Finite Character-Trace Saturation and the Limits of Vertex-Based Class-Function Cascades

Abstract

The Cosmochrony relaxation cascade generates an inter-generational mass hierarchy through the contraction of the Kesten–McKay spectral support as the effective valence $p(n)$ grows. The cascade exponent $\beta$ in $p(n) \sim n^\beta$ is the sole remaining free parameter: O3 constrains ${\beta^*} \in (0.09, 0.13)$ from lepton masses, but no structural mechanism fixing $\beta$ is established; the companion note O4 audits one candidate route (bounded flux plus Cheeger expansion) and finds no structural bound follows from it. The present paper tests a different, representation-theoretic route, importing no bound from O4. The LPS quaternion generators produce $G = {\mathrm{PSL}}(2,{\mathbb{F}}_q)$ or $\mathrm{PGL}(2,{\mathbb{F}}_q)$ depending on the Legendre symbol of $p$ mod $q$; canonicalizing group elements by their induced permutation of $\mathbb{P}^1({\mathbb{F}}_q)$, rather than an ad hoc matrix-scalar quotient, is verified to reproduce the exact closed-form group order, a symmetric Cayley graph, and shell growth matching the free tree before any collision, with the character table computed by Dixon's algorithm and verified via Burnside's identity. For a sector $\rho$, write $A_\rho=\sum_{s\in\mathcal S_p}\rho(s)$ and ${\bar\mu_\rho}={\operatorname{tr}}(A_\rho)/\dim\rho$: since the six generators are only a small part of their conjugacy class, $A_\rho$ is generally not central and ${\bar\mu_\rho}$ is a trace average, not an eigenvalue ($q=13$: $55$ distinct graph eigenvalues against only $9$ distinct trace averages across $15$ sectors), so every sector weight, window, and fingerprint below is a character-trace object, never a Laplacian eigenvalue or Ramanujan-admissible mode. We prove this suffices to bound novelty regardless: for any class-function encoding ${\pi_A}(g)=({\bar\kappa_\rho}\chi_\rho(g))_{\rho\in{\widehat{G}^{{\mathrm{tr}}}}}$, the span $\mathcal R_A=\mathrm{span}\{{\pi_A}(g):g\in G\}$ has dimension $r_A={\operatorname{rank}}({M_{{\mathrm{tr}}}} D_\kappa)\le{\operatorname{rank}}({M_{{\mathrm{tr}}}})\le|{\mathrm{Cl}}(G)|=O(q)\ll|G|=O(q^3)$, because ${\pi_A}$ is constant on conjugacy classes regardless of the weights chosen; the middle inequality is strict whenever a selected sector has ${\bar\kappa_\rho}=0$, observed for several sectors at once in every case tested. A finite spanning witness $T$ with $|T|=r_A$ therefore exists, but an explicit shell-by-shell exploration of $X^{5,13}$ needs far more than $r_A$ vertices to find one. A shell-layered transitional novelty, testing shell $n$ only against strictly earlier shells (avoiding both self-reference and traversal-order dependence), shows that character-trace transition fingerprints collapse to a fixed, low-dimensional span, fixed-matrix proxies saturate their own ambient dimension within a handful of shells, and a Steinberg-based fingerprint on $\mathbb{P}^1({\mathbb{F}}_q)$ saturates its full ambient space within the first two to three graph-distance shells for every tested $q\in\{13,17,29\}$, leaving no pre-saturation window from which to extract an exponent. None of these constructions yields a viable mechanism for ${\beta^*}$. A matrix-level redundancy law of the form ${\beta_{\mathrm{eff}}}=1/(1/2+\alpha)$ is not supported by the constructions examined here: no construction here supplies a regime in which $\alpha$ could even be measured, and its functional form coincides with a conversion law the companion Span-Growth Note proves does not transfer natively to a different admissibility substrate. O5's contribution is an obstruction result: finite character-trace encodings of the vertex boundary cannot supply the rich, mode-resolved novelty a viable mechanism would need; the genuine eigenvalue-level frontier remains unconstructed, left open for a future paper.