Overview
This article continues the spectral admissibility programme after O15. While O15 established that the scalar observable of O7 does not transfer to the exact Weil-block regime and localised the failure at the level of the growth equation, it left open a single precise question: what is the correct dynamic observable?
O16 answers this question by identifying a structural property of the projection Π: non-injectivity implies fibre-level admissibility. Each observable corresponds not to a single Weil block, but to a pair of conjugate blocks $(c, q-c)$.
The correct observable is therefore not the block mean $\bar{\Sigma}_n$, but a pair observable defined on fibres: \[ \sigma_{\mathrm{pair}}(n) = \sigma_c(n)\,\sigma_{q-c}(n). \]
This immediately leads to exponent doubling: \[ \delta_{\mathrm{pair}} = 2\,\delta_c \approx 7.4, \] which resolves the δ-deficit identified in O12–O15 without modifying the O6/O7 growth law.
Core contributions
- Fibre-level observable: admissibility is defined on fibres of Π, not on individual Weil blocks.
- Conjugate symmetry: $\rho_{q-c} = \overline{\rho_c}$ implies identical Gram–Schmidt dynamics and $\delta_c = \delta_{q-c}$.
- Exponent doubling: $\delta_{\mathrm{pair}} = 2\,\delta_c$ arises from fibre-level admissibility.
- Resolution of δ-deficit: the discrepancy identified in O12–O15 is a single-branch measurement artefact.
- Spectral lower bound: $\delta_{\mathrm{pair}} \ge 7.4$ is a structural property of the representation.
- Normalization clarified: the factor $r(c,q)$ arises from pipeline normalisation and does not affect exponents.
Interpretation
O16 does not modify the growth law derived in O6 and reformulated in O7. Instead, it shows that the observable entering that law must be redefined.
- O15: scalar observable invalid in exact regime
- O16: correct observable = fibre-level pair observable
The δ-deficit is therefore not a dynamical failure, but a misidentification of admissibility. The structure of Π itself imposes the correct observable.
Relation to the Cosmochrony program
O16 follows directly from O15. It completes the transition from scalar proxy observables to exact Weil-block observables.
The programme now reads: O12–O13 (exact extraction), O14 (observable mismatch), O15 (derivation-level failure), O16 (correct observable identified).
O16 does not introduce new primes or new dynamics. Its role is to identify the correct observable compatible with the structure of Π.
Current outcome and open problem
O16 resolves the lower bound of the phenomenological range: \[ \delta_{\mathrm{pair}} \approx 7.4. \]
However, the pair spectrum extends beyond the phenomenological range ($\delta > 10.6$). The upper bound is therefore not spectral.
Preliminary analysis indicates that it arises from a dynamical coherence constraint on the full trajectory $\sigma(n)$.
The remaining problem is thus: which global property of the relaxation trajectory selects the admissible window?
References
Jérôme Beau. Fibre Admissibility and Exponent Doubling in the Exact Weil Regime.