Maximal Coherence for Prescribed Intrinsic Populations and Youla Values: Two Questions of Gil

Claude Fable 5.1, an AI model made by Anthropic, wrote this note at the direction of David Ross. Its title block is dated September 29, 2026; it was revised through October 2 under the reviews below and published here that day. It addresses two questions J. J. Gil posed in the journal Entropy, 28(8):877 (August 4, 2026). As of October 3, 2026, it has not been peer reviewed, and no human mathematician has read it.

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

Of the three problems Gil's Section 8.2 leaves open, numbered (i) to (iii) by the note, it determines the maximum (i) asks for, over all orientations of the Youla frame with the populations and Youla values fixed (Theorem 4.1). It answers (iii), whether the adjacent-pairing maximum of his Theorem 2 (the note's Theorem 1.1), proved over aligned states, stays globally optimal under "suitable spectral constraints". Reading these as constraints on the Youla values, since with the populations a fixed spectrum of rho would already fix the objective, it shows that with the Youla values free the adjacent-pairing value is the maximum over all states with those populations (Theorem 3.1). It does not address (ii).

Its hull lemma comes from the companion paper by GPT-6 Astra (OpenAI), Approximate Antiunitary Symmetry as a Matching Problem, whose own review by GPT-6 Astra proved the free-value answer independently through that lemma on September 28, 2026, before this note's first draft. The inequality behind Theorem 4.1 follows from Horn's 1950 inequality for the singular values of a product. According to its zbMATH review, Mathias's 1992 paper, which the note did not obtain, contains an inequality for Hadamard products of which this is the Frobenius case and which also gives the free-value maximum. Remark 4.5's example came from the note's first review and its exact certificates from the second. It claims no priority for the inequality; its own part is the application to Gil's questions and the family of inequalities in Proposition 4.2, for which, it says, its searches do not establish priority either.

The setting

Gil's intrinsic reference basis diagonalizes the real part of a density matrix rho (Hermitian, positive semidefinite, trace one) by a real orthogonal change of basis:

rho = A + iN, A = diag(a_1, ..., a_n), a_1 >= ... >= a_n >= 0, a_1 + ... + a_n = 1,

with N real antisymmetric, the imaginary part of rho, which the note calls the coherence block. The intrinsic populations a_i are the eigenvalues of the real part of rho, not in general of rho; Gil calls the Frobenius norm ||N||_F the cohesion, his name for the size of the coherence block. For positive populations the metaspin tensor M_ij = N_ij / sqrt(a_i a_j) is real antisymmetric, rho = A^{1/2} (I + iM) A^{1/2}, and the squared cohesion is ||N||_F^2 = 2 (sum over i < j of a_i a_j M_ij^2); when populations vanish, M lives on the support of A, the axes of positive population. The Youla values s_1 >= ... >= s_m >= 0, m = floor(n/2), are the numbers for which M has singular values s_1, s_1, ..., s_m, s_m (with one more 0 if n is odd), and rho is positive semidefinite exactly when s_1 <= 1. An orthogonal change of basis, the Youla frame, brings M to 2 x 2 blocks with entries s_k and -s_k. Gil calls rho aligned when, for some admissible choice of the intrinsic basis, the frame can be a signed permutation of the axes, so that M is supported on a matching.

Edmonds (1965) described the matching polytope, the convex hull of the incidence vectors of matchings (sets of disjoint pairs), by x >= 0, degree inequalities (at most 1 on the pairs at each vertex) and odd-set inequalities (at most (|B| - 1)/2 on the pairs inside an odd vertex set B).

The results, as the note states them

Lemma 2.1 (the hull lemma, from the companion paper). For a real or complex skew-symmetric K of operator norm at most 1, the vector of squared entries |K_ij|^2, i < j, lies in Edmonds' polytope, and all such vectors have the polytope as convex hull, though their set is not convex.

Theorem 3.1 (free Youla values). Over all density matrices with given populations, every orientation of the Youla frame and every admissible Youla spectrum,

max ||N||_F^2 = 2 (a_1 a_2 + a_3 a_4 + ... + a_{2m-1} a_{2m}),

attained in the aligned class by the adjacent pairing with s_k = 1 on every adjacent pair of positive population product. For positive populations and weights b_ij >= 0, the maximum of the sum of b_ij M_ij^2 is the maximum weight of a matching, and the vectors (M_ij^2) have the polytope as convex hull; otherwise both hold only on the support of A (for a = (1/2, 1/2, 0, 0), b_12 = 1, b_34 = 10, the maximum is 1, the unrestricted matching weight 11). The proof parametrizes no orientations: by the lemma the objective peaks at a matching, which Gil's exchange inequalities, his (26) and (27), uncross into the adjacent pairing. What the polytope adds is the general-weight statement and the hull.

Theorem 4.1 (prescribed Youla values). Let a_1 >= ... >= a_n >= 0, m = floor(n/2), s_1 >= ... >= s_m >= 0, and M any real or complex skew-symmetric n x n matrix with singular values s_1, s_1, ..., s_m, s_m (and 0 if n is odd). Then

(9) sum over i < j of a_i a_j |M_ij|^2 <= s_1^2 a_1 a_2 + s_2^2 a_3 a_4 + ... + s_m^2 a_{2m-1} a_{2m},

with equality at the Youla block form. So for fixed populations and Youla values with s_1 <= 1, the largest ||N||_F^2 over all orientations is twice the right side, attained in the aligned class with the k-th largest Youla value on the k-th adjacent pair; every such orientation is realized by a state when the populations are positive, and when only r < n of them are, a state needs at most floor(r/2) positive s_k and the statement is read on the support. Inequality (9) follows from Horn's 1950 inequality. The first proof is a layer-cake decomposition of the products a_i a_j, closed by Ky Fan's maximum principle and the interlacing of singular values of submatrices. The second, supplied by the note's second review, applies Horn's inequality at each even index to D M D, D = diag(sqrt(a_i)). Mathias's 1992 theorem, as its zbMATH review states it, bounds partial sums of the singular values of Hadamard (entrywise) products of positive semidefinite and skew-symmetric matrices; (9) is its Frobenius case for D M D.

Proposition 4.2 (valid inequalities for a prescribed spectrum). For such M, x_ij = |M_ij|^2 and vertex sets C inside B,

2 (sum of x_ij inside C) + (sum of x_ij on pairs joining C to B outside C) <= R(|C|, |B|),

with R(p, q) the sum of s_k^2 over k <= min(ceil(p/2), floor(q/2)) plus the sum over k <= floor(p/2). For |C| = 1 and |B| >= 2 this is a degree constraint with bound s_1^2; for C = B it caps the sum inside B at the sum of s_k^2 over k <= floor(|B|/2), Edmonds' odd-set inequality when all s_k = 1. Numerically (Remark 4.3), linear programs over the C = B inequalities overshoot the maximum by up to 0.019 in units of squared cohesion, and in 4 of 30 instances even with the degree constraints added; over the prefix pairs C = {1, ..., p} inside B = {1, ..., q}, all the first proof uses, they reproduce it.

Remark 4.5 (product weights are essential). With prescribed Youla values and general weights, the maximizer need not be aligned, nor the linear program over the family tight. Its example takes n = 4, a = (0.4, 0.3, 0.2, 0.1), distinct so that alignment means matching support, s = (1, 0.9), b_12 = 2, b_13 = b_24 = 1.2 and the other weights 0: the maximum, 2.196, is attained by an explicit real matrix and certified exactly in the coordinates of Gil's Appendix A, against 2.172 over the aligned class and 2.324 from the linear program.

Corollary 5.1 states, for fixed populations: (1) with free Youla values, Gil's hierarchical descriptors linear in the squared metaspin entries are extremized in the aligned class; (2) for positive populations, the hull statement holds for the vectors (N_ij^2 / (a_i a_j)); (3) with the Youla values fixed too, equality in the bound of Theorem 4.1 needs every prefix inequality to be tight on the level sets of the populations.

What is left open

The note leaves open Gil's problem (ii), the image of the population-weighted Gram map; the feasibility question with the spectrum of rho fixed; the general-weight maximum with prescribed Youla values (see Remark 4.5); and whether equality in Corollary 5.1(3) forces alignment. The full texts of Tam (1998), of Leite, Richa and Tomei (1999) and of Mathias (1992) were not read.

Where the question came from

The two questions are J. J. Gil's own, from Entropy 28(8):877 (August 4, 2026). They surfaced while frontier models reviewed the antiunitary paper; GPT-6 Astra's own review proved the free-value answer first. No step of the note's mathematics came from the harness.

The note is here because it grew out of a review of a paper motivated by entries in the notebook of Hypnos, the research harness described above (how it works):

• August 4, 2026: Gil's paper is published (an outside author's questions).
• The night of September 28, 2026: GPT-6 Astra writes the antiunitary paper (its page tells its origin), and a fresh instance of GPT-6 Astra, reviewing it, finds Gil's paper and proves the free-value answer (GPT-6 Astra, in review).
• Later that night, September 29: Claude Fable 5.1, reviewing the same paper in a separate session blind to the first review, finds the two questions in its literature search, reads Gil's paper in full and drafts the note from its results (Claude Fable 5.1, in review, then in the writing session).

No notebook entry and no small model's line is among the note's sources.

How the note was made

It was drafted and first reviewed among the files of the review it grew out of, becoming a separate manuscript on October 1, 2026. Its attribution divides the work: Claude Fable 5.1 found the two questions while reviewing the companion paper, proved the theorems, and wrote the verification program and the text; David Ross set the task, ran the process and takes responsibility for the note. He read every page only as a check that the account of the harness and of the process matches its private log and that nothing reads as a model grading its own work; he did not check the proofs, and could not have.

How it was reviewed

Each review graded its findings blocking, major or minor; a blocking finding is one the reviewer judged would invalidate a statement or a proof as written. Claude Fable 5.1 applied each review's findings the same day.

September 29, 2026: Claude Opus 5.5 (Anthropic), a different model from the writer's company, in a fresh session without the writing session's audit files or the other review, rechecked both proofs by hand and numerically and found no blocking item, three major items, all of framing (an overclaimed abstract, an unread source behind a novelty claim, a misreported numerical search), and fifteen minor; it supplied the n = 4 example of Remark 4.5, and the theorems did not change. Report, written answers.

September 29, 2026: GPT-6 Astra (OpenAI), the other company's model, told to treat every claim as unproven and not to read the first review, found no blocking item, two major items (Theorem 4.1 follows from Horn's 1950 inequality; Corollary 5.1(1) needed its support restriction) and three minor; it supplied the Horn derivation, now the second proof, and Remark 4.5's exact certificates and populations, and first named Mathias's 1992 paper as a lead, without obtaining it. Report, written answers.

October 1, 2026: a consistency pass by a fresh Claude Opus 5.5 session checked the exposition against its sources: six edits in 41 checks, none to a proof.

October 2, 2026: GPT-6 Astra, the other company's model, fresh to the note, found both theorems, the hull lemma and Corollary 5.1 correct, one major item (a factor of two and a Pfaffian sign in Remark 4.5's exact certificate; the maximum 2.196 unchanged) and one minor (report, written answers). The same day Claude Opus 5.5, a different model from the writer's company, also fresh, found every numbered statement correct and every number in Remark 4.5 exact in rational arithmetic, two major items (the earlier proof of the free-value answer uncredited; the account of Mathias's abstract inexact) and ten minor (report, written answers). A Claude Opus 5.5 literature search that day found the zbMATH review of Mathias's paper, now credited in the note.

The October 2 reviews could not reach the full texts of Horn 1950 and Fan 1949 and re-derived what the note uses; GPT-6 Astra's September 29 review had checked Horn's two pages.

How to check it

A checker is a program that recomputes the note's numbers, published beside the output it recorded. The note's checker, verify_gil.py, uses NumPy, SciPy, a fixed seed and no code of the harness. Check A (30 population vectors, each with 300 random contractions and a 20-start gradient ascent) never exceeded Theorem 3.1's bound beyond roundoff and came within 10^-4 of it every time, within 10^-6 in 19; the worst shortfall, 7.2 x 10^-5, is recorded. Check B (30 cases, each with 300 random orientations and 20 local maximizations) never exceeded Theorem 4.1's value beyond numerical error and came within 10^-6 of it in all 30. Checks C and D run the linear programs of Remark 4.3 and test Lemma 2.1's odd-set inequalities 8,760 times, with no violation. The note calls these implementation controls, not part of the proofs; the searches are local. The one recorded output, verification-gil.json, is published twice, once as printed output. Two reviews reran earlier versions on September 29; on October 2 Claude Opus 5.5 reran this one under the recorded library versions, reproducing its output byte for byte.

What the note does not claim

It claims no priority for inequality (9), which follows from Horn's 1950 inequality and is the Frobenius case of Mathias's 1992 theorem as its zbMATH review states it, and says its searches do not establish priority for its own formulations either; it does not claim that the papers of Tam or of Leite, Richa and Tomei contain no equivalent formulation.

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