Overview
This article continues the spectral admissibility programme after the geometric obstruction identified in O8. While O8 showed that the $k=3$ permutation-path fingerprint escapes the fixed finite-dimensional obstruction of O6, it also established that exponential shell growth on LPS graphs compresses the entire pre-saturation window into only $O(\log q)$ BFS steps.
O9 performs the next necessary step: it replaces LPS expanders with a family of polynomial-growth Cayley graphs, namely the discrete Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, and transports the O8 fingerprint and O7 projective-capacity framework to this new geometric setting.
The result is structurally decisive. The obstruction identified in O8 is shown to be purely geometric. On Heisenberg Cayley graphs, an $O(q^2)$ vertex window spans at least $\Omega(q^{1/2})$ BFS steps, and in practice $\Theta(q)$ steps, giving polynomially many observable points instead of only logarithmically many on LPS graphs.
Core contributions
- Formal LPS compression theorem: proof that an $O(q^2)$ vertex window on LPS graphs is exhausted in only $O(\log q)$ BFS steps.
- Polynomial-growth replacement: introduction of $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}),S_q)$ as the correct discrete family for probing the capacity exponent.
- Window-Depth Theorem: proof that the same $O(q^2)$ vertex window now spans $\Omega(q^{1/2})$ BFS steps, with actual diameter scaling as $\Theta(q)$.
- Geometric separation: demonstration that graph geometry controls observability, while projective dynamics and the fingerprint construction remain universal.
- Numerical validation: exact BFS confirms diameter $\Theta(q)$, effective growth exponents $\hat{D}$ increasing toward $4$, and window-depth improvements up to $5.4\times$ over LPS at $q=29$.
Interpretation
O9 shows that the failure of exponent extraction in O8 was not caused by insufficient fingerprint complexity. The $k=3$ permutation-path fingerprint was already structurally adequate. What failed was the geometry of the graph family used to measure it.
- O6 ruled out fixed finite-dimensional fingerprints for algebraic reasons.
- O8 escaped that no-go but revealed a new obstruction: exponential shell growth compresses the observable window.
- O9 removes that obstruction by moving to polynomial-growth geometry.
- Conclusion: the remaining open problem is no longer geometric, but the actual extraction and derivation of the asymptotic exponent $\delta$.
In this picture, $\delta$ is not a special property of LPS graphs or Heisenberg graphs as such. It is a property of the underlying projective dynamics, provided the embedding graph does not obstruct its measurement.
Relation to the Cosmochrony program
O9 follows directly from O8. 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, and O8 identifies geometric compression on LPS graphs.
The present paper resolves that obstruction. It establishes polynomial-growth Cayley graphs as the minimal discrete setting in which the capacity exponent becomes observable, and hands off to O10 the tasks of extracting $\delta$ at larger $q$ and deriving it from first principles.
References
Jérôme Beau. Projective Capacity Beyond Expanders: Polynomial-Growth Relaxation Graphs and the Capacity Exponent. Preprint.