Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$

For rational-valued signals $f,g:C_{30}\to\mathbb Q$ with exact Fourier support $U_{30}$ , equality of autocorrelations through order five is completely classified. After translating $g$ , there are $\alpha\in\mathbb Q(\zeta_{30})^\times$ and $z\in\mathbb Q(\zeta_6)^\times$ , with $z\bar z=1$ , such that $\widehat f(u)=\sigma_u(\alpha),\qquad \widehat g(u)=\sigma_u(z\alpha)$ for every $u\in U_{30}$ . Conversely, every such pair, extended by zero off $U_{30}$ , is rational-valued and agrees through order five. Normalized parameters are translation-equivalent exactly modulo $\mu_6$ , and the sixth-order data agree exactly when $z^6=1$ .

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Finite harmonic analysis / phase retrieval on cyclic groups
Posed by: Aaron Agulnick, Toby Busick-Warner
Year posed: 2026
Years open: 0y
Solved: 2026-08-07
Model: GPT-5.6 Sol (Codex)
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Significance: 4 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support $U_{30}$ and conjectured that it was not a complete classification because it did not use the full field $\mathbb Q(\zeta_{30})$ . This work proves the complete classification. The larger field enlarges the common amplitude $\alpha$ , while every relative ambiguity remains a norm-one parameter in $\mathbb Q(\zeta_6)$ . The result is a specialization of a theorem for every exact unit support $U_{6m}$ .

The entry does not claim a complete parametrization for arbitrary supports: on the 255 automorphism-stable supports treated elsewhere in the paper, the broader result is a closing-degree classification. It does not treat noisy data or noncyclic groups, and it makes no novelty, priority, or firstness claim.

What the AI did

Following the self created protocol PAPP, that you can find here:https://github.com/aconsciousfractal/Gate-Disciplined-Computational-Mathematics
GPT-5.6 Sol, operating through Codex under my direction, developed the central relation-lattice reduction, the phase-ratio normal form, and the Galois-descent argument that confines the relative phase to the norm-one torus in $\mathbb Q(\zeta_6)$ . It also generated exact symbolic replays and the verification package. I selected the problem and scope, controlled the literature and claim boundary, directed repeated adversarial reviews, checked the mathematical outputs, and revised the manuscript after each finding. An earlier AI-agent package supplied preliminary computational observations; the workflow independently rederived and checked them before use.

Verification

No named independent domain expert has endorsed the theorem. The repository carries exact symbolic checks, fail-closed verification scripts, frozen manifests and an adversarial review report, but those are author-side and agent-side assurance, so this stays Unreviewed and Candidate.

This site ran its own checks, written from the statement rather than from the repository's scripts. Building a pair from a chosen $\alpha$ and $z$ : both inverse transforms are rational at all 30 points, the support is exactly $U_{30}$ , the pair agrees through order five and differs at order six, and replacing $z$ by a sixth root of unity restores order-six agreement - the claimed $z^6=1$ boundary, exactly. The phase lattice was recomputed independently by Smith normal form: $\mathbb Z^8/L$ has free rank 1 through order five and rank 0 at order six, so a one-parameter ambiguity survives order five and dies at six. And the Agulnick-Busick-Warner pair itself fits the classification - its Fourier ratio is Galois-equivariant with $z+\bar z=13/7$ and $z\bar z=1$ , so $z=(13\pm3\sqrt{-3})/14$ lies in $\mathbb Q(\zeta_6)$ .

Not checked: completeness itself, which is the novelty. The converse direction, the lattice skeleton and the known example are all consistent with it without establishing it.

Sources

Submitted by VividMarten473 on 17 Aug 2026

Complete rational classification of fifth-order autocorrelation ambiguities on U30U_{30}U30​

What was actually shown

What the AI did

Verification

Sources

Discussion

Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$