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

The Cosmochrony spectral relaxation programme has progressively constrained the cascade exponent governing the growth law p (n) n^ of the effective relational valence. O3 established the phenomenological window ^* (0. 09, 0. 13) from charged lepton masses. O4 derived the structural upper bound 1 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 correct class of mechanisms. The present paper constructs and tests the first O (q²) -dimensional fingerprint in this programme: the Steinberg-matrix fingerprint ₌₀ₓ (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, ₐ 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 ₄₅₅. 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 (q^2k) 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.