Quadratic Completion of Admissible Spectral Pairs via Binary-Icosahedral Representation

Overview

Following the numerical consolidation achieved in O25, which established the robustness and normalization structure of \(\delta_{\mathrm{pair}}\), O26 addresses the next structural question: what is the intrinsic mathematical nature of the pair observable itself?

The central aim of O26 is: to identify \(\sigma_{\mathrm{pair}}(n)\) as a canonical quadratic object in a representation space, and to connect the admissible cascade to a specific representation-theoretic sector.

The paper constructs a dictionary between conjugate Weil blocks and matrix coefficients in an isotypic representation space, and shows that in the pre-saturation regime the observable satisfies a growth equivalence with a Hilbert–Schmidt norm trajectory.

This leads to a hierarchy of identifications:

Level I (proved): growth equivalence
Level II: quotient identification modulo normalization
Level III: canonical identification in a binary-icosahedral sector

Scope statement. This page summarizes the structural contribution of O26: the representation-theoretic dictionary, the three-level identification hierarchy, and the falsifiability framework linking spectral admissibility to SU(2)-type sectors.

Main contributions

Representation-theoretic dictionary: explicit mapping between conjugate Weil blocks and isotypic sectors of the binary icosahedral group.
Hilbert–Schmidt identification (Level I): \(\sigma_{\mathrm{pair}}(n)\) shares its growth exponent with a matrix trajectory norm.
Three-level hierarchy: progressive strengthening from growth equivalence to canonical identification.
Admissible SU(2) sector: identification of the candidate sector along the chain \(Q_8 \subset 2I \subset SU(2)\).
Falsifiability framework: five concrete tests, including effective dimension and universality across pairs.

Interpretation

O26 shifts the interpretation of the pair observable from a bilinear construction to a canonical quadratic object.

Before O26: \(\sigma_{\mathrm{pair}}\) is a constructed observable
After O26: it behaves as a restriction of a canonical norm

This suggests that the admissible cascade exponent \(\beta^*\) may admit a representation-theoretic interpretation, rather than being purely phenomenological.

Relation to the Cosmochrony programme

O26 follows O25 by moving from numerical structure to intrinsic interpretation.

The sequence now reads: O16–O19 (pair construction and normalization), O20–O23 (persistence, saturation, shell locking, threshold), O24 (rank stability), O25 (numerical campaign), O26 (representation-theoretic identification).

It provides the first candidate for embedding admissible dynamics into a canonical representation framework.

Current result and open directions

Level III validation: test the admissible embedding hypothesis \(\Phi_{q,\rho}\).
Effective dimension: measure trajectory dimension in \(\mathrm{End}(V_\rho)\).
Universality: verify consistency across primes and pairs.
Analytical link: derive \(\beta^*\) from representation structure.

Reference

Jérôme Beau. Quadratic Completion of Admissible Spectral Pairs via Binary-Icosahedral Representation.