Overview
This article continues the spectral hierarchy programme by addressing the mechanism identified but not resolved in O5: matrix-level admissible redundancy as the possible origin of the small cascade exponent.
The key question is whether a sufficiently refined transition-level encoding can produce a long pre-saturation regime in which the admissible frontier decays as a power law. O6 provides the first explicit construction of such a matrix-level fingerprint and tests it on LPS graphs.
The main result is negative but decisive. While matrix-level fingerprints can produce $q$-dependent saturation scales, they still exhibit a bounded exploration depth that prevents any genuine power-law regime from emerging. This reveals a structural obstruction: the cascade exponent cannot arise from any fixed finite-dimensional representation.
Core contributions
- First matrix-level admissible fingerprint: introduction of a Steinberg-based transition fingerprint acting on $\mathbb{P}^1(\mathbb{F}_q)$ with ambient dimension $O(q^2)$.
- $q$-structural saturation: the admissible saturation scale grows with $q$, with $|S^*|/|G| \to 0$, identifying the correct class of candidate mechanisms.
- Bounded-depth saturation: despite increasing dimension, saturation occurs at a fixed BFS depth independent of $q$, preventing any extended pre-saturation regime.
- No-go theorem: any fingerprint based on a fixed finite-dimensional representation exhibits bounded-depth saturation and cannot generate a power-law decay of the admissible frontier.
- Separation of mechanisms: distinction between representation-theoretic saturation (dimension-driven) and genuinely dynamical saturation (depth-driven).
Interpretation
The article shows that increasing the dimension of the representation is not sufficient to explain the cascade exponent. Even at the matrix level, a fixed-dimensional encoding explores only a bounded number of independent admissible directions.
- Vertex-level mechanisms fail due to early representation-theoretic saturation.
- Matrix-level fingerprints improve structural resolution but still saturate at bounded depth.
- Fixed-dimensional encodings cannot sustain a long enough regime to produce $R_n \sim p(n)^{-\alpha}$.
- Depth, not dimension is the key missing ingredient.
The cascade exponent $\beta$ therefore cannot be interpreted as a static representation-theoretic quantity. It must arise from a dynamical process in which the effective space of admissible directions grows along the cascade.
Relation to the Cosmochrony program
O6 follows from the structural localisation achieved in O5. Spectral admissibility, capacity, rigidity, and stratigraphy fix the spectral backbone of the theory. O1 restores ordering through projective dynamics, O3 amplifies the hierarchy via valence growth, and O5 isolates matrix-level redundancy as the relevant mechanism.
The present paper performs the next necessary step: it proves that even matrix-level mechanisms fail if they remain finite-dimensional. This establishes a general obstruction and forces the next stage of the programme toward dynamically growing constructions.
References
Jérôme Beau. Matrix-Level Dynamic Redundancy and the Structural Confinement of the Cascade Exponent β. Preprint.