Overview
This article continues the spectral admissibility programme after O13. While O13 ruled out the strong finite-size explanation of the exact δ–β* tension, it left open the remaining structural issue, identified there as Source S2: the proxy-level derivation of the map $\delta \mapsto \beta^*$ does not apply unchanged to the exact Weil-block observable.
O14 addresses that issue directly by identifying the observable-class mismatch between the scalar proxy capacity of O7 and the exact block-wise capacity $\Sigma_n^{(c)} = \Delta r_n^{(c)} / |S_n|$ measured in O12 and O13. The mismatch is decomposed into three structural failures: block heterogeneity, block normalisation, and the central-phase contribution.
The paper then derives a corrected exact-block relation $\beta^* = 1 / (\delta_{\mathrm{eff}} + \tfrac{1}{2}) + \epsilon(q)$ and evaluates it using the real BFS pipeline outputs of O12/O13. The main numerical outcome is that the normalisation correction is dominant, while the shell-level central-coordinate coherence remains high and the inferred central-phase bias is negligible in the fitting window.
Core contributions
- Observable-class mismatch identified: O14 shows that the O7 derivation applies to a scalar proxy observable, whereas the exact observable of O12/O13 is a block-wise mean over irreducible Weil representations of internal dimension $q$.
- Three structural corrections: the exact/proxy mismatch is decomposed into block heterogeneity (A1), block normalisation by $q$ (A2), and the central-phase contribution $\psi_c(\gamma + bx)$ (A3).
- Corrected structural relation: the paper derives $\beta^* = 1 / (\delta_{\mathrm{eff}} + \tfrac{1}{2}) + \epsilon(q)$ with $\delta_{\mathrm{eff}}(q) = \hat{\delta}_{\mathrm{exact}}(q) - \delta_\gamma(q) - \eta \log q / \log n^*(q)$.
- Block-normalisation exponent: O14 identifies $\eta = 1/2$ as the unique value compatible with Weil amplitude scaling, block multiplicity, and square-root accumulation under biased phase mixing.
- Pipeline-based numerical check: using the real BFS outputs of O12/O13, the paper measures $C(q)=\bar{\ell}_\gamma$, computes the associated circular variance, and evaluates the corrected exponent in the actual fit windows.
- Dominant measurable correction: the shell-level central-coordinate coherence remains high, so the inferred $\delta_\gamma$ is negligible in pipeline mode, while the normalisation term $\eta \log q / \log n_1$ is the dominant correction.
- Structural outcome: even after correction, the effective exponent stays below the phenomenological target regime, supporting Scenario S2-B rather than a recovery of the O7 relation in the exact-block setting.
Interpretation
O14 does not revisit the finite-size question closed by O13. Instead, it explains why the exact Weil-block exponent and the proxy-level structural relation belong to different observable classes. This is the central conceptual step of the paper.
- O11 restored a representation-adapted proxy observable and extracted a proxy-level decay exponent.
- O12 replaced that proxy by the exact Weil observable and revealed a higher decay exponent.
- O13 showed that the discrepancy persists and is not a finite-size artefact.
- O14 derives the corrected exact-block correspondence and shows that the dominant measurable correction is block normalisation, not a shell-level phase decoherence.
The pipeline analysis also reveals an important conceptual distinction: the O12 observable $\ell_\gamma(n)$ is shell-level, whereas the phase bias entering the O14 correction acts at block level inside individual Weil representations. This distinction matters because the current pipeline directly measures only the former.
Relation to the Cosmochrony program
O14 follows directly from O13. Spectral admissibility, capacity, rigidity, and stratigraphy define the spectral backbone of the theory. O1 restores ordering through projective dynamics, O3 amplifies the hierarchy via valence growth, O4 constrains the cascade exponent from bounded relational flux, O5 localises the failure of vertex-level mechanisms, O6 proves the no-go for fixed finite-dimensional fingerprints, O7 reformulates the observable in terms of projective capacity, O8 identifies geometric compression on LPS graphs, O9 removes that compression by moving to polynomial-growth Heisenberg geometry, O10 isolates the dense-sketch bottleneck, O11 restores observability at the proxy representation level, O12 implements the exact Weil-block observable, and O13 closes the finite-size explanation.
The present paper is the structural bridge between exact measurement and phenomenological interpretation. It does not add new prime ranges, but re-derives the map from decay exponent to cascade exponent in the only setting that remains physically relevant: the exact-block regime.
Current outcome and open problem
In pipeline mode, the measured shell-level coherence satisfies $C(q) \in [0.92,0.96]$, the associated circular variance remains small, and the inferred central-phase correction is negligible compared with the exact decay exponent. By contrast, the normalisation correction $\eta \log q / \log n_1 \approx 1.00\text{--}1.06$ is large and directly measurable in the real O12/O13 fit windows.
The corrected exponent therefore remains below $5.0$ for all tested primes, and the corresponding values of $\beta^*$ stay outside the phenomenological window $\beta^* \in (0.09,0.13)$. O14 thus supports Scenario S2-B: observable-class correction alone does not recover the proxy-level relation in the exact regime.
The remaining open problem is now sharply localised. To test the central-phase ansatz directly, the programme must extract block-level phase observables before Gram-Schmidt orthogonalisation in the O12 pipeline. Larger-prime numerics remain useful, but the main conceptual bottleneck is now the block-level phase observable itself.
References
Jérôme Beau. Observable-Class Mismatch and the Corrected δ→β* Relation on Heisenberg Graphs: Block Normalisation, Central-Phase Contribution, and Structural Correspondence. Preprint.