This repository contains the source of the O12 Cosmochrony paper
Exact Weil-Block Projective Capacity on Heisenberg Graphs: Resolving the Final Obstruction to δ Extraction.

This work extends the spectral admissibility sub-programme by replacing the proxy-level Weil-block construction of O11 with the exact Weil projection, and by providing the first exact representation-level measurement of the projective capacity decay exponent.

While O11 established that the representation-level obstruction identified in O10 could be overcome at the proxy level, it left open the central question of whether the exact metaplectic dynamics, including the central coordinate $\gamma$ of the discrete Heisenberg group, would yield the same exponent.

The present work answers this question by introducing an exact blockwise projective capacity based on the irreducible Weil blocks of the three-step fingerprint on Heisenberg Cayley graphs.

The central observable is now the exact incremental block capacity [ \Sigma_n^{(c)} = \frac{\Delta r_n^{(c)}}{|S_n|}, ] where $\Delta r_n^{(c)}$ is the rank increment contributed by shell $S_n$ inside an exact Weil block $H_c$.

The paper measures the exact Weil observable at $q\in{29,53,61}$, and separately derives its intrinsic asymptotic exponent in closed form.

Starting from:

• the O7 capacity observable $\Sigma_n$
• the O9 Heisenberg graph setting
• the O10 diagnosis of dense representation failure
• the O11 proxy-level block construction
• the exact Weil representation including the central phase $\gamma$

the analysis shows that:

• a measurable pre-saturation decay regime is present already at accessible primes
• the exact mean block capacity satisfies $\bar\Sigma_n \sim C,n^{-\hat\delta_{\mathrm{exact}}}$ over a short but structurally meaningful fitting window, with fitted slopes:
  • $\hat\delta_{\mathrm{exact}} = 4.37 \pm 0.08$ at $q = 29$
  • $\hat\delta_{\mathrm{exact}} = 4.28$ at $q = 53$
  • $\hat\delta_{\mathrm{exact}} = 4.80$ at $q = 61$ ($R^2 > 0.98$ at all tested primes), independently reproduced by pure combinatorial set arithmetic (no Weil phase, no Gram–Schmidt, no SVD)
• these fitted slopes admit an exact closed form: $\Sigma_n^{(c)}$ depends on the block $c$ only through a projective pair $(\alpha,\beta)$, and is therefore, by algebraic identity, exactly independent of the central coordinate $\gamma$ and of the translational coordinate $a$ — no coherence or metaplectic-phase mechanism can affect it
• using this closed form, the fitted slopes above are shown to be finite-$q$ crossover statistics of the block saturation-depth distribution, not estimates of an intrinsic asymptotic exponent
• the intrinsic unfolded exponent is derived exactly: [ \delta_{\mathrm{exact}} = 3 \qquad (q\to\infty,\ n\to\infty,\ n=o(\sqrt q)), ] from Heisenberg shell growth alone, holding block-by-block (not only in the mean)

Thus:

• the central coordinate $\gamma$, contrary to the paper's original motivation for building the exact projection, is provably inert for this observable
• the gap between the finite-$q$ fitted slopes and the O11 proxy value $\hat\delta_{\mathrm{cap}}\approx 3.39$ is real and reproducible but is not evidence of a central-coordinate/coherence mechanism
• the remaining issue is no longer observability or asymptotic convergence (both are now settled) but determining which regime the O3–O7 $\delta\mapsto\beta^*$ relation legitimately takes as input

O12 completes the extraction chain initiated in O8:

• O8: geometric obstruction from exponential shell growth
• O9: polynomial-growth geometry restores the observable window
• O10: dense fingerprints fail at the representation level
• O11: proxy Weil-block construction restores observability
• O12: exact Weil projection reveals the true exponent regime

O12 is the first step where the exponent is measured in the exact irreducible representation.

O12 introduces several decisive advances:

• the exact Weil-block observable replacing the O11 proxy
• a blockwise rank-tracking algorithm in irreducible dimension $q$
• an exact closed form for that observable (Proposition on the projective pair $(\alpha,\beta)$), proving it independent of $\gamma$ rather than merely diagnosing its effect empirically — the central coherence observable $\ell_\gamma(n)$ is retained as a diagnostic of the BFS geometry, not of the capacity observable itself
• the first exact extraction of the finite-$q$ capacity slope, cross-checked by pure combinatorial set arithmetic
• the first derivation of the intrinsic unfolded asymptotic exponent, $\delta_{\mathrm{exact}}=3$, in closed form
• a decomposition showing the finite-$q$ fitted slopes are crossover statistics of the saturation-depth distribution, not a departure from a shared per-block exponent

The main conceptual outcome is that the central coordinate is exactly irrelevant to this rank-based capacity observable — a stronger and more surprising finding than the paper originally set out to test.

The exact Weil projection shows that:

• the metaplectic phase $\psi_c(\gamma)$ is present in the representation and in the group action, but is provably invisible to the rank observable $\Sigma_n^{(c)}$ computed here
• the gap between the finite-$q$ exact and proxy fitted slopes reflects different ambient dimensions and covering geometries, not different sensitivity to $\gamma$
• the finite-$q$ fitted slopes are crossover statistics, not evidence of a steeper intrinsic decay: the intrinsic unfolded exponent is exactly 3ドル$, block-by-block, not merely in the mean

This leads to:

• shorter finite-$q$ fitting windows than the proxy, because the exact ambient block dimension is $q$ rather than $q^2$ — a real structural feature, unrelated to $\gamma$
• a refined picture of universality: block-by-block universality of the intrinsic exponent is now established analytically (not merely tested for at the mean level), with the observed finite-$q$ inter-block variance attributable to the saturation-depth distribution rather than to a failure of universality

In particular:

• O11 identified the correct proxy-level observable
• O12 shows the exact observable's finite-$q$ fitted slope is not directly comparable to it as a measure of $\gamma$-dynamics, and derives the exact observable's own asymptotic exponent independently

O12 preserves all previous structural results:

• spectral admissibility from Step 1
• binary-polyhedral maximality from Step 2
• three-level ADE stratigraphy from Step 3
• projective ordering via O1
• hierarchical amplification via O3
• structural upper bound on $\beta$ from O4
• admissible-frontier saturation from O5
• fixed finite-dimensional no-go from O6
• capacity formulation and state law from O7
• growing fingerprint and geometric obstruction from O8
• polynomial-growth resolution from O9
• algorithmic obstruction diagnosis from O10
• proxy-level representation-adapted extraction from O11

It does not replace the capacity definition of O7, but replaces the O11 proxy by the exact Weil representation.

O12 advances the structural chain as follows:

1. Spectral admissibility → mode selection
2. Spectral capacity → binary-polyhedral maximality
3. Spectral stratigraphy → discrete ADE levels
4. O1 → ordering via support contraction
5. O3 → amplification via valence growth
6. O4 → structural upper bound on $\beta$
7. O5 → admissible-frontier saturation
8. O6 → fixed finite-dimensional no-go
9. O7 → projective capacity formulation
10. O8 → growing fingerprint + geometric obstruction
11. O9 → polynomial-growth geometry
12. O10 → algorithmic obstruction
13. O11 → proxy-level representation-adapted extraction
14. O12 → exact Weil-block extraction of $\delta$

The programme now identifies:

• the correct observable class (O7)
• the correct geometry (O9)
• the correct proxy-level representation (O11)
• the first exact representation-level regime shift (O12)

O12 provides:

• the exact implementation left open in O11
• a proof, not merely a numerical comparison, that the finite-$q$ exact and proxy fitted slopes cannot differ because of $\gamma$-dynamics
• the exact combinatorial mechanism behind the finite-$q$ fitted slopes (saturation-depth crossover, not an estimate of the asymptotic exponent), and the intrinsic asymptotic exponent itself, derived independently in closed form ($\delta_{\mathrm{exact}}=3$)
• a determination that no $\beta^*$ constraint can currently be drawn from this observable

No constraint on $\beta^$ is drawn from O12. The O3–O7 structural relation [ \beta^ = \frac{1}{\delta + \tfrac12} ] takes a capacity decay exponent $\delta$ as input, but O12 shows this construction carries three structurally distinct exponent-like quantities: the finite-$q$ fitted slope, the unfolded intrinsic exponent ($=3$), and a saturation-scale scaling function with no exponent at all. The O3–O7 derivation does not, on its own text, specify which of these is the intended input. Substituting any of them into the relation without first settling this would repeat the same category error the present paper corrects. Resolving it requires re-auditing O3–O7 itself.

What remains is no longer the existence of an exact observable, nor its asymptotic exponent — both are now settled.

The remaining tasks are:

• re-auditing the O3–O7 structural relation to determine which regime its $\delta$ input is drawn from, before any $\beta^*$ constraint can be stated
• auditing O13 and O15, which share this pipeline, for the same regime confusion
• classifying O16 and O17's own use of this construction

1. Larger-prime exact computation (O12-O1), as an empirical check
   Extend the exact Weil-block computation to $q \geq 101$ to confirm empirically that the fitted crossover slope moves toward the analytic prediction $\delta_{\mathrm{exact}}=3$ as the fitting window is pushed further into the unfolded regime.

2. Re-audit of the structural relation $\delta \mapsto \beta^*$
   Determine which of the three exponent-like regimes identified above O3–O7's derivation actually presupposes.

3. Full block survey
   Test the analytically-derived block-by-block universality at larger sample sizes.

This framework is now:

• free of algebraic obstruction (O6)
• free of geometric obstruction (O8 → O9)
• free of representation obstruction at the proxy level (O10 → O11)
• extended to the exact Weil representation (O12)

It does not assume:

• that the proxy and exact observables coincide
• that blockwise universality must hold exactly
• that the exact finite-$q$ regime already matches the asymptotic phenomenological target

cd code
python3 -m venv .venv && source .venv/bin/activate
pip install -r requirements.txt
python3 spectral_O12.py --mode paper

Reproduces the published figures (fig2_variance.pdf–fig5_coherence.pdf) and Table~3 (δ̂_exact = 4.37, 4.28, 4.80 at q = 29, 53, 61) at fixed RNG_BASE = 42. Three further scripts in code/, self-contained in this repository, reproduce the paper's independent checks without running spectral_O12.py's full pipeline:

• code/o12_set_arithmetic.py — the exact (α,β) closed-form cross-check of Section 5: reproduces δ̂_exact at all three primes by pure combinatorial set arithmetic, with no Weil phase, Gram–Schmidt, or SVD (python3 o12_set_arithmetic.py).
• code/o12_saturation_law.py — the exact saturation-depth law of Section 6.3–6.4 (python3 o12_saturation_law.py 29 53 61 101 211 401 809).
• code/o12_bfs_truncation_check.py — confirms the BFS mid-shell-truncation scheme does not corrupt any shell used in the published tables (python3 o12_bfs_truncation_check.py).

o12/
├── tex/ # LaTeX sources
├── code/ # Reproduction script, figures, requirements.txt
└── README.md

out/ (compiled PDF) is generated locally by compile.sh and is git-ignored, not part of the repository; the citable PDF is the one attached to the Zenodo record (see Citation below).

If you reference this work, please cite:

J. Beau, Exact Weil-Block Projective Capacity on Heisenberg Graphs: Resolving the Final Obstruction to δ Extraction, 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.