This repository contains the source of the O13 Cosmochrony paper
Asymptotic Stability of Exact Weil-Block Capacity on Heisenberg Graphs: Extended Prime Range, Variance Reduction, and Requalification of the δ–β Tension*.

This work extends the spectral admissibility sub-programme by testing the finite-size hypothesis left open in O12, through an extension of the exact Weil-block computation to larger primes.

O12 measured a higher finite-$q$ decay exponent than the proxy and identified a tension with the phenomenological target for $\beta^*$; its corrective revision shows that finite-$q$ value is a crossover statistic, not an intrinsic asymptotic exponent, and derives the intrinsic unfolded exponent exactly ($\delta_{\mathrm{exact}}=3$), independent of the central coordinate $\gamma$. O12 originally left open whether the finite-$q$ mismatch could be explained as a finite-size effect.

The present work answers this question by extending the computation to [ q \in {101, 151, 211}, ] and by analysing both convergence behaviour and variance structure.

The central observable remains the exact incremental block capacity [ \Sigma_n^{(c)} = \frac{\Delta r_n^{(c)}}{|S_n|}, ] measured within irreducible Weil blocks of the Heisenberg Cayley graph.

The paper establishes that the exact decay exponent does not drift upward toward the phenomenological target range at larger primes.

Starting from:

• the O7 capacity observable $\Sigma_n$
• the O9 Heisenberg graph setting
• the O11 proxy-level extraction
• the O12 exact Weil-block observable

the analysis shows that:

• the sequence of exact exponents [ \hat\delta_{\mathrm{exact}} = 4.42,,4.80,,4.51,,4.27,,3.59 ] at $q \in {29,61,101,151,211}$ exhibits one non-monotone step (4ドル.42 \to 4.80$, from $q=29$ to $q=61$), followed by a strict monotone decrease from $q = 61$ to $q = 211$ (4ドル.80\to4.51\to4.27\to3.59$) — the sequence as a whole is not monotone
• the measurement quality improves with $q$:
  • fitting window length increases from 4 to 11 points
  • inter-block variance decreases from $V_n^{\max} = 5.20$ to 0ドル.30$
  • $R^2 > 0.993$ throughout
• condition (E2) is satisfied at $q \geq 151$, establishing block-by-block universality
• no fitting strategy produces $\delta_\infty > 5.0$ — and O12's corrective revision independently derives, in closed form, the exact value this sequence converges toward: $\delta_\infty = \delta_{\mathrm{exact}} = 3$

Thus:

• the exponent decreases precisely where the measurement becomes most reliable
• the observed trend is not a numerical artefact
• the finite-size hypothesis is ruled out in its strong form

O13 performs the decisive test left open by O12:

• O11: proxy-level extraction
• O12: exact Weil-block extraction and tension identification
• O13: asymptotic test and falsification of the finite-size explanation

O13 is the first step where the discrepancy between theory and phenomenology is shown not to be a finite-size artefact; whether it is otherwise "structural" awaits the regime determination in the O3–O7 relation described below, since no $\beta^*$ constraint is currently drawn at all.

O13 introduces several decisive advances:

• extension of the exact computation to larger primes
• identification of a strict monotone decreasing regime of $\hat\delta_{\mathrm{exact}}$ from $q=61$ to $q=211$ (not from $q=29$: the sequence rises from $q=29$ to $q=61$ before decreasing)
• a systematic analysis of convergence:
  • failure of 1ドル/q$ scaling
  • log-linear behaviour without plateau
• a quantitative law for variance reduction: [ V_n^{\max}(q) \sim 754 \cdot q^{-1.41} ]
• identification of the correlation:
  • measurement quality increases while $\delta$ decreases

The main conceptual outcome is that the mismatch with the phenomenological window for $\beta^*$ cannot be explained by finite-size effects — and, per O12's corrective revision, cannot be explained by a central-coordinate mechanism either, since the observable is exactly, algebraically blind to $\gamma$.

No numeric $\beta^$ value is propagated from O13's own $\hat\delta_{\mathrm{exact}}$ sequence. O12's revision shows this construction carries several structurally distinct quantities — the finite-$q$ fitted slope, the unfolded intrinsic exponent $\delta_{\mathrm{exact}}=3,ドル and a saturation-scale scaling function — any of which the $\delta\mapsto\beta^$ relation might have intended as its input, and the O3–O7 derivation does not specify which. Substituting any one of them, including 3ドル$, would repeat the same error.

The paper shows that:

• no upward drift toward the target range occurs
• improved statistics reinforce the discrepancy
• the mismatch is structural, but not for the reason O13 originally proposed

O13 reclassifies the tension identified in O12:

• not a finite-size artefact (S1 rejected)
• not a central-coordinate effect either — O12's correction proves $\gamma$ contributes nothing to this observable, at every prime including the ones tested here
• but an unresolved regime mismatch: which of the construction's several structurally distinct quantities the $\delta\mapsto\beta^*$ relation takes as input (S2, correctly stated)

The source of the mismatch is:

• the normalisation of $\Sigma_n^{(c)}$ by $|S_n|$, not by the block dimension $q$
• the unresolved choice of regime (finite-$q$ crossover slope vs.\ the unfolded exponent $\delta_{\mathrm{exact}}=3$ vs.\ the saturation-scale scaling function) as the $\delta\mapsto\beta^*$ relation's intended input

O13 preserves all previous structural results:

• spectral admissibility (Step 1)
• binary-polyhedral maximality (Step 2)
• ADE stratigraphy (Step 3)
• ordering via O1
• amplification via O3
• structural bound via O4
• admissible-frontier dynamics via O5
• no-go via O6
• capacity formulation via O7
• geometric obstruction via O8
• polynomial-growth resolution via O9
• algorithmic obstruction via O10
• proxy-level extraction via O11
• exact Weil-block regime via O12

It does not modify these steps, but tests their asymptotic compatibility.

O13 completes the following chain:

1. Observable defined (O7)
2. Geometry resolved (O9)
3. Representation adapted (O11)
4. Exact regime identified (O12)
5. Asymptotic behaviour tested (O13)

The programme now establishes:

• the exact observable is correctly defined
• the exact exponent is reliably measurable
• the asymptotic trend is downward, toward the now-exact value $\delta_{\mathrm{exact}}=3$
• the mismatch with phenomenology is not a finite-size artefact and not a central-coordinate effect; whether it survives as a genuine tension depends on an unresolved regime question, not yet on established physics

O13 provides:

• a direct test of the finite-size hypothesis
• a falsification of its strong form
• a robust asymptotic trend for $\delta_{\mathrm{exact}}$
• a precise localisation of the structural gap

The remaining problem is now clearly identified:

• the mapping $\delta \mapsto \beta^*$ is not yet valid in the exact-block setting, because it is unclear which of three structurally distinct quantities it should take as input

The task is to:

• re-audit the O3–O7 derivation to determine which regime — the finite-$q$ crossover slope, the unfolded exponent $\delta_{\mathrm{exact}}=3$, or the saturation-scale scaling function — is the intended input
• understand the role of block normalisation ($|S_n|$ vs.\ block dimension $q$)

1. Exact-block δ → β relation (O13-O1)*
   Determine which regime the structural mapping should take as input, and account for the normalisation difference

2. Scaling exponent α (O13-O2)
   Derive the correction law in: [ \delta(q) = \delta_\infty + \frac{a}{q^\alpha} ] from the combinatorial shell-growth/saturation-depth mechanism of O12, not from any central-phase effect

3. Second-seed validation (O13-O3)
   Quantify inter-seed variability at $q \in {151, 211}$

4. Larger-prime extension
   Empirically confirm the fitted slope's continued convergence toward the now-exact value $\delta_{\mathrm{exact}}=3$

5. Analytical derivation of δ — resolved by O12
   The exponent is derived from Heisenberg shell growth and the block saturation-depth law, not from metaplectic structure, which plays no role in this observable

This framework is now:

• free of algebraic obstruction (O6)
• free of geometric obstruction (O8 → O9)
• free of representation obstruction (O10 → O11 → O12)
• tested asymptotically (O13)

It does not assume:

• convergence toward the phenomenological target
• equivalence between proxy and exact observables
• validity of the O7 relation in the exact setting

paper/
├── out/ # Compiled O13 PDF
├── tex/ # LaTeX sources
└── README.md

If you reference this work, please cite:

J. Beau, Asymptotic Stability of Exact Weil-Block Capacity on Heisenberg Graphs: Extended Prime Range, Variance Reduction, and Requalification of the δ–β* Tension, Zenodo, 2026.

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.

This repository is intended as a research reference.

Critical feedback, independent verification, alternative implementations of exact Weil-block tracking, and asymptotic extensions to larger primes are welcome.

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