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

The O25 programme identified the ratio $n_1(q)/q$ as the central open asymptotic variable controlling the finite-size drift of $\bar{\delta}_{\mathrm{pair}}(q)$, and O26–O27 identified the effective dimension $r_{\mathrm{eff}} = d_\rho^2$ of the per-pair covariance operator in $\mathrm{End}(V_\rho)$ 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 $n_1(q)$ from the auto-calibrated fitting windows and fit $n_1(q) = \hat{\alpha}\,q + \hat{\beta}$ by ordinary least squares, obtaining $\hat{\alpha} \approx 0.053$ with $R^2 \approx 0.879$ over $q \le 211$. Extending the window-depth measurement out of sample to $q \in \{307, 401, 503, 601\}$ makes this calibrated linear extrapolation fail (it overpredicts $n_1$ by up to $+88\%$). The exact law is established in the companion Critical Coverage note: $n_1(q) \to 22$ in probability for five independently and uniformly sampled generic blocks, with critical coverage $x_1(q) = |B_{n_1}|/q^2 \asymp q^{-2}$; the measured decrease of $n_1(q)/q$ belongs to the transitional resonance regime of that law. The declared finite-size adjustment $\delta_{\mathrm{corr}}(q) = \bar{\delta}_{\mathrm{pair}}(q) - \eta\log q/\log n_1(q)$ lies in $[7.4, 10.6]$ for all $q \in \{29, 61, 101, 151, 211\}$; 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 $\pi_c(v)$ stored in the Q5a-O5 checkpoints, we perform the formal effective-dimension computation of O26 Criterion 5.4 in $\mathrm{End}(H_{\mathrm{eff}})$, where $H_{\mathrm{eff}} = \mathbb{C}^3$ is the admissible projection space ($\mathrm{HEFF\_DIM} = 3$, consistent with the supplied selection rule $\Sigma_c(n_3) = 3$ of O23). Over the samples of the present perimeter, the covariance $\mathcal{C}_c$ has $1\%$ threshold rank $r_{\mathrm{eff}}^{1\%} = 3$ for every conjugate pair at $q \in \{61, 101, 151\}$, with normalised resolved spectrum $[\lambda_1: \lambda_2: \lambda_3] = [1: \tfrac{1}{2}: \tfrac{1}{2}]$. 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 $H_{\mathrm{eff}}$ at the stated threshold. The gap from the spin-$\tfrac{1}{2}$ prediction $d_\rho^2 = 4$ reflects that $H_{\mathrm{eff}} \neq V_\rho$, and stays open. O29 audits what the measurement identifies: the projected-coordinate relation $\pi_{q-c}(v) = \overline{\pi_c(v)}$ is motivated by the Weil-block identity $\rho_{q-c} = \bar\rho_c$ without following from it, the finite-precision diagnostics are compatible with an adjoint reading of $H_{\mathrm{eff}}$ without deriving that carrier, and $V_\rho$ 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-$\tfrac{1}{2}$ dimension unmeasured by the conjugate-pair covariance.