This repository contains the source of the O11 Cosmochrony paper
Weil-Block Projective Capacity on Heisenberg Graphs: Representation-Adapted Extraction of the Decay Exponent.

This work extends the spectral admissibility sub-programme by resolving the representation-level obstruction identified in O10, and by providing the first stable numerical extraction of the projective capacity decay exponent on polynomial-growth graphs.

While O10 demonstrated that the failure to extract the exponent $\delta$ was not due to geometry but to a mismatch between the dense fingerprint representation and the irreducible structure of the dynamics, it left open whether a representation-adapted construction could recover the asymptotic regime.

The present work answers this question by introducing a Weil-block proxy fingerprint, based on a bidimensional Fourier-character embedding aligned with the abelianised structure of the Heisenberg group.

The central object is now the blockwise incremental projective capacity [ \Sigma_n^{(\tau,\sigma)} = \frac{\Delta r_n^{(\tau,\sigma)}}{|S_n|}, ] which measures the effective novelty contributed by each BFS shell relative to the subspace generated by previous shells.

Core Result

The paper establishes that the algorithmic obstruction identified in O10 can be resolved by moving from dense fingerprint representations to a representation-adapted blockwise construction.

Starting from:

• the O7 capacity observable $\Sigma_n$
• the O9 Heisenberg graph setting
• the O10 diagnosis of dense TensorSketch failure
• a bidimensional Fourier-character proxy on $(a,b)$ coordinates

the analysis shows that:

• a stable pre-saturation decay regime is observed across all tested primes
• the mean block capacity satisfies: [ \bar\Sigma_n \sim C,n^{-\delta_{\mathrm{cap}}} ] on a well-defined BFS window
• the extracted decay exponent converges:
  • $\hat{\delta}_{\mathrm{cap}} \approx 3.3$--$3.4$ for $q = 211, 307$
• log-log fits are robust:
  • $R^2 > 0.94$ for all tested primes
  • $R^2 = 0.987$ at $q = 307$
• inter-block variance remains controlled in the pre-saturation window:
  • $\mathrm{Var}(\Sigma_n)/\bar\Sigma_n^2 \ll 1$

Thus:

• the O10 algorithmic obstruction is resolved
• the exponent becomes numerically accessible
• the observable exhibits a stable asymptotic regime

Structural Role of O11

O11 completes the resolution chain initiated in O8:

• O8: geometric obstruction from exponential shell growth
• O9: polynomial-growth geometry restores the observable window
• O10: dense representations fail to resolve the asymptotic regime
• O11: representation-adapted blockwise construction restores observability

The obstruction is now fully removed at:

• algebraic level (O6)
• geometric level (O8 → O9)
• representation level (O10 → O11)

O11 is the first step where the exponent becomes numerically measurable.

What O11 Adds

O11 introduces several decisive advances:

• a representation-adapted fingerprint based on Weil-block structure (proxy level)
• a blockwise incremental capacity observable aligned with O7
• a bidimensional embedding capturing both non-central coordinates $(a,b)$
• the first stable extraction of a decay exponent across multiple primes
• numerical evidence of:
  • a well-defined pre-saturation decay window
  • low inter-block variance in the relevant regime
  • convergence of the exponent at large $q$

Interpretation of the Result

The main conceptual outcome is the identification of the correct observable:

• not the cumulative span
• not the raw number of distinct modes
• but the incremental novelty of each BFS shell

This leads to a decay law rather than a growth law:

• geometry controls shell size
• projective dynamics controls redundancy accumulation
• the exponent $\delta_{\mathrm{cap}}$ measures the rate at which new relational directions become redundant

In particular:

• O10 showed that dense encodings cannot access the asymptotic regime
• O11 shows that a representation-aligned observable restores it

Thus, exponent extraction is not only possible, but structurally constrained.

Relation to Previous Steps

O11 preserves all previous structural results:

• spectral admissibility from Step 1
• binary-polyhedral maximality from Step 2
• three-level ADE stratigraphy from Step 3
• projective dynamics and support contraction from O1
• hierarchical amplification via growing valence from O3
• structural upper bound on the cascade exponent 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

It does not modify the observable definition of O7, but provides the first implementation that makes it measurable.

Conceptual Structure

O11 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 → representation-adapted extraction of $\delta$

The programme now identifies:

• the correct observable (O7)
• the correct geometry (O9)
• the correct representation level (O11)

What O11 Resolves

O11 provides:

• a complete resolution of the representation-level obstruction
• a numerically stable extraction of the decay exponent
• a validated observable aligned with the theoretical framework
• evidence that the asymptotic regime is accessible at finite $q$

Residual Open Problem

What remains is no longer numerical extraction.

The remaining tasks are:

• replacing the proxy with the exact Weil representation
• deriving $\delta$ analytically from representation theory
• establishing universality across blocks and primes

Open Directions

1. Exact Weil-block implementation (O12)
   Replace the proxy by the full metaplectic representation

2. Analytical derivation of $\delta$
   Link the decay exponent to ADE spectral structure

3. Universality tests
   Verify stability of $\delta$ across blocks and primes

4. Continuum connection
   Connect discrete decay dynamics to the O7 continuum limit

5. Higher-order fingerprints
   Extend beyond $k=3$ paths and test stability of the exponent

Status

This framework is now:

• free of algebraic obstruction (O6)
• free of geometric obstruction (O8 → O9)
• free of representation obstruction (O10 → O11)
• validated numerically with stable exponent extraction

It does not assume:

• that dense representations are sufficient
• that cumulative observables capture the dynamics
• that finite-size effects dominate the observed exponent

Repository Structure

paper/
├── out/ # Compiled O11 PDF
├── tex/ # LaTeX sources
└── README.md

Citation

If you reference this work, please cite:

J. Beau, Weil-Block Projective Capacity on Heisenberg Graphs: Representation-Adapted Extraction of the Decay Exponent, Zenodo, 2026.

Acknowledgements

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

Contributions

This repository is intended as a research reference.

Critical feedback, independent verification, and alternative implementations of representation-adapted fingerprints are welcome.

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