Quadratic Completion of Admissible Spectral Pairs via Binary-Icosahedral Representation

Current Zenodo release: version 2.1 (2026-09-17). Official title: 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 (conjectural): quotient identification, with depth-independent normalization open
Level III (conditional): 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 scope of diagnostics on supplied or conjectured carriers.

Status of the embedding hypothesis. Hypothesis 4.4, the canonical admissible embedding of the pair vectors into \(V_\rho\), remains open. O27 supplies the conditional adjoint-sector reading under O23's supplied carrier and O27's admissible-saturation hypothesis; it does not furnish the required vector lift. An equivariant adjoint-factorising realisation is excluded, since \(\mathrm{Hom}_{\mathfrak{su}(2)}(\mathbf{3},\mathbf{2}) = 0\); a quotient-factorising linear lift that is not equivariant is not excluded. The final arrow of the implication chain is the conditionally supplied carrier, not a consequence of the three constraints.

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: conditional reading of the candidate sector along the chain \(Q_8 \subset 2I \subset SU(2)\).
Diagnostic scope: Tests 1 and 2 are implementation checks; Test 1 holds by construction for arbitrary vectors. Test 4 on existing data measures a covariance in \(\mathrm{End}(H_{\mathrm{eff}})\), not in \(\mathrm{End}(V_\rho)\). The embedded decision table requires the open embedding; existing O28/O29 measurements occupy no row of that table.
Symmetric-square bound (Proposition 4.7): under Hypothesis 4.4(a), with conjugation in a fixed basis, the Level III products are \(uu^{\mathsf T}\). Their span has dimension at most \(d(d+1)/2\), hence at most 3 for \(d=2\). The rank-four target is excluded under this contract; other embeddings remain open.

Interpretation

O26 proposes a conditional interpretation of the pair observable as a canonical quadratic object. This reading requires the conjectural depth-independent proportionality of Level II and the admissible embedding of Level III; neither is established.

Proved: pre-saturation growth-exponent equivalence with a Hilbert–Schmidt norm
Conditional prospect: identification with a canonical norm under those additional premises

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: construct the missing embedding before measuring its covariance in \(\mathrm{End}(V_\rho)\). Existing ranks are measured in \(\mathrm{End}(H_{\mathrm{eff}})\): three is modal, with seed-sensitive exceptions of 4–9 at \(q=101\) and 5 at \(q=211\). They do not identify \(V_\rho\); \(d_\rho=2\) is a minimality selection.
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.