Overview
This article extends the spectral hierarchy programme by addressing the remaining gap left open by O4: bounded flux constrains the cascade exponent only at the level of a quadratic upper bound, while the phenomenological window required by O3 remains far smaller.
The central question is therefore not whether the raw LPS frontier expands, since it does, but which part of that frontier remains admissibly productive once one filters out directions that add no genuinely new content in the admissible spectral subspace. O5 introduces this notion explicitly and studies several candidate definitions.
The main result is twofold. First, the vertex-based admissible frontier saturates rapidly by representation-theoretic necessity, at scale $O(|\mathrm{Cl}(G)|)=O(q)$, far below $|G|=O(q^3)$. Second, the paper shows that this low-dimensional saturation is not yet the physical origin of the small exponent: the true mechanism must be a matrix-level redundancy phenomenon, not a mere dimension bound.
Core contributions
- Admissible frontier concept: the paper introduces the admissible frontier as the subset of the raw BFS boundary whose elements add genuinely new directions in the admissible spectral subspace.
- Representation-theoretic saturation theorem: the vertex-based admissible span $\Pi_A(S)$ satisfies $\dim \Pi_A(S)\le \mathrm{rank}(M_{\mathrm{adm}}) \le |\mathrm{Cl}(G)|=O(q)$, so the corresponding admissible frontier vanishes after a saturation scale $|S^*|=O(q)$.
- Negative structural result for character-based refinements: transition fingerprints based only on characters remain too coarse on LPS graphs and do not produce the needed structural decay.
- Negative structural result for fixed-dimensional proxies: small matrix fingerprints can generate extinction of the admissible frontier, but only through a $q$-independent ambient dimension effect, which disqualifies them as explanations of the hierarchy.
- First structurally valid matrix candidate: a Steinberg-based transition fingerprint on $\mathbb{P}^1(\mathbb{F}_q)$ yields the first saturation scale that grows with $q$, with $|S^*|/|G|\to 0$, thereby identifying the correct class of candidate mechanisms.
Interpretation
The article shows that the missing mechanism behind the small cascade exponent is not a better Cheeger estimate, nor a simple representation-theoretic counting effect. The raw combinatorial frontier of the LPS graph remains expansive, but the admissibly productive part of that frontier can collapse much earlier.
- Vertex-based admissibility is too rigid: it saturates exactly, but too early and for purely representation-theoretic reasons.
- Character-based transition fingerprints are still too coarse: they fail to resolve enough directional information to distinguish genuinely new admissible channels.
- Low-dimensional matrix proxies are too small: they produce extinction, but only by filling a fixed ambient vector space.
- Growing matrix spaces are the first structurally credible candidates: they allow $q$-dependent saturation while keeping $|S^*|/|G| \to 0$.
Within this perspective, O5 is not yet the derivation of $\beta^*$ itself. It is the paper that isolates which mechanisms cannot explain it, proves the first exact admissible-saturation theorem, and localises the remaining problem in a genuinely dynamical matrix-level redundancy effect.
Relation to the Cosmochrony program
O5 follows directly from the open problem identified in O3 and sharpened by O4. Spectral admissibility selects the relevant sectors, spectral capacity and Gram rigidity constrain the admissible binary group structure, spectral stratigraphy fixes the three-level ADE organisation, spectral relaxation and O1 restore the ordering by support dynamics, O3 amplifies the hierarchy through dynamic valence growth, and O4 excludes super-quadratic growth laws while leaving the smallness of the exponent unexplained.
The present paper does not close the cascade-exponent problem quantitatively. Instead, it performs the next logically necessary step: it proves that low-dimensional admissible quotients are insufficient, and shows that the remaining mechanism must live in a transition-level matrix space of growing dimension. In that sense, O5 is the obstruction-and-localisation paper of the programme, and it opens naturally toward O6.