Overview
After O25 identified the ratio \(n_1(q)/q\) as the key asymptotic variable and O26 introduced the effective-dimension test, the next step becomes concrete: measure both quantities directly from the spectral pipeline.
The central aim of O28 is: to calibrate the BFS fitting window asymptotically and to compute the effective dimension of the admissible trajectory via the covariance operator.
The paper extracts \(n_1(q)\) from auto-calibrated windows, establishes a stable linear scaling, and performs the covariance computation in the admissible projection space \(H_{\mathrm{eff}}\).
This reveals a fully saturated rank-3 structure and isolates the precise obstruction to the representation-theoretic test.
Main contributions
- BFS window calibration: extraction of \(n_1(q)\) and identification of linear scaling \(n_1(q) \approx \hat{\alpha} q + \hat{\beta}\).
- Asymptotic variable identified: validation of \(n_1(q)/q\) as the correct scaling parameter.
- Effective-dimension computation: covariance operator evaluated in \(H_{\mathrm{eff}}\).
- Rank-3 universality: \(r_{\mathrm{eff}} = 3\) for all primes and all conjugate pairs.
- Invariant eigenvalue structure: \([1 : 1/2 : 1/2]\) across all datasets.
Interpretation
O28 clarifies the structure of the admissible trajectory but also exposes a gap.
- Observed: the admissible trajectory fills a 3-dimensional space \(H_{\mathrm{eff}}\)
- Expected: the representation-theoretic prediction is \(r_{\mathrm{eff}} = d_\rho^2 = 4\)
The gap is structural: the covariance test is performed in \(H_{\mathrm{eff}}\), not in the representation space \(V_\rho\).
O28 therefore does not contradict the SU(2) prediction; it identifies the missing step: the embedding \(V_\rho \subset H_{\mathrm{eff}}\).
Relation to the Cosmochrony programme
O28 follows O27 by moving from representation rigidity to measurable spectral structure.
The sequence now reads: O16–O19 (pair construction), O20–O23 (stability and quaternionic minimality), O24 (rank stability), O25 (numerical campaign), O26 (quadratic completion), O27 (rigidity), O28 (asymptotic calibration and effective dimension).
It provides the final numerical layer before the representation identification step.
Current result and next step
- Established: asymptotic scaling of \(n_1(q)\) and universal rank-3 covariance structure.
- Identified gap: \(H_{\mathrm{eff}} \neq V_\rho\).
- Next step: restrict the covariance operator to \(V_\rho\) and test \(r_{\mathrm{eff}} = d_\rho^2\).
Reference
Jérôme Beau. Asymptotic Calibration of the BFS Window and Effective Dimension of the Admissible Trajectory.