Overview
This article continues the spectral admissibility programme after the geometric repair established in O9. While O9 proved that polynomial-growth Heisenberg Cayley graphs remove the shell-compression obstruction identified in O8, it left open the first actual large-q test of the capacity observable and the extraction of the exponent $\delta$.
O10 performs that next step at the first computationally accessible large prime, $q = 101$, on the Cayley graph of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ using the $k=3$ permutation-path fingerprint together with a dense TensorSketch approximation.
The result is mixed but structurally decisive. Three positive signals are confirmed at large scale: ball growth remains consistent with the Bass--Guivarc'h law, the mean capacity decays monotonically over a visible pre-saturation window, and the O7 state law remains valid. The negative result is equally important: the log-log fit does not yet yield a reliable $\delta$, not because the Heisenberg setting fails, but because the dense TensorSketch remains computationally misaligned with the irreducible structure of the fingerprint.
Core contributions
- First large-q computation: implementation of the $k=3$ permutation-path capacity fingerprint on $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}),S_q)$ at $q = 101$.
- Ball-growth confirmation: numerical recovery of polynomial expansion with measured exponent $\hat{D} = 3.05$, continuing the convergence toward the homogeneous dimension $D_{\mathrm{hom}} = 4$.
- Direct projective depletion signal: monotone decay of the mean capacity by $23.5\%$ over a pre-saturation window of $n^* = 13$ BFS steps.
- State-law validation at large scale: confirmation that $R_n^{(3)} \approx \Phi(\bar{\eta}_n)$ still holds at $q = 101$, extending the O9 verification beyond the small-q regime.
- Algorithmic obstruction diagnosis: proof that reliable power-law extraction within $|B_n| \leq q^2$ would require a dense sketch dimension $D \gg q^4$, which is computationally impractical.
- Weil-block roadmap: identification of the decomposition of $\rho_{\mathrm{aff}}^{\otimes 3}$ into $O(q)$-dimensional Weil blocks as the correct route toward a tractable extraction of $\delta$ in the next step of the programme.
Interpretation
O10 shows that the Heisenberg replacement introduced in O9 is not merely geometrically elegant but physically productive: at large $q$, the projective signal is genuinely present. The capacity decreases, the state law survives, and the polynomial-growth background behaves as expected.
- O8 showed that exponential shell growth on LPS graphs geometrically compresses the entire observable window.
- O9 replaced LPS by polynomial-growth Heisenberg graphs and restored a scalable BFS window in principle.
- O10 confirms that this restored window carries a real projective signal at large $q$, but also reveals that the dense TensorSketch is the wrong computational representation for extracting the asymptotic slope.
- Conclusion: the remaining obstruction is no longer geometric. It is representational and algorithmic.
In this picture, the capacity exponent $\delta$ remains a property of the underlying projective dynamics rather than of a particular graph family. But measuring it requires not only a non-obstructive geometry, as established by O9, but also a fingerprint representation aligned with the irreducible block structure of the Heisenberg action.
Relation to the Cosmochrony program
O10 follows directly from O9. 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, and O9 removes that compression by moving to polynomial-growth Heisenberg geometry.
The present paper performs the first large-q test in that repaired setting. It confirms that the projective signal is measurable, isolates the final dense-sketch bottleneck, and hands off to O11 the representation-adapted extraction of $\delta$ through Weil blocks.
References
Jérôme Beau. Polynomial-Growth Capacity Dynamics at Large q: Ball Growth Confirmation, Capacity Decay, and the Weil Decomposition Roadmap. Preprint.