Approximate Antiunitary Symmetry as a Matching Problem

GPT-6 Astra, an AI model made by OpenAI, wrote this paper at the direction of David Ross. Its title block is dated September 28, 2026; it was revised through October 2 under the four reviews below, with the review edits made by Claude Fable 5.1 (Anthropic), and published here on October 2, 2026. As of October 3, 2026, it has not been peer reviewed, and no human mathematician has read it.

Read the 11-page paper (PDF)

What the paper settles, and what in it is earlier work

The paper determines exactly the least combined error in antiunitary commutation with commuting Hermitian matrices and in the relation T^2 = -I: a weighted matching value of their joint eigenvalues. Its stated contribution is this reduction, "together with the resulting certificates and stability statements."

It claims none of its ingredients as new: Wigner's normal form (1960) and Loring's constructive treatment (2026), Higham's matrix nearness (1989), Kramers' theorem (1930), Mirsky's inequality (1960), the parity facts, and Edmonds' matching polytope (1965), the proof's one substantial outside result; nor the single-observable case, which follows from Wigner, Kramers and Mirsky and reduces to Proposition 4.1. Its substance is the general cost array, hence commuting tuples of two or more observables, carried by Lemma 2.2.

On priority, the writing session's search did not locate identity (3) of Theorem 2.1 in the sources examined, which "supports presenting it as the contribution of this manuscript, while not establishing an exhaustive priority claim." Two reviews and two later searches found no statement of it, and four adjacent works: Gil (Entropy 28(8):877, 2026) maximizes coherence for prescribed intrinsic populations over an aligned class, a restriction Lemma 2.2 removes, as the companion note on Gil's questions develops; Looi (arXiv 2609.37133) proves stability of Wigner symmetries for maps on states; Miyazaki, Kuroiwa and Murao (arXiv 2609.36408) quantify time-reversal violation of quantum channels against a fixed antiunitary; and older work on congruence orbits with prescribed singular values, such as Tam (1998) and Leite, Richa and Tomei (1999), concerns selected entries of one orbit.

The setting

An antiunitary operator on complex n-space is T = U C, with C coordinatewise conjugation and U unitary, so T v = U conj(v) and T^2 = U conj(U). A Hermitian H commutes with T exactly when H U = U conj(H); the conjugate in the commutation error H U - U conj(H) is load-bearing: without it the formula fails for complex matrices. Exact commutation with T^2 = -I forces even multiplicities (Kramers pairing).

For a tuple H = (H_1, ..., H_d) of Hermitian n-by-n matrices and a penalty weight τ ≥ 0, the paper defines

Φ_τ(H) = min over unitary U of [ Σ_r ||H_r U - U conj(H_r)||_F^2 + τ ||U conj(U) + I||_F^2 ],

with unnormalized Frobenius norms, fixed observable scales, and no assumption that the competing antiunitaries square to a scalar.

A matching M of {1, ..., n} is a set of disjoint pairs, possibly empty, with |M| pairs. By Edmonds' theorem, the matching polytope P_n, the convex hull of the matchings' indicator vectors, is the set of nonnegative x on pairs summing to at most 1 over the pairs at each index and to at most (|A| - 1)/2 over the pairs inside each odd set A. A contraction has operator norm at most 1; skew-symmetric means K^T = -K (plain transpose, complex entries allowed). A unitary U has symmetric part S = (U + U^T)/2 and skew part K = (U - U^T)/2, a skew-symmetric contraction.

A cost array c is real symmetric with zero diagonal and nonnegative entries; E_{c,τ}(U) = Σ_{i,j} c_ij |U_ij|^2 + τ ||U conj(U) + I||_F^2. For commuting H_1, ..., H_d, a common eigenvector's d eigenvalues form its joint eigenvalue vector; the n vectors λ_1, ..., λ_n are repeated with joint multiplicity. The benefits b_ij = 8τ - 2 c_ij make the least energy 4τn minus the largest matching benefit, so the minimum is a maximum-weight matching problem.

The results, as the paper states them

Theorem 2.1 (exact matching reduction). For every such c, every τ ≥ 0 and every n ≥ 1, the least E_{c,τ}(U) over unitary U is

min over matchings M of [ 2 Σ_{{i,j}∈M} c_ij + 4τ (n - 2|M|) ], (3)

attained by the real orthogonal matrix with the block [[0, 1], [-1, 0]] on each pair of a minimizing M and 1 on each unmatched index, so complex unitaries do no better. In the paper's words, "a continuous matrix cannot gain an advantage by distributing incompatible pairing weights around an odd cycle."

Lemma 2.2 (squared-entry convex hull). If K^T = -K and ||K||_op ≤ 1, the vector x with x_ij = |K_ij|^2 (i < j) lies in P_n, and the convex hull of all such vectors is P_n. This is a convex-hull identity; it does not assert that every point of P_n is realized by one contraction.

The proof uses ||U conj(U) + I||_F^2 = 4 ||S||_F^2 (6) to write the exact decomposition of the objective, with x_ij = |K_ij|^2,

E_{c,τ}(U) = 4τn + 2 Σ_{i<j} (c_ij - 4τ) x_ij + 2 Σ_{i<j} c_ij |S_ij|^2, (7)

drops the nonnegative last term, and minimizes the affine part over P_n, which contains x by Lemma 2.2, at a matching whose signed-swap matrix attains it. The odd-set inequalities hold because an odd-order skew-symmetric block K_A is singular (det K_A = det K_A^T = det(-K_A) = -det K_A), so, as a contraction, ||K_A||_F^2 ≤ rank K_A ≤ |A| - 1. Remark 2.3 shows they are needed: at n = 3, c = 0, τ = 1, x_ij = 1/2 on all three pairs meets every degree inequality and would give energy 0, not the true minimum 4.

Corollary 3.1 (commuting observables). If H_1, ..., H_d commute, with joint eigenvalue vectors λ_1, ..., λ_n counted with multiplicity, then

Φ_τ(H) = min over matchings M of [ 2 Σ_{{i,j}∈M} ||λ_i - λ_j||^2 + 4τ (n - 2|M|) ]. (8)

The proof diagonalizes H_r = V D_r V^* and substitutes the congruence U = V W V^T, not a similarity. The multiplicity clause matters: for the 2-by-2 zero matrix, read over its one distinct eigenvalue, the formula gives 8τ, not the true 0 (the final review's example).

Proposition 3.2 (exact and near-minimizer structure). If the minimizing matching M* is unique, every minimizer consists, in the given coordinates, of the signed-swap blocks of M* and of diagonal entries on unmatched indices, each times an arbitrary unit scalar. For n ≥ 2, with Δ > 0 the gap between the second-best and best matching costs and ε a unitary's excess energy, ||x - 1_{M*}||_1 ≤ nε/Δ, and, if the least off-diagonal cost c_min is positive, ||S - diag(S)||_F^2 ≤ ε/c_min (9). Uniqueness is needed here (at n = 3, c = 0, τ = 1 some minimizers are not signed swaps), but the theorem assumes no uniqueness or simple spectrum.

Corollary 3.3 (parity and exact commutation). The least ||U conj(U) + I||_F^2 over unitaries is 4 (n mod 2) (10); for odd n, ||U conj(U) + I||_op = 2 for every unitary (11); under exact commutation with a commuting tuple, the least ||U conj(U) + I||_F^2 is 4 times its number of odd-dimensional joint eigenspaces (12). The paper calls these classical, "included to calibrate residuals, not as priority claims", and a determinant-only bound such as 2 sin(π/(2n)) "not the sharp operator-norm answer."

Corollary 3.4 (the exact negative-square constraint). For even n and every real symmetric c with zero diagonal, with no sign condition on the costs, the least Σ c_ij |U_ij|^2 over unitaries with U conj(U) = -I is twice the least cost of a perfect matching (13).

Proposition 3.5 (a Lipschitz bound). For Hermitian tuples H and A of the same length and dimension, the same τ ≥ 0, and η = (Σ_r ||H_r - A_r||_F^2)^{1/2}, |sqrt(Φ_τ(H)) - sqrt(Φ_τ(A))| ≤ 2η (14); if A commutes with matching value p, then (max{0, sqrt(p) - 2η})^2 ≤ Φ_τ(H) ≤ (sqrt(p) + 2η)^2.

Section 4.1 (a polynomial-size optimization problem). Edmonds' algorithm solves the matching problem in polynomial time. The matching dual gives exact certificates, checkable in rational arithmetic for rational c and τ; z_{1,2,3} = 8 and y = 0 certify the three-index minimum 4. In τ the optimum is continuous, nondecreasing, concave and piecewise affine; the number of matched pairs never decreases as τ grows, though the pairs can change.

Proposition 4.1 (sorted-spectrum recurrence). For one real spectrum λ_1 ≤ ... ≤ λ_n and c_ij = (λ_i - λ_j)^2, an optimal matching can use only adjacent pairs, so the minimum F_k over the first k values obeys F_k = min{F_{k-1} + 4τ, F_{k-2} + 2 (λ_k - λ_{k-1})^2}, F_0 = 0, F_1 = 4τ (17), computable in O(n) operations once sorted; for even n the exact negative-square minimum is 2 Σ_j (λ_{2j} - λ_{2j-1})^2 (18). For (0, 2, 3, 5), Φ_τ = min{16τ, 2 + 8τ, 16} (19): nothing is paired below τ = 1/4, then 2 with 3, and above 7/4, 0 with 2 and 3 with 5, a rearrangement "independent gap thresholding would miss."

What is left open

The paper leaves open an exact formula for noncommuting tuples: Proposition 3.5 gives a bound only if a commuting approximant is supplied, which the paper does not construct. It does not solve arbitrary unitary optimization or assert the reduction for other penalties. The reviews left further questions: the exact value with a fixed defect budget (for (0, 2, 3, 5) and ||U conj(U) + I||_F^2 ≤ 4, the first review showed 9 < value ≤ 12), and the four open questions the second reviewer's note lists.

Where the question came from

Its motivation is five notebook entries of late September 2026: two entries the judge wrote on small-model lines (one a recall of the classical fact that a skew-symmetric matrix of odd order has determinant zero, one a false conjecture) and three results of programs. The paper calls them motivation, not premises; its main theorem is a reformulation the writing session made; the record shows no source for its key tool, the matching polytope.

The notebook is that of Hypnos, the research harness described above (how it works); the judge, Claude Opus 5 (Anthropic), writes each small-model line it keeps into the notebook as an entry. The harness had been screening hypothetical operators, whose spectrum would be the imaginary parts of the Riemann zeta zeros, for time-reversal symmetry; the paper says no claim about the Riemann hypothesis, zeta-zero simplicity or infinite-dimensional operators follows. The chain, dated:

• August 22, 2026: a small model conjectured that a random Hermitian matrix with those eigenvalues has no commuting antiunitary with square +I (a small model's line); the entry on it says the conjecture is false for every finite Hermitian matrix, since an eigenbasis yields one (the judge's entry, Claude Opus 5).
• September 27: a small model recalled that an odd-order skew-symmetric matrix has determinant zero (a small model's line); the entry identified the unitaries with U conj(U) = -I as the skew-symmetric ones and left open how U conj(U) + I and U + U^T compare (the judge's entry); equation (6) settles it.
• September 28: a small model conjectured that the least ||U conj(U) + I||_op over 3-by-3 unitaries is exactly 1 (a small model's line); the entry derived the floor 2 sin(π/(2n)) for odd n, called the sharpness refutable, and asked for the true value (the judge's entry). Equation (11) gives 2.
• September 27 and 28: three programs found the least Frobenius norms of U conj(U) + I and (U + U^T)/2 to be 2 and 1 in dimensions 3 and 5, and built a commuting antiunitary with square +I from an eigenbasis for 140 random matrices (programs by the judge's model on judge-written entries, graded by code).

On September 28 David Ross asked GPT-6 Astra (OpenAI) to read the harness's recent work and prove something from it. The session saved a read-only copy of 17 entries and 3 program results and that night wrote the paper, replacing, as the paper says the observations suggested, an existence screen by an optimization with spectral and square errors, and bringing in the matching polytope (GPT-6 Astra, in the writing session). On September 29, handed the September 28 entry's own pairs, the small models went the wrong way, predicting a value below 2 (small models' lines). The harness supplied the question's material and three measurements; the theorem is the session's.

How the paper was made

The writing session ran in OpenAI's Codex agent, the command-line program through which GPT-6 Astra runs. As the paper's attribution states, the model chose the problem from the harness's work, developed the derivation, wrote and ran the verification programs, searched the literature and drafted the text; David Ross set the task, ran the process and takes responsibility for the manuscript. He read every page only as a check for red flags in the account of the harness and the process, not as a review, and did not check the proofs.

Claude Fable 5.1 (Anthropic) made the review edits: eight on September 29, each "a review-driven change of exposition or hypothesis, not new mathematics" in the written answers' words; the paragraph on Looi's paper on October 1; and the October 2 items. So the published text contains writing by the other company's model, which also wrote one of the reviews.

How it was reviewed

Its four reviews are each published with the written answer to each finding. The October reviews graded findings blocking, major or minor; a blocking finding is one the reviewer judged would invalidate a statement or a proof as written.

September 28, 2026: GPT-6 Astra, a fresh instance of the writer's own model. It asked for two presentation changes and no repair to the mathematics, and found that the paper had missed Gil's work. It also proved that Proposition 3.2 needs only a unique optimal matching, and proved the unrestricted maximum of coherence in Gil's setting, which the companion note later developed and credits to it. Report.

September 29: Claude Fable 5.1 (Anthropic), the other company's model, in a session that had not read the first review. It found no statement that fails, but two hypotheses stronger than needed, one typo, and a framing to correct: the single-observable case is classical. Its own results, in a 7-page note, include a quantitative Kramers inequality and two theorems on Gil's questions; one, for prescribed Youla values, the companion note later found to follow from Horn's 1950 inequality (by a derivation from GPT-6 Astra's review of that note) and to be the Frobenius case of Mathias's 1992 inequality as its zbMATH review states it (found by a Claude Opus 5.5 literature search on October 2; the paper itself was not obtained), and the note claims no priority for it. Neither September review's results entered the paper. Report; written answers to both.

October 1: a fresh Claude Opus 5.5 session made a consistency pass of 46 checks and two edits, not a review of the mathematics.

October 1: GPT-6 Astra, a fresh instance of the writer's own model, with the earlier reviews available, found all eight numbered statements correct and made 2 minor findings, none blocking or major (τ ≥ 0 in the abstract; the source-access sentence), both applied. Report, written answers.

October 2: Claude Opus 5.5 (Anthropic), the other company's model, in a fresh session with the earlier reviews available, ran its own numerical battery, reran both programs (below), found every numbered statement correct, and made 12 minor findings, none blocking or major: ten applied, among them the abstract's multiplicity clause, and two declined with reasons. Report, written answers.

No search or review obtained the full texts of Wigner (1960) or Mirsky (1960); the final review obtained Kramers (1930).

How to check it

A checker is a program that recomputes the paper's numbers, published beside the output it recorded. This paper has two, unchanged since the writing session.

verify.py runs seven numerical checks with a fixed seed, reading nothing from the harness; its recorded output is verification.json and its library versions are in requirements.txt. It checks the matching value against exhaustive search (1,000 cases), random unitaries against the minimum (4,800, none below beyond roundoff), the odd-set inequalities (240 skew contractions), the recurrence (1,500 cases), a local optimizer (120 starts: 96 reached the minimum, 80 set its success flag, none went below, which the paper says illustrates "why a local optimizer is not a certificate"), the theorem's optimizer (70 commuting triples) and the two exact examples.

verify_exact.py uses only Python's standard library, in exact rational arithmetic on real orthogonal matrices of size 1 to 7, not all complex unitaries. Its recorded output, verification-exact.json, holds 210 exact checks of the decomposition (7), 1,270 odd-set inequalities (990 on three or more indices) and 177 cases of both stability estimates, three tied cases excluded, with no failures.

The final review reran both on October 2 under the recorded versions: the exact output matched byte for byte apart from line endings, the numerical one in every count, four roundoff-level maxima aside. September reruns under other versions got 82 success flags instead of 80 and the same 96 accurate outcomes. The paper calls these "implementation and falsification checks"; the theorem rests on its arguments and Edmonds' theorem.

What the paper does not claim

The antiunitary is optimized over, so the value measures distance to an algebraic symmetry condition and does not test a prescribed physical time-reversal operator; for a single observable it certifies spectral pairing, not the symmetry of any prescribed antiunitary. The paper makes no exhaustive priority claim, and the theorem has not been formalized in a proof assistant.

If you read this paper

The owner would like to know what you make of it. A few words are enough: right, wrong, known, minor, worth a look, or not your area. In his words: "If it is right I want it in the record." If it is wrong, the page is corrected the same day and you are told; if it is already known, the page credits the earlier work.

Write to david@hypnosmath.org. He reads and answers replies himself; an error you find is fixed in the published files and on this site the same day, and you are told; nothing you send is given to any AI model without your permission.

Both checkers can be rerun from the published files, one under its recorded library versions, the other with Python's standard library alone.

Every file published with the paper

Every file is readable on this site in the paper's folder, each with its SHA-256, and in the same folder of the public repository, except the PDF, which sits at the repository's root. The review files are as the reviewing models delivered them, in the project's internal names, except that one line of one review, naming a private folder, is withheld from its public copy and marked where it stood. The read-only copy of the entries the paper cites and the reviews' supporting files are not published. A SHA-256 is a fingerprint of a file's bytes: two files with the same fingerprint are the same file.

The PDF has 11 pages and SHA-256:

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