Overview
This article continues the spectral admissibility programme after O20. While O20 introduced a persistence criterion selecting the admissible exponent window, it still relied on an external threshold and a fit parameter.
O21 removes this dependence entirely. Its central objective is: to derive the persistence criterion intrinsically from the observable itself, without introducing any external parameter.
The paper introduces a canonical observable crossing rank \[ n_{3}^{\mathrm{obs}}, \] defined directly from the canonical pair observable \[ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n), \] without any amplitude normalisation or threshold input.
The key structural statement is that the saturation threshold is not external: it is encoded in the observable through a canonical crossing condition.
Core contributions
- Intrinsic crossing rank: definition of a canonical rank \[ n_{3}^{\mathrm{obs}} \] directly from the observable sequence, without amplitude parameters.
- Elimination of external thresholds: the saturation threshold \(\sigma_{\mathrm{BI}}\) is no longer postulated, but encoded structurally in the observable.
- Removal of fit parameter: the parameter \(C\) of O20 disappears entirely from the formulation.
- Scale invariance: the construction is invariant under \[ \sigma(n)\mapsto \alpha\,\sigma(n), \] ensuring independence from pipeline normalisation.
- Geometric interpretation: the intrinsic rank corresponds to a distinguished BFS shell, reflecting alignment between decay and admissibility.
- Structural exponent selection: the exponent \(\delta_{\mathrm{pair}}\) is determined by compatibility with the intrinsic crossing condition rather than a window constraint.
- Restoration of the cascade relation: under the cascade interpretation, \[ \beta^* = \frac{1}{\delta_{\mathrm{pair}} + \frac{1}{2}}, \] resolving the obstruction identified in O15.
Interpretation
O21 completes the transition from externally defined admissibility to intrinsic structural selection.
- O20: admissibility via external threshold
- O21: admissibility encoded in the observable itself
The observable is no longer evaluated against a prescribed scale. Instead, it carries within itself the signal of saturation through a canonical crossing condition.
This yields a fully intrinsic formulation of the persistence criterion, removing the last remaining ambiguity inherited from earlier stages of the programme.
Relation to the Cosmochrony program
O21 occupies a decisive position in the O-series. It follows the canonicalisation of the observable in O19 and the persistence criterion of O20, and removes their remaining external dependencies.
The programme now reads: O16 (pair observable), O17 (pair dynamics), O18 (fibre structure), O19 (canonical normalisation), O20 (persistence criterion), O21 (intrinsic persistence).
After O21, the admissibility condition is no longer defined phenomenologically, but structurally.
Current outcome and open directions
O21 establishes a parameter-free formulation of admissibility. The observable, its normalisation, and the persistence criterion are now fully intrinsic.
Remaining directions include:
- Substrate-level derivation: derive the intrinsic condition directly from χ-dynamics.
- Universality: test the intrinsic criterion beyond Heisenberg graphs.
- Asymptotics: determine the scaling of \(n_{3}^{\mathrm{obs}}\) at large \(q\).
- Full cascade derivation: establish the \(\delta_{\mathrm{pair}}\to\beta^*\) relation without additional assumptions.
References
Jérôme Beau. Intrinsic Saturation and the Canonical Observable: A Parameter-Free Selection Criterion for the Capacity Exponent.