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

Abstract

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 O14 correction yields $\delta_{\mathrm{corr}}(q) \in [7.4, 10.6]$ for all $q \in \{29, 61, 101, 151, 211\}$, confirming the structural conclusion of O25. 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). The covariance $\mathcal{C}_c$ has $1\%$ threshold rank $r_{\mathrm{eff}}^{1\%} = 3$ for every conjugate pair and every prime $q \in \{61, 101, 151\}$ (100\ $[\lambda_1: \lambda_2: \lambda_3] = [1: \tfrac{1}{2}: \tfrac{1}{2}]$ invariant across all pairs and primes. These three resolved modes fill $H_{\mathrm{eff}}$ completely. The gap from the spin-$\tfrac{1}{2}$ prediction $d_\rho^2 = 4$ reflects that $H_{\mathrm{eff}} \neq V_\rho$. The structural resolution of this gap is established in O29: the anti-linear Born–Infeld parity makes the target $d_\rho^2 = 4$ inaccessible from conjugate-pair data, and $r_{\mathrm{eff}} = 3$ identifies the irreducible adjoint carrier $H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)$ (spin-$1$); the spin-$\tfrac{1}{2}$ space $V_\rho$ ($d_\rho = 2$) is recovered as the Veronese square-root structure, not as the rank measured by Test 4. Interpretively, the combined result separates a finite-$q$ calibration diagnostic from its now-proved asymptotic law and separates a covariance-carrier measurement from a direct measurement of the spin-$\tfrac{1}{2}$ dimension.