Overview
This article continues the spectral admissibility programme after O19. While O19 removed the last pipeline-dependent ambiguity at the amplitude level, one structural problem remained open: how to select the physically relevant exponent window from the canonical pair observable itself.
O20 addresses this issue by introducing a persistence criterion. Its role is not to modify the observable class, nor to change the measured exponent, but to determine which decay exponents correspond to physically admissible, projectively persistent behaviour.
The paper starts from the canonical pair observable \[ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n) \] established in O19, assumes a power-law decay regime \[ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n)\sim C\,n^{-\delta_{\mathrm{pair}}}, \] and formulates a threshold-crossing condition expressing when the observable remains above an effective projective saturation scale.
This leads to a crossing rank \[ n^*=\left(\frac{C}{\sigma_{\mathrm{BI}}}\right)^{1/\delta_{\mathrm{pair}}}, \] whose admissible range defines a physical sub-spectrum of exponents.
The main structural outcome is that the persistence requirement selects a nontrivial window of admissible decay exponents, \[ \delta_{\mathrm{pair}}\in[7.4,10.6], \] thereby linking the observable-level dynamics to the phenomenological target range of the programme.
Core contributions
- Persistence criterion: the paper formulates admissibility as a persistence condition for the canonical pair observable, rather than as a purely descriptive fit to a decay law.
- Threshold-crossing rank: it introduces a crossing rank \[ n^*=\left(\frac{C}{\sigma_{\mathrm{BI}}}\right)^{1/\delta_{\mathrm{pair}}}, \] marking the shell at which the observable reaches the effective saturation threshold.
- Physical sub-spectrum: the persistence condition selects a nontrivial admissible interval \[ \delta_{\mathrm{pair}}\in[7.4,10.6]. \]
- Observable-level selection: O20 is the first step in the programme where the canonical observable is used not only for measurement, but also for structural selection.
- Projective interpretation: the threshold is interpreted as an effective saturation scale associated with Born–Infeld-type admissibility at the fibre level.
- Conditional recovery of the cascade map: once the exponent window is selected, the relation \[ \beta^*=\frac{1}{\delta_{\mathrm{pair}}+\frac{1}{2}} \] becomes physically meaningful as a candidate fibre-level cascade law.
- Isolation of the remaining non-intrinsic ingredients: O20 makes clear that the persistence criterion still depends on an external threshold \(\sigma_{\mathrm{BI}}\) and on the fit amplitude \(C\).
- Structural transition: the paper shifts the programme from canonicalisation of the observable to dynamical selection of the physically relevant exponent sector.
Interpretation
O20 does not alter the canonical observable defined in O19. Instead, it assigns that observable a new role: not only to represent the fibre-level dynamics canonically, but also to test which decay exponents correspond to persistent physical behaviour.
- O16: the physically relevant observable is the conjugate-pair observable
- O17: conjugate Weil blocks carry identical dynamical information
- O18: the minimal fibre structure is derived from Born–Infeld indiscernability
- O19: the observable is canonically normalised at the amplitude level
- O20: admissibility is reformulated as persistence of that canonical observable
The conceptual gain is that the exponent window is no longer introduced only as an external phenomenological target. It becomes the output of a structural persistence condition applied to the canonical observable itself.
At the same time, O20 shows precisely what remains unresolved: the threshold \(\sigma_{\mathrm{BI}}\) is still external, and the crossing rank still depends on the fitted amplitude \(C\). The criterion is therefore structurally meaningful, but not yet intrinsic.
Relation to the Cosmochrony program
O20 occupies the transition point between observable canonicalisation and intrinsic admissibility. It does not revisit the fibre structure of O18 and does not modify the canonical normalisation of O19. Its role is to formulate the first dynamical selection criterion for the physical exponent range.
The programme now reads: O12–O13 (exact block extraction), O14 (observable mismatch), O15 (block-level derivation failure), O16 (pair observable identified), O17 (pair dynamics derived), O18 (fibre structure derived), O19 (canonical amplitude normalisation), O20 (persistence criterion).
O20 therefore opens the admissibility step of the spectral programme. After this paper, the remaining problem is no longer to define the observable correctly, but to eliminate the last external ingredients of the persistence condition and derive it in a fully intrinsic form.
Current outcome and open directions
O20 establishes that the canonical pair observable selects a physical exponent window through a persistence condition. This yields a structural basis for the interval \([7.4,10.6]\), and makes the \(\delta_{\mathrm{pair}}\to\beta^*\) pathway meaningful at the fibre level.
The result is conditional in a precise sense: the admissibility criterion still relies on an externally specified saturation scale and on the fitted amplitude entering the crossing rank.
Two immediate follow-up directions are identified:
- O21: replace the external threshold by an intrinsic crossing condition defined directly from the observable
- Beyond O21: derive the persistence criterion directly from substrate dynamics and Born–Infeld admissibility
More broadly, an open problem is to determine whether the persistence-based selection mechanism extends beyond the current Weil/Heisenberg setting and remains stable under other admissible graph families.
References
Jérôme Beau. Projective Persistence and the Physical Sub-Spectrum: A Dynamical Selection Criterion for the Capacity Exponent.