Overview
The companion note O4 audits a candidate route to a structural bound on the cascade exponent $\beta$ — bounded flux translated into a front-size estimate via the Cheeger inequality — and finds that route establishes no such bound: the argument contains three independent defects. O5 imports nothing from O4 and tests a different, representation-theoretic route on the same LPS relaxation cascade.
The group generated by the LPS quaternion matrices is verified to be exactly $\mathrm{PSL}(2,\mathbb F_q)$ or $\mathrm{PGL}(2,\mathbb F_q)$ (both occur among the tested $q$), with a symmetric Cayley graph and a character table computed by Dixon's algorithm and verified via Burnside's identity, rather than transcribed from the classical classification.
The paper's central caution is that the natural sector weight $\bar\mu_\rho$ — a trace average over the six LPS generators — is not an eigenvalue of the corresponding representation block, since those six generators are only a small part of their conjugacy class. For $q=13$, the graph's true adjacency spectrum has 55 distinct eigenvalues against only 9 distinct trace averages. Every construction in the paper is therefore named and treated as a character-trace object, never identified with a proven spectral eigenvalue.
Core contributions
- Verified group and graph construction: group elements are canonicalized by their induced permutation of $\mathbb{P}^1(\mathbb F_q)$ rather than an ad hoc matrix-scalar quotient, reproducing the exact closed-form group order, a symmetric Cayley graph, and shell growth matching the free tree before any collision.
- Class-function saturation theorem: for any weighted class-function encoding $\pi_A(g)=(\kappa_\rho\chi_\rho(g))_{\rho}$, the span $\mathcal R_A=\mathrm{span}\{\pi_A(g):g\in G\}$ has dimension $r_A=\mathrm{rank}(M_{\mathrm{tr}}D_\kappa)\le\mathrm{rank}(M_{\mathrm{tr}})\le|\mathrm{Cl}(G)|=O(q)$, far below $|G|=O(q^3)$, because $\pi_A$ is constant on conjugacy classes regardless of the weights chosen. A finite spanning witness exists, but an explicit graph exploration needs far more vertices to find one.
- Negative result for character-based transition fingerprints: they collapse to a fixed, low-dimensional span well short of the trace-selected ambient dimension.
- Negative result for fixed-dimensional matrix proxies: they saturate their own small ambient dimension within a handful of shells, independent of $q$.
- Steinberg-based fingerprint: the one construction with genuinely $q$-structural ambient growth, but it saturates its full ambient rank within the first two to three graph-distance shells for every tested $q$, leaving no pre-saturation window from which an exponent could be extracted.
Interpretation
O5 is an obstruction result, not a construction of the genuine spectral frontier. It excludes finite class-function (character-trace) encodings of the vertex boundary as a source of the rich, mode-resolved novelty a viable cascade mechanism would need — without constructing or selecting that mode-resolved, eigenvalue-level frontier itself.
- Vertex-based class functions are provably too rigid: the class-function saturation bound confines them to an $O(q)$-dimensional abstract span, whatever the weights.
- Character-based transition fingerprints plateau well short of their own ambient dimension.
- Fixed-dimensional matrix proxies saturate a small ambient space too quickly to be $q$-structural.
- The Steinberg fingerprint, the one $q$-structural construction, saturates so early that no pre-saturation window survives.
A matrix-level redundancy law of the form $\beta_{\mathrm{eff}}=1/(1/2+\alpha)$ is not supported by the constructions examined here either: no construction tested supplies a regime in which its exponent could 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.
A genuine mode-resolved, eigenvalue-level frontier — built from the true representation blocks $A_\rho=\sum_{s}\rho(s)$ rather than their trace average — is a distinct and harder question, left open for a future paper under its own contract.
Relation to the Cosmochrony program
O5 follows the open problem left by O3's phenomenological window $\beta^*\in(0.09,0.13)$ and by O4's finding that its audited candidate route to a structural bound does not succeed. Spectral admissibility selects the relevant sectors, spectral capacity and Gram rigidity constrain the admissible binary group structure, spectral stratigraphy fixes the three-level ADE organisation, and O1 and O3 restore the ordering and amplify the hierarchy through valence growth.
O5 does not close the cascade-exponent problem. It rules out finite class-function encodings of the vertex boundary as the source of the missing mechanism, verified against a corrected and independently reviewed group/character-table foundation, and narrows the search toward a genuinely mode-resolved, eigenvalue-level construction — opening naturally toward O6.
References
Jérôme Beau. Finite Character-Trace Saturation and the Limits of Vertex-Based Class-Function Cascades. Preprint.