Matrix-Level Dynamic Redundancy and the Structural Confinement of the Cascade Exponent β

The Cosmochrony spectral relaxation programme has progressively constrained the cascade exponent β governing the growth law p(n) ~ n^β of the effective relational valence. O3 established the phenomenological window β^* ∈ (0.09, 0.13) from charged lepton masses. O4 derived the upper bound β ≤ 1 within its growing-degree LPS closure from bounded Born–Infeld flux. O5 proved that this gap cannot be closed by any purely representation-theoretic or fixed-dimensional mechanism, and identified dynamic redundancy in a matrix-level transition space of growing dimension O(q²) as the next candidate class. The present paper constructs and tests the first O(q²)-dimensional fingerprint in this programme: the Steinberg-matrix fingerprint πₘₐₜ(v) = vec(Pᵥ - J/(q+1)), where Pᵥ is the permutation matrix of the M\"obius action of v on P¹(Fq). We define the redundancy functional Rₙ as the effective-to-raw frontier ratio and prove that Rₙ → 0 (q-structurally) for this fingerprint. Numerical experiments on LPS families X5,q for q ∈ \29, 41, 61\ reveal a universal BFS-depth rigidity: steps 1–3 always contribute exactly 186 independent Steinberg directions regardless of q, and full saturation occurs at BFS depth 6 for all tested q. The relative threshold |S^*|/|G| decreases from 0.82 to 0.19 as q grows from 29 to 61, confirming q-structural saturation. However, the saturation depth remains bounded (step 6), indicating that the Steinberg-matrix fingerprint is a representation-theoretic rather than a genuinely dynamical saturation. The pre-saturation window is too short to extract a stable effective exponent βeff. We give a structural explanation via the Hecke operator action of LPS generators on the Steinberg module, and identify the multi-step path fingerprint in a space of dimension O(q²ᵏ) as the next construction required for a growing pre-saturation window. These results extend the hierarchy of O5 by one level, confirm q-structural saturation at the O(q²) Steinberg level, and precisely characterise the obstruction to exponent extraction at this level.