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 and performs the covariance computation in the admissible projection space \(H_{\mathrm{eff}}\). An out-of-sample extension (to \(q=601\)) makes the calibrated linear extrapolation fail (overprediction up to \(+88\%\)). The exact law is the interval theorem of the Critical Coverage note: \(n_1(q)\to 22\) in probability, with critical coverage \(x_1(q)\asymp q^{-2}\); O28's out-of-sample values are the transitional-regime data of that law.
Over the measured samples this gives a rank-3 structure, finite-data rather than invariant, and isolates the obstruction to the representation-theoretic test.
Main contributions
- BFS window calibration: extraction of \(n_1(q)\); the linear law \(n_1(q) \approx \hat{\alpha} q + \hat{\beta}\) calibrated on \(q\le 211\) fails out of sample (to \(q=601\), overprediction up to \(+88\%\)).
- Exact law (Critical Coverage note): \(n_1(q)\to 22\) in probability and \(x_1(q)=|B_{n_1}|/q^2\asymp q^{-2}\); O28's measured depths are the transitional-regime data of that theorem.
- Finite-size adjustment: \(\delta_{\mathrm{corr}}(q) = \bar\delta_{\mathrm{pair}}(q) - \eta\log q/\log n_1(q)\) lies in the target window \([7.4, 10.6]\) over \(q \in \{29, 61, 101, 151, 211\}\). O14 establishes that this term does not correct the intra-\(q\) fitted exponent, so the stability is a declared diagnostic; O25 reports the same adjustment overcorrecting beyond \(q = 211\), to \(6.53\) at \(q = 601\).
- Effective-dimension computation: covariance operator evaluated in \(H_{\mathrm{eff}}\).
- Rank-3 measurement: \(r_{\mathrm{eff}} = 3\) for every conjugate pair at \(q \in \{61, 101, 151\}\) in the samples of the paper's perimeter. 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 \(q = 211\).
- Resolved eigenvalue structure: \([1 : 1/2 : 1/2]\) across the measured pairs, with no analytical derivation.
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 \(r_{\mathrm{eff}} = d_\rho^2 = 4\); under the projected-coordinate relation of O29, motivated but unproved, that target is not attainable from conjugate-pair data
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) structure; it identifies what is missing. No subspace of \(H_{\mathrm{eff}}\) selected from these data may be identified with \(V_\rho\), which is supplied elsewhere, so the missing step is a typed bridge to \(V_\rho\) together with an independent observable.
Relation to the Cosmochrony programme
O28 follows O27 by moving from the quotient factorisation of admissible morphisms to measurable spectral structure.
The sequence now reads: O16–O19 (pair construction), O20–O23 (stability and conditional adjoint dimension), O24 (rank stability), O25 (numerical campaign), O26 (quadratic completion), O27 (quotient factorisation, open vector lift), O28 (asymptotic calibration and effective dimension).
It supplies the rank measurement that O28's successors audit; the representation identification is not reached.
Current result and next step
- Established: out-of-sample failure of the calibrated linear extrapolation of \(n_1(q)\), and the rank-3 covariance measurement over the paper's perimeter.
- Open gap: \(H_{\mathrm{eff}} \neq V_\rho\), and the adjoint reading of \(H_{\mathrm{eff}}\) is supplied rather than derived.
- Next step: an independent observable and a typed bridge to \(V_\rho\), which the conjugate-pair covariance does not reach.
Reference
Jérôme Beau. Asymptotic Calibration of the BFS Window and Effective Dimension of the Admissible Trajectory.