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

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 β in p(n) ~ n^β is the sole remaining free parameter: O3 constrains β^* ∈ (0.09, 0.13) from lepton masses, but no structural mechanism fixing β 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 = PSL(2,Fq) or PGL(2,Fq) depending on the Legendre symbol of p mod q; canonicalizing group elements by their induced permutation of P¹(Fq), 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 ρ, write A_ρ=∑s∈ Sₚρ(s) and μ_ρ=tr(A_ρ)/ρ: since the six generators are only a small part of their conjugacy class, A_ρ is generally not central and μ_ρ 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 πA(g)=(κ_ρχ_ρ(g))_ρ∈G^tr, the span RA=span\πA(g):g∈ G\ has dimension rA=rank(M_tr D_κ)≤rank(M_tr)≤|Cl(G)|=O(q)|G|=O(q³), because πA is constant on conjugacy classes regardless of the weights chosen; the middle inequality is strict whenever a selected sector has κ_ρ=0, observed for several sectors at once in every case tested. A finite spanning witness T with |T|=rA therefore exists, but an explicit shell-by-shell exploration of X5,13 needs far more than rA 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 P¹(Fq) saturates its full ambient space within the first two to three graph-distance shells for every tested q∈\13,17,29\, leaving no pre-saturation window from which to extract an exponent. None of these constructions yields a viable mechanism for β^*. A matrix-level redundancy law of the form βeff=1/(1/2+α) is not supported by the constructions examined here: no construction here supplies a regime in which α 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.