Overview
This article continues the spectral admissibility programme after the reformulation developed in O7. While O6 proved that no fixed finite-dimensional fingerprint can sustain the power-law regime required to explain the cascade exponent, O7 identified the correct observable as the discrete projective capacity $\Sigma_n$.
O8 performs the next necessary step: it introduces the first explicit q-growing path fingerprint, namely a three-step permutation-path encoding in ambient dimension $O(q^3)$, and tests whether this is sufficient to produce a measurable pre-saturation regime for the capacity growth exponent $\delta$.
The answer is structurally sharp. The new fingerprint genuinely escapes the algebraic obstruction of O6 and opens the first pre-saturation window growing as $O(q^2)$ in vertex count. But on LPS Ramanujan graphs this window is compressed by exponential shell growth into only $O(\log q)$ BFS steps, which is too short to support a reliable exponent extraction.
Core contributions
- Three-step permutation-path fingerprint: introduction of the fingerprint $\pi_3(\gamma)=\rho_{\mathrm{perm}}(v_1)\otimes\rho_{\mathrm{perm}}(v_2)\otimes \rho_{\mathrm{perm}}(v_3)$ for non-backtracking paths on $X_{p,q}$.
- First q-growing admissible window: identification of an effective pre-saturation window of order $O(q^2)$ in vertex count, the first growing window in the O-series.
- Escape from the O6 no-go: explicit demonstration that a growing fingerprint space can evade the bounded-depth saturation universal for fixed finite-dimensional representations.
- Geometric compression mechanism: proof that exponential shell growth on LPS graphs compresses the $O(q^2)$-vertex window into only $O(\log q)$ BFS steps.
- New obstruction class: identification of a geometric, rather than representation-theoretic, reason why the target exponent cannot be extracted on the LPS family at accessible sizes.
Interpretation
The article shows that escaping the algebraic obstruction of O6 is not enough. A growing admissible state space is necessary, but not sufficient. Once the fingerprint dimension grows with $q$, the decisive question becomes whether the underlying graph geometry provides a long enough observable cascade.
- Fixed-dimensional fingerprints fail because they saturate at bounded BFS depth.
- Three-step path fingerprints avoid that algebraic no-go and continue to generate new directions up to $O(q^2)$ vertices.
- LPS shell growth is too rapid: the observable window in BFS depth remains only logarithmic.
- The obstruction is geometric: the graph family itself prevents reliable exponent extraction, even though the fingerprint is structurally adequate.
In this picture, the unresolved problem is no longer merely to enlarge the fingerprint. It is to find a graph-growth regime in which a growing admissible state space unfolds over a sufficiently long cascade.
Relation to the Cosmochrony program
O8 follows directly from O7. 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, and O7 reformulates the relevant observable in terms of projective capacity.
The present paper performs the first explicit test beyond that no-go. It shows that the remaining obstruction is not algebraic but geometric: on LPS expanders, exponential shell growth compresses the observable window before the asymptotic capacity-scaling regime can be resolved.