Abstract
We study symmetric conference matrices of order \(334\) whose \(333\)-vertex core admits a semiregular cyclic group of order \(37\), with nine vertex orbits.
The zero-frequency equations have \(196{,}560{,}000\) labelled orbit-sum matrices, \(625\) classes under simultaneous permutation, and \(314\) after also identifying global sign. Their reductions have two parity types up to complement, and every integral lift obeys a universal \(6/3\) quadratic-residue incidence law on the diagonal circulant blocks.
Both parity types have explicit exact-margin binary supports satisfying every graph equation modulo two, but neither retained support satisfies the next modulo-four adjacency layer. Constant generators through rank three and two complete natural families at the first nonconstant rank-two layer are exactly excluded.
Main results
Complete quotient layer. The integral zero-frequency quotient system has exactly \(625\) permutation classes and \(314\) sign-permutation classes.
Universal diagonal law. No quotient has zero diagonal. Fourier inversion forces each diagonal circulant block into the same \(6/3\) quadratic-residue incidence pattern.
Modular realizations. Exact-margin binary supports for both complement classes satisfy the full graph equation modulo two, equivalently the conference-core equation modulo eight.
Formal obstructions. All constant symmetric generators of ranks at most three are excluded across all \(625\) quotients, together with two explicitly defined first-nonconstant rank-two families.
In characteristic two, the orientation-fixed relaxation is a rank-four Hermitian projection in a nine-dimensional unitary space and has between \(2^{719}\) and \(2^{720}\) points. Characteristic three and characteristic \(37\) expose further formal flexibility, but physical binary realizability remains the central obstruction.
Claim-to-evidence map
| Claim | Evidence |
|---|---|
| 625 quotient classes | Canonical machine-readable census, digest, and orderly-enumeration replay |
| Parity types and diagonal law | Exact quotient verifier and independent symbolic identities |
| Characteristic-two supports | Explicit block supports checked against every margin and convolution equation |
| Rank-two conjugation obstruction | Exact finite-field verifier |
| Rank-two Jordan obstruction | Separate exact verifier |
| First nonconstant families | Normal-form and exceptional-plane verifiers |
Novelty and scope
Conference doubling, orbit matrices, finite-field unitary counts, and multicirculant methods are established tools. The candidate contributions are the complete \(625\)-class quotient census at these parameters, its universal diagonal consequence, the exact modular supports, and the specified low-rank obstructions.
The semiregular hypothesis is a genuine restriction. The paper does not classify arbitrary conference matrices of order \(334\), and a failure of the tested low-rank or switching families is not evidence of global nonexistence. The priority audit is incomplete and cannot establish worldwide novelty.
AI-assistance and verification disclosure
The mathematical exploration, program development, certificate analysis, manuscript drafting, and publication materials used heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is a non-specialist and cannot independently certify every derivation or implementation. Exact artifacts improve auditability; they are not peer review or proof-assistant formalization.
Suggested citation
Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol, “Semiregular \(C_{37}\) conference lifts at order \(334\): quotient classification, modular realizations, and low-rank obstructions,” provisional research paper, 25 July 2026. Permanent page.