³ is the admissible projection space (HEFFDIM = 3, consistent with the supplied selection rule Σc(n₃) = 3 of O23). Over the samples of the present perimeter, the covariance Cc has $1%$ threshold rank reff1% = 3 for every conjugate pair at q ∈ \61, 101, 151\, with normalised resolved spectrum [λ₁: λ₂: λ₃] = [1: 1/2: 1/2]. This is a finite-data value of the stated protocol and not an invariant: the extended recomputation of O29 finds rank three modal, at $32/50$ pairs for $q = 101$ and $103/105$ for $q = 211$. These three resolved modes fill Heff at the stated threshold. The gap from the spin-1/2 prediction d_ρ² = 4 reflects that Heff ≠ V_ρ, and stays open. O29 audits what the measurement identifies: the projected-coordinate relation πq-c(v) = πc(v)̄ is motivated by the Weil-block identity ρq-c = ρc without following from it, the finite-precision diagnostics are compatible with an adjoint reading of Heff without deriving that carrier, and V_ρ is supplied elsewhere rather than recovered from these data. Interpretively, the combined result separates a finite-q calibration diagnostic from its proved asymptotic law, and leaves the spin-1/2 dimension unmeasured by the conjugate-pair covariance.