This repository contains the source of the O20 Cosmochrony paper
Projective Persistence and the Physical Sub-Spectrum: A Dynamical Selection Criterion for the Capacity Exponent.

This work extends the spectral admissibility sub-programme by addressing the next structural issue left open by O19:

Once the physically relevant observable has been fixed at the canonical fibre level, what selects the physical sub-spectrum (\delta \in [7.4, 10.6])?

Context

O19 established that:

• the physically relevant observable is the canonical pair-level quantity [ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n) ]
• the residual amplitude ambiguity inherited from the O12/O13 Gram–Schmidt pipeline can be removed canonically
• the exponent (\delta_{\mathrm{pair}}) is invariant under this canonicalisation
• the measured fibre-level exponent remains [ \delta_{\mathrm{pair}} \approx 7.44 ]

This closed the normalisation problem of O19:

• the observable class had already been fixed by O16–O18
• the amplitude is now canonically normalised
• the remaining question is no longer what the correct observable is, but why the physical sub-spectrum is restricted to ([7.4, 10.6])

This defines the scope of O20.

Core Result

The paper proposes a projective persistence criterion.

A fibre [ {c,q-c} ] is declared physically admissible if and only if the canonical observable [ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n) ] reaches a Born–Infeld saturation threshold [ \sigma_{\mathrm{BI}} ] within the effective cascade window [ [n_0,n_1] ] defined by the pre-saturation regime.

Under two structural hypotheses,

• [H1] (\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)) is a monotone function of an effective fibre-level amplitude
• [H2] this amplitude inherits the Born–Infeld saturation bound [ A_n^{\max}=\frac{c_{\mathrm{BI}}}{\sqrt{\lambda_n}} ]

the persistence condition becomes equivalent to an admissible exponent window [ \delta_{\mathrm{pair}} \in [\delta_{\min},\delta_{\max}] ] with endpoints determined by (\sigma_{\mathrm{BI}}), (n_0), and (n_1).

The contribution of O20 is therefore precise:

it does not yet derive the saturation threshold from first principles,
it identifies the structural criterion that a physical fibre-level observable must satisfy in order to belong to the admissible sub-spectrum.

Main Structural Results

1. Observable hierarchy fixed

The paper starts from the hierarchy established by O12–O19:

• raw cumulative span [ \Sigma^{(c)}(n) ]
• normalised block observable [ \sigma_c(n) ]
• pair observable [ \sigma_{\mathrm{pair}}(n)=\sigma_c(n),\sigma_{q-c}(n) ]
• canonical pair observable [ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n) ]

The key point is:

O20 does not redefine the observable.
It uses the canonical fibre-level observable already fixed by O19.

2. Block-level mismatch is now definitively bypassed

The paper recalls that:

• the block-level exponent [ \hat{\delta}_{\mathrm{exact}} \approx 3.72 ] is structurally below the physical target range
• O15 proved that no reweighting of block data can recover the target
• O16–O18 showed that the correct observable lives at the fibre level
• O19 made this fibre-level observable canonical at the amplitude level

Thus O20 works entirely beyond the block-level mismatch.

3. Conditional persistence criterion

The central proposition states that, under [H1] and [H2], a fibre is projectively persistent if and only if there exists [ n^* \in [n_0,n_1] ] such that [ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n^*)=\sigma_{\mathrm{BI}}. ]

Under the empirically established power-law ansatz [ \sigma_{\mathrm{pair}}^{\mathrm{can}}(n)\sim C,n^{-\delta_{\mathrm{pair}}}, ] this gives [ n^*(\delta_{\mathrm{pair}})

\left(\frac{C}{\sigma_{\mathrm{BI}}}\right)^{1/\delta_{\mathrm{pair}}} ] and therefore the admissible interval [ \delta_{\mathrm{pair}} \in \left[ \frac{\log(C/\sigma_{\mathrm{BI}})}{\log n_1}, \frac{\log(C/\sigma_{\mathrm{BI}})}{\log n_0} \right]. ]

4. Structural interpretation of the lower bound

The paper shows that the lower bound [ \delta_{\min}\approx 7.4 ] is structurally consistent with the previously established fibre-level result [ \delta_{\mathrm{pair}}=2,\delta_c \approx 7.44. ]

So the lower edge of the physical window is not introduced ad hoc: it coincides with the minimal fibre-level exponent already realised by the pair construction of O16–O18.

5. The upper bound remains only partially derived

The upper bound [ \delta_{\max}\approx 10.6 ] is not yet derived from substrate dynamics alone.

At this stage, it still carries a phenomenological component imported through the O7 structural relation [ \beta^=\frac{1}{\delta+\tfrac{1}{2}} ] from the lepton-mass window [ \beta^ \in (0.09,0.13). ]

O20 is explicit about this status.

6. The central open problem is identified precisely

The main unresolved issue is now:

derive (\sigma_{\mathrm{BI}}) directly from the Born–Infeld saturation constraint [ |\partial_t \chi_v|\le c_{\mathrm{BI}} ] without using the phenomenological window as input.

This is the key problem that separates a conditionally consistent persistence window from a fully derived prediction.

Mathematical Role of O20

The mathematical contribution of O20 is to convert the physical window ([7.4,10.6]) from a mere compatibility range into the image of a structural selection criterion.

More precisely, the paper does the following:

• imports the canonical observable from O19
• formulates two explicit structural hypotheses
• derives a necessary and sufficient threshold-crossing condition
• translates that condition into an admissible interval for (\delta_{\mathrm{pair}})
• shows that the measured value [ \delta_{\mathrm{pair}} \approx 7.44 ] sits exactly at the lower edge of this window in the way expected from the minimal fibre construction

The central point is:

O20 is not a paper about changing the exponent,
but about assigning a structural admissibility criterion to it.

Epistemic Structure of the Paper

A major feature of O20 is that it separates clearly:

Established input

• the existence of the power-law regime
• the measured value (\delta_{\mathrm{pair}}\approx 7.44)
• the necessity of the fibre-level observable
• the canonical amplitude normalisation of O19

Conditional proposal

• the identification of persistence with threshold crossing
• the interpretation of (\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)) as a monotone image of a fibre-level amplitude
• the inheritance of a Born–Infeld saturation threshold at the fibre level

Open problems

• derive the threshold (\sigma_{\mathrm{BI}}) intrinsically
• derive the monotone correspondence between span-based redundancy and fibre-level amplitude
• determine the large-(q) stability of the persistence window

This epistemic separation is part of the contribution of O20 itself.

Interpretation of the Result

The conceptual significance of O20 is that the question is no longer:

Why is (\delta_{\mathrm{pair}}) numerically close to the target window?

but instead:

Under what structural condition does a fibre-level observable belong to the physical sub-spectrum?

This shifts the programme:

• from exponent extraction
• to admissibility selection
• from empirical compatibility
• to projective persistence

O20 therefore plays the role of a structural bridge between:

• the observable identification programme of O16–O19
• and the dynamical (\delta_{\mathrm{pair}}\to\beta^*) programme of O21

Structural Role of O20

O20 occupies a very specific place in the O-series:

• O16: pair observable identified
• O17: pair dynamics derived
• O18: fibre structure derived
• O19: canonical amplitude normalisation derived
• O20: persistence criterion for the physical sub-spectrum derived

Thus:

• the observable is physically justified
• the fibre structure is structurally grounded
• the amplitude is canonical
• the admissible sub-spectrum is now associated with a structural threshold criterion

This is the last structural step before the full fibre-level (\delta_{\mathrm{pair}}\to\beta^*) derivation.

What O20 Adds

• explicit formulation of the projective persistence criterion
• explicit statement of hypotheses [H1] and [H2]
• derivation of the crossing-rank condition [ \exists,n^\in[n_0,n_1]: \sigma_{\mathrm{pair}}^{\mathrm{can}}(n^)=\sigma_{\mathrm{BI}} ]
• derivation of the induced exponent window [ \delta_{\mathrm{pair}} \in [\delta_{\min},\delta_{\max}] ]
• structural interpretation of the lower bound [ \delta_{\min}\approx 7.4 ]
• explicit statement that the upper bound [ \delta_{\max}\approx 10.6 ] still retains a phenomenological component
• explicit identification of the intrinsic derivation of (\sigma_{\mathrm{BI}}) as the central open problem
• a fully explicit separation between:
  • what is established
  • what is proposed
  • what remains open

Outcome

The spectral admissibility framework is now:

• fibre-level grounded (O18)
• amplitude-level canonical (O19)
• persistence-level structured (O20)

The pair observable is now:

• physically justified
• structurally derived
• canonically normalised
• equipped with a conditional physical admissibility criterion

Residual Open Problems

The main remaining questions are now:

1. Intrinsic derivation of (\sigma_{\mathrm{BI}})
   Derive the saturation threshold directly from the Born–Infeld constraint and substrate dynamics

2. Derivation of the monotone amplitude correspondence
   Prove the link between Gram–Schmidt redundancy growth and a fibre-level saturating amplitude

3. Large-(q) stability of the persistence window
   Determine whether the selected sub-spectrum remains stable in the large-prime limit

4. Full fibre-level (\delta_{\mathrm{pair}}\to\beta^*) derivation (O21)
   Verify that the O7 structural growth law applies at the fibre level and derive [ \delta_{\mathrm{pair}} \approx 7.44 \mapsto \beta^* \approx 0.126 ]

5. Extension beyond the current Weil/Heisenberg setting
   Determine how universal the persistence criterion is across broader admissibility models

Status

The programme is now:

• observationally well-defined (O17)
• structurally grounded (O18)
• canonically normalised (O19)
• conditionally selection-based (O20)

Remaining work is now concentrated on:

• deriving the persistence threshold from substrate dynamics
• deriving the monotone amplitude correspondence
• completing the fibre-level cascade interpretation
• extending the analysis beyond the present spectral realisation

Repository Structure

paper/
├── out/      # Compiled O20 PDF
├── tex/      # LaTeX sources
└── README.md

Citation

If you reference this work, please cite:

J. Beau, Projective Persistence and the Physical Sub-Spectrum: A Dynamical Selection Criterion for the Capacity Exponent, Zenodo, 2026.

Acknowledgements

Portions of the derivations, conceptual synthesis, numerical strategy, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All theoretical results, computations, and interpretations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference.

Critical feedback, independent verification, and further analysis of:

• projective persistence
• fibre-level admissibility
• Born–Infeld saturation thresholds
• canonical pair observables

are welcome.

Please open an issue to discuss conceptual points, technical details, or possible extensions.