Weil-Block Projective Capacity on Heisenberg Graphs: Representation-Adapted Extraction of the Decay Exponent

Core contributions

• Representation-adapted observable: replacement of the dense TensorSketch by a bidimensional Fourier-character proxy indexed by generic sextuples $(\tau,\sigma) \in (\mathbb{Z}/q\mathbb{Z})^6$.
• Incremental capacity definition: introduction of the block-wise observable $\Sigma_n^{(\tau,\sigma)} = \Delta r_n^{(\tau,\sigma)} / |S_n|$, which measures the genuinely new proxy directions contributed by shell $S_n$.
• Stable pre-saturation decay regime: direct numerical observation of a monotone log-log decay window on Heisenberg Cayley graphs for all tested primes.
• Cross-prime exponent extraction: fitted decay exponents $\hat{\delta}_{\mathrm{cap}} = 5.78,\ 4.29,\ 3.33,\ 3.39$ for $q = 101,\ 151,\ 211,\ 307$, with convergence at large $q$.
• High-quality fits: robust regressions with $R^2 > 0.94$ for all runs and $R^2 = 0.987$ at $q = 307$, showing that the decay window is structurally clean.
• Controlled inter-channel variance: validation of the O11 condition (B2), showing that generic channels remain coherent throughout the fitted window.
• Quantitative outcome: extraction of a convergent proxy-level decay exponent near $\hat{\delta}_{\mathrm{cap}} \approx 3.4$, together with a precise diagnosis of why this value may still differ from the exact Weil result.

Interpretation

O11 shows that the remaining obstruction isolated in O10 is not a failure of the Heisenberg setting itself, but of the fingerprint representation used to probe it. Once the observable is aligned with the block structure of the dynamics, the asymptotic regime becomes numerically visible.

• O8 showed that exponential shell growth on LPS graphs geometrically compresses the observable window.
• O9 replaced LPS by polynomial-growth Heisenberg graphs and restored a scalable BFS window.
• O10 confirmed the large-$q$ signal but proved that dense TensorSketch fingerprints remain algorithmically misaligned.
• O11 replaces that dense encoding by a bidimensional Weil-block proxy and recovers a stable decay law for the incremental capacity.
• Conclusion: the observable window is now open both geometrically and representationally, at least at the proxy level.

In this picture, the fitted exponent is not a raw growth rate of cumulative span, but the decay exponent governing how quickly new BFS shells become redundant relative to the proxy subspace already generated. The key signal is therefore a law of diminishing novelty rather than of direct accumulation.

Relation to the Cosmochrony program

O11 follows directly from O10. Spectral admissibility, capacity, rigidity, and stratigraphy define the spectral backbone of the theory. O1 restores ordering through projective dynamics, O3 amplifies the hierarchy via valence growth, O4 constrains the cascade exponent from bounded relational flux, O5 localises the failure of vertex-level mechanisms, O6 proves the no-go for fixed finite-dimensional fingerprints, O7 reformulates the observable in terms of projective capacity, O8 identifies geometric compression on LPS graphs, O9 removes that compression by moving to polynomial-growth Heisenberg geometry, and O10 isolates the final dense-sketch bottleneck at large $q$.

The present paper resolves that bottleneck at the proxy level. It shows that a representation-adapted block-wise observable can be implemented efficiently, that the resulting decay window is numerically stable, and that cross-prime fits converge toward a coherent asymptotic exponent. In that sense, O11 is the first step of the programme where the exponent becomes practically measurable rather than only structurally defined.

References

Jérôme Beau. Weil-Block Projective Capacity on Heisenberg Graphs: Representation-Adapted Extraction of the Decay Exponent. Preprint.