Overview
This article develops the first explicit step of the spectral admissibility programme. Its central question is whether a local bounded-flux constraint on a relational substrate can select preferred spectral sectors without introducing any additional dynamical principle.
The answer is positive. On Cayley graphs built from SU(2)-type finite subgroups, the local constraint $|\partial_t \chi_v| \le c_\chi$ induces an admissibility envelope $A_n^{\max} = c_\chi/\sqrt{\lambda_n}$, so sectors with smaller Laplacian eigenvalue acquire larger admissible effective amplitudes.
The paper establishes this mechanism explicitly on the quaternion group $Q_8$, then extends it to the binary icosahedral group $2I$, and finally to the Lubotzky--Phillips--Sarnak graph family approaching the SU(2) continuum. At every finite stage, the most admissible sectors are the low-spin non-abelian ones.
Core contributions
- Admissibility envelope: the paper identifies the effective amplitude $A_n = a_n \lVert \psi_n \rVert_\infty$ as the relevant local variable and derives the universal bound $A_n^{\max} = c_\chi/\sqrt{\lambda_n}$.
- Q8 result: on the Cayley graph $\Gamma(Q_8,\{\pm i,\pm j,\pm k\})$, the quaternionic non-abelian sector carries the smallest non-trivial eigenvalue and therefore the widest admissibility window.
- Non-linear robustness: the weakly non-linear and exact one-mode DBI analyses on $Q_8$ confirm that the same spectral envelope remains the true saturation boundary.
- 2I hierarchy: on the binary icosahedral group, the admissibility structure resolves into an explicit low-spin hierarchy, sharpening the separation already visible on $Q_8$.
- LPS extension: the mechanism extends to the Ramanujan LPS family, where the spin-$1/2$ sector remains the most admissible for every finite $p$ examined.
Interpretation
The article shows that bounded relational capacity does not merely impose a cut-off. It acts as a genuine spectral selector.
- Local bounded flux translates into a mode-by-mode admissibility geometry in the $(\lambda_n, A_n)$ plane.
- Spectral position determines which sectors survive for the largest range of effective amplitudes.
- Non-abelian low-spin sectors are favoured because the generating sets have stronger character coherence with them than with higher-spin sectors.
Within this perspective, spectral admissibility is the first structural filter of the whole programme: before any hierarchy of generations or masses can emerge, bounded flux already selects which sectors remain dynamically resilient.
Relation to the Cosmochrony program
This paper is the first concrete realisation of the spectral admissibility programme. It makes explicit how bounded flux acts on the substrate spectrum and establishes that non-abelian low-spin sectors are structurally preferred at finite resolution.
Later papers build on this result. Spectral capacity aggregates these admissibility windows at the level of whole groups, spectral Gram rigidity classifies the neutral-generator structures compatible with the mechanism, and spectral stratigraphy uses discrete spectral levels to construct generation structure. This first paper provides the selection principle on which the rest of the chain depends.