Overview
This article develops the third step of the spectral admissibility programme. After identifying locally admissible sectors and globally preferred groups, the question becomes: which generating sets can actually realise these structures under bounded flux?
The paper shows that admissibility requires neutrality constraints on the generating set, expressed through vanishing projection on certain representation sectors. These constraints translate into geometric conditions on the Gram matrix associated with the generators.
The central result is a rigidity phenomenon: once neutrality is imposed, the space of admissible generating sets collapses to a small number of discrete configurations. This eliminates large classes of otherwise combinatorially valid Cayley graphs.
Core contributions
- Neutrality constraints: admissibility imposes vanishing conditions on representation projections of the generating set.
- Gram formulation: these constraints are expressed as algebraic conditions on the Gram matrix of generators.
- Rigidity theorem: admissible configurations form a discrete set rather than a continuous family.
- Structural elimination: many combinatorially valid generating sets are excluded by neutrality.
- Binary compatibility: the surviving configurations align with the binary polyhedral chain identified in previous steps.
Interpretation
Admissibility is not only spectral but geometric. It constrains not just which sectors are allowed, but how the underlying relational structure can be generated.
- Neutrality removes dynamically unstable directions.
- Gram rigidity enforces geometric consistency of generators.
- Discrete solutions replace continuous design freedom.
The result is a strong reduction of structural freedom: admissible substrates are not generic but highly constrained objects.
Relation to the Cosmochrony program
This paper follows spectral capacity by addressing the realisability problem: given a preferred group, which generating sets can implement it under admissibility constraints?
It shows that neutrality imposes rigidity, drastically reducing the admissible configurations. This prepares the next stage of the programme, where discrete stratification and generation structure emerge from these constrained substrates.
References
Jérôme Beau. Spectral Gram Rigidity and Neutral Generator Constraints in SU(2) Subgroups. Preprint, Zenodo.