O28 — Asymptotic Calibration of the BFS Window and Effective Dimension of the Admissible Trajectory

Paper of the Cosmochrony spectral admissibility sub-programme. Citable version: Zenodo concept DOI 10.5281/zenodo.19767801. Web page: https://cosmochrony.org/science/spectral/program/o28/.

Quick summary

O28 reports two measurements from the Q5a–O5 checkpoints:

• Part A — BFS window calibration. The auto-calibrated window depth $n_1(q)$ is measured over $q \in {29, 61, 101, 151, 211}$ and extended out of sample to $q \in {307, 401, 503, 601}$. The linear law $n_1 \approx \hat{\alpha} q$ ($\hat{\alpha} \approx 0.053$, calibrated on $q \le 211$) fails out of sample (overprediction up to $+88%$ at $q = 601$), while $n_1(q)/q$ decreases to $0.032$. The O14-corrected exponent satisfies $\delta_{\mathrm{corr}}(q) \in [7.4, 10.6]$ across the full range, confirming the structural conclusion of O25.
• Part B — effective dimension. The per-pair covariance operator $\mathcal{C}c \in \mathrm{End}(H{\mathrm{eff}})$, $H_{\mathrm{eff}} = \mathbb{C}^3$, has $1%$ threshold rank $r_{\mathrm{eff}}^{1%} = 3$ for every conjugate pair and every tested prime, with invariant resolved spectrum $[1 : 1/2 : 1/2]$. The gap with the spin-½ prediction $d_\rho^2 = 4$ is resolved in O29: the anti-linear Born–Infeld parity makes $d_\rho^2 = 4$ inaccessible from conjugate-pair data, and $r_{\mathrm{eff}} = 3$ identifies the adjoint (spin-1) carrier $H_{\mathrm{eff}} \simeq \mathrm{Sym}^2(V_\rho)$.

The exact window-depth law

The measurements of Part A reject the calibrated linear extrapolation over the tested range; they do not by themselves establish an asymptotic limit. The exact asymptotic law is the interval theorem of the companion Critical Coverage note (10.5281/zenodo.21049163): $n_1(q) \to 22$ in probability under uniform generic block sampling, with critical coverage $x_1(q) = |B_{n_1}|/q^2 \asymp q^{-2}$ and deterministic bounds for every prime $q \ge 311$. The out-of-sample values reported in O28 ($n_1 = 16 \to 19$ over $q = 307 \to 601$) are the transitional-regime data of that law: block resonances keep $n_1$ below its limit $22$ at accessible $q$.

Context

O28 follows the chain:

• O25: $\delta_{\mathrm{pair}}$ as a structural invariant; the role of $n_1(q)/q$.
• O26: representation-theoretic dictionary and the effective-dimension test (Criterion 5.4).
• O27: rigidity — all admissible morphisms factor through $\mathfrak{su}(2)$.
• O29: identification of the adjoint carrier resolving the $3$-vs-$4$ dimension gap.

Repository structure

• tex/ — LaTeX source (SpectralO28.tex, cosmochrony-bibliography.bib)
• code/ — analysis scripts (o28_analysis.py, o28_reff_extract.py, o28_reff_figures.py) and script-generated figures
• out/ — compiled PDF, generated by compile.sh (pdflatex → bibtex → pdflatex ×2, git-ignored)

Checkpoint datasets (q{q}_o25.npz, o28_reff_summary.npz) are referenced from the paper's Data and Code Availability section.

Keywords

spectral admissibility, BFS window depth, pair observable, Weil representation, Heisenberg graphs, capacity exponent, normalisation correction, covariance operator, effective dimension, admissible trajectory, Born–Infeld saturation