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

The Cosmochrony spectral relaxation programme has progressively constrained the cascade exponent $\beta$ governing the growth law $p(n) \sim n^\beta$ of the effective relational valence. O3 established the phenomenological window $\beta^* \in (0.09, 0.13)$ from charged lepton masses. O4 derived the upper bound $\beta \leq 1$ 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 $O(q^2)$ as the next candidate class. The present paper constructs and tests the first $O(q^2)$-dimensional fingerprint in this programme: the Steinberg-matrix fingerprint $\pi_{\mathrm{mat}}(v) = \mathrm{vec}(P_v - J/(q+1))$, where $P_v$ is the permutation matrix of the M"obius action of $v$ on $\mathbb{P}^1({\mathbb{F}}_q)$. We define the redundancy functional ${\mathcal{R}_n}$ as the effective-to-raw frontier ratio and prove that ${\mathcal{R}_n} \to 0$ ($q$-structurally) for this fingerprint. Numerical experiments on LPS families $X_{5,q}$ for $q \in \{29, 41, 61\}$ 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 ${\beta_{\mathrm{eff}}}$. 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 $O(q^2)$ Steinberg level, and precisely characterise the obstruction to exponent extraction at this level.