This repository contains the source of the O26 Cosmochrony paper
Quadratic Completion in Admissible Spectral Pairs: σpair as Pullback of a Hermitian Form on the Binary-Icosahedral Thread.

This work extends the spectral admissibility sub-programme by providing the first representation-theoretic interpretation of the pair observable introduced and numerically validated in O16–O25.

It addresses the next structural question after O25:

Is the pair observable σpair an intrinsic quadratic object, and does it arise from a canonical Hermitian structure?

Quick Summary

σpair is not merely a product of two observables.

It behaves as the pullback of a Hermitian quadratic form.

More precisely:

• its growth matches a Hilbert–Schmidt norm (proved)
• its structure is consistent with a rank-one matrix construction
• it may arise from a canonical representation-theoretic sector (conjectural)

This turns σpair from a constructed observable into a candidate intrinsic norm.

Context

O16–O25 established that:

• the correct observable is the canonical pair quantity
  $\sigma_{\mathrm{pair}}^{\mathrm{can}}(n)$
• the exponent $\delta_{\mathrm{pair}} \approx 7.44$ lies in the admissible window
  $[7.4, 10.6]$
• the transfer chain
  $c_{\mathrm{BI}} \to \delta_{\mathrm{pair}} \to \beta^*$
  holds unconditionally (O24)
• δpair is numerically stable and structurally invariant (O25)

However:

• σpair remained a bilinear construction
• no intrinsic quadratic interpretation was available
• no representation-theoretic embedding was identified

This defines the scope of O26.

Core Result

The paper establishes that:

The pair observable σpair behaves as the pullback of a Hermitian quadratic form on a representation space, with a precise dictionary linking Weil blocks to rank-one matrix coefficients.

Three levels of identification are introduced:

• Level I (proved): exponent equivalence
• Level II (structural): identification modulo normalisation
• Level III (conjectural): canonical representation-theoretic realisation

Main Structural Results

1. Dictionary between Weil blocks and matrix coefficients

Result. Each conjugate pair {c, q−c} defines a rank-one operator:

$$ \widetilde{M}_n = v_c^{(n)} \otimes v_{q-c}^{(n)} $$

with:

$$ |\widetilde{M}_n|_{\mathrm{HS}}^2 \sim \sigma_{\mathrm{pair}}(n) $$

Thus:

• σpair corresponds to a Hilbert–Schmidt norm
• the observable is naturally quadratic

2. Growth equivalence (Level I)

Result. The exponent satisfies:

$$ \delta_{\mathrm{pair}} = \delta_{\mathrm{HS}} $$

Thus:

• σpair and the matrix norm share identical scaling
• the identification is rigorously established

3. Quotient identification (Level II)

Result. Up to normalisation:

$$ \sigma_{\mathrm{pair}}(n) \approx |\widetilde{M}_n|_{\mathrm{HS}}^2 $$

Thus:

• σpair is the pullback of a quadratic form
• remaining freedom is purely pipeline normalisation

4. Canonical identification (Level III)

Conjecture. There exists an irrep ρ of 2I such that:

• $v_c^{(n)} \in V_\rho$
• $\widetilde{M}n \in \mathrm{End}(V\rho)$
• σpair equals the canonical norm up to universal factors

Candidate:

• spin-$\tfrac{1}{2}$ sector (dimension 2)
• $\dim \mathrm{End}(V_\rho) = 4$

5. Falsifiability via effective dimension

Result. The conjecture is testable via:

• covariance rank of $\widetilde{M}_n$
• universality across pairs
• exponent stability

Thus:

• the representation dimension can be directly measured
• the conjecture is fully falsifiable

Foundational Chain from the Substrate

The derivation is fully internal:

Born–Infeld admissibility
$\to$ pair observable (O16–O21)
$\to$ projection locking (O22)
$\to$ quaternionic structure (O23)
$\to$ rank stability (O24)
$\to$ numerical validation (O25)
$\to$ quadratic completion (O26)

No external structure is imposed.

Mathematical Role of O26

O26 provides the structural completion of the observable:

• identifies σpair as a quadratic object
• links it to Hilbert–Schmidt geometry
• introduces a representation-theoretic embedding
• formulates a hierarchy of identification levels
• provides falsifiability criteria

More precisely, the paper:

• constructs a dictionary between Weil blocks and matrix coefficients
• proves exponent equivalence (Level I)
• isolates normalisation freedom (Level II)
• proposes a canonical embedding (Level III)
• defines tests for representation selection

Epistemic Structure of the Paper

Established input

• pair observable (O16–O21)
• fibre structure and normalisation (O17–O19)
• projection locking (O22)
• quaternionic admissibility (O23)
• rank stability (O24)
• numerical validation (O25)

New results

• matrix dictionary construction
• Hilbert–Schmidt identification
• Level I proof
• Level II structural equivalence
• Level III conjecture
• falsifiability framework

Remaining open problems

• validation of Level III
• identification of the correct irrep
• embedding of $v_c^{(n)}$ into $V_\rho$
• large-q validation of dimension
• analytical derivation of the sector

Interpretation of the Result

The conceptual shift is:

• previous view: σpair is a product
• O26: σpair is a quadratic norm

Thus:

• the observable is intrinsic, not constructed
• quadratic structure emerges from projection
• representation theory becomes physically relevant

Structural Role of O26

O26 completes the observable hierarchy:

• O16: pair observable
• O17–O19: fibre structure
• O20–O21: persistence
• O22: projection locking
• O23: quaternionic structure
• O24: rank stability
• O25: numerical validation
• O26: quadratic completion

Thus:

• the observable is identified
• its scaling is validated
• its structure is interpreted

What O26 Adds

• quadratic interpretation of σpair
• dictionary with matrix coefficients
• Hilbert–Schmidt framework
• representation-theoretic embedding
• falsifiability of the sector
• bridge between spectral data and representation theory

Outcome

The spectral admissibility framework is now:

• structurally grounded (O24)
• numerically validated (O25)
• geometrically interpreted (O26)

The observable is:

• stable
• intrinsic
• quadratic
• potentially representation-theoretic

Residual Open Problems

Representation selection

Identify the correct irrep ρ of 2I.

Level III validation

Test whether $r_{\mathrm{eff}} = d_\rho^2$.

Embedding

Construct the explicit map:

$$ v_c^{(n)} \mapsto V_\rho $$

Large-q regime

Verify stability of dimension at larger primes.

Analytical derivation

Derive the representation structure from admissibility.

Status

The programme is now:

• structurally closed (O24)
• numerically validated (O25)
• geometrically lifted (O26)
• ready for representation-theoretic resolution

Repository Structure

paper/
├── out/      # Compiled O26 PDF
├── tex/      # LaTeX sources
└── README.md

Citation

If you reference this work, please cite:

J. Beau Quadratic Completion in Admissible Spectral Pairs: σpair as Pullback of a Hermitian Form on the Binary-Icosahedral Thread Zenodo, 2026.

Acknowledgements

Portions of the derivations, conceptual synthesis, structural organisation, and editorial refinement benefited from iterative interactions with large language models used as analytical assistants.

All theoretical results, computations, and interpretations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference.

Critical feedback, independent verification, and further analysis of:

• pair observables
• quadratic structures
• representation selection
• admissible sectors
• Hilbert–Schmidt dynamics

are welcome.

Please open an issue to discuss conceptual points, technical details, or possible extensions.