Papers and reports

The unfolded zeros of the Riemann zeta function do not form a Riesz basis of exponentials

The unfolded zeros of the Riemann zeta function, counted with multiplicity, form no Riesz basis of exponentials on any bounded interval, for every unfolding constant and every finite modification; for the set of distinct frequencies this is proved when all but finitely many ordinates are simple. The obstruction is Selberg's growth of the variance of the argument of zeta against the bounded-mean-oscillation condition in Pavlov's characterization of exponential Riesz bases.

Where the question came from. The only one whose chain begins with a small model's own line: a ten-word recall of Kadec's theorem on August 12, 2026, kept by the judge as a question; more lines and sharper judge-written questions over five weeks; ten programs; on September 28 a session the owner started scored six chains of entries and chose this one; the paper says its main theorem answers the question the notebook posed.

Approximate Antiunitary Symmetry as a Matching Problem

For commuting Hermitian matrices, the least combined error in antiunitary commutation and in the relation T^2 = -I is an exact weighted-matching value over the joint eigenvalues, counted with multiplicity, attained by signed swaps on the matched pairs; the single-observable case is classical.

Where the question came from. Its motivation is five notebook entries of late September: 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.

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

For fixed intrinsic populations, Gil's adjacent-pairing value is the largest squared cohesion over all states, and with the Youla values prescribed the maximum over all orientations is 2 (s_1^2 a_1 a_2 + s_2^2 a_3 a_4 + ...); the second rests on an inequality, for which the note claims no priority, that follows from Horn's 1950 inequality and is the Frobenius case of a theorem that Mathias's 1992 paper contains according to its zbMATH review.

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.

A dividing-plane barrier in the OpenAI forced Navier-Stokes blow-up construction

A 13-page note that proves, within the OpenAI manuscript's leading-order profile equations, the explanation the manuscript gives in words for making its axial profile slightly asymmetric: a smooth stress-free core with positive swirl and no axial velocity on the dividing plane keeps its swirl shear below 2 on that plane, so the manuscript's Theorem 4.6(iii) fails where the inner edge of the annulus meets it.

Where the question came from. A separate line of work the owner ran with frontier-model sessions on the OpenAI forced Navier-Stokes blow-up manuscript: Claude Opus sessions wrote a step-by-step account of the construction's moves and eleven section explanations; the explanation of the manuscript's Appendix B asked the question and sketched the answer; a Claude Fable 5.1 session ranked it first on its map of open questions and proved it. No small model's line is among its sources, and the material has not been filed into the notebook.

Self-Grading Without Self-Deception: Verdict Discipline for a Long-Running Mathematical Research Agent

The rules under which the harness measures its own parts, and what they caught in its first 18 days; the PDF is the paper's text of August 29, 2026.