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

Claude Fable 5.1, an AI model made by Anthropic, wrote this paper at the direction of David Ross, the owner of Hypnos. Its title block is dated September 29, 2026; the text was revised through October 2, 2026 under the four reviews below, with that date unchanged, and was 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 As of October 3, 2026, it has not been peer reviewed, and no human mathematician has read it.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 18-page paper (PDF)

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

The paper proves three things about exponentials whose frequencies are the unfolded, symmetrized zeta ordinates. At each simple ordinate, the displacement of the unfolded ordinate from the integers is the argument of zeta (Theorem 3.1), a two-sentence rearrangement of the Riemann-von Mangoldt formula. The sufficient conditions of Kadec (1964) and Avdonin (1974, in the form of Alemany and Nitzan, arXiv 2501.11598) fail for every centering and block length (Theorem 4.1). And the main result, Theorem 5.1, is the statement under the title above. In the paper's words, "a Riesz basis forces the deviation of the counting function from linearity into BMO(ℝ), and Selberg's variance growth is incompatible with bounded mean oscillation."

The inputs are classical: Selberg (1946), Littlewood (1924), Pavlov (1979) in the form of Hruščev, Nikol'skii and Pavlov (1981), John-Nirenberg (1961), Landau (1967), Beurling-Malliavin (1967) and, for one bound, Conrey (1989). The paper distinguishes two adjacent works: "Unfolded Gaussian Gram stability for zeta zeros", Mikulik's preprint of April 2026, seen only in abstract, and Burnol's complete and minimal systems in de Branges' Sonine spaces (2004). Neither accessible account states Theorem 5.1, and its searches found no earlier proof: "a bounded search, not a certification of priority."

The setting

Let ρ = β + iγ run over the nontrivial zeros of ζ, with positive ordinates γ_1 ≤ γ_2 ≤ ..., counted with multiplicity. The number N(t) of zeros with 0 < γ ≤ t, with multiplicity, satisfies the Riemann-von Mangoldt formula

N(t) = θ(t)/π + 1 + S(t), S(t) = (1/π) arg ζ(1/2 + it),

with θ(t) = (t/2) log(t/2π) − t/2 − π/8 + O(1/t) the Riemann-Siegel theta function, and S and N the means of their one-sided limits at an ordinate. An ordinate's multiplicity is the jump of N there, simple if 1; a simple zero need not lie on a simple ordinate, since β + iγ and 1 − β + iγ share one.

The ordinates' density (1/2π) log(t/2π) tends to infinity, so they cannot themselves give a Riesz basis or frame on any interval. Unfolded, x_n = θ(γ_n)/π + 3/2 ≈ n, with displacement d_n = x_n − n. The paper studies Λ_a = {±(θ(γ_n)/π + a)}, a real, counted with multiplicity, and writes Λ for Λ_{3/2}: Λ_{3/2} with 0 is modeled on the integers and Λ_1 on the half-integers, both matched to (−π, π). Changing a is not a translation, since it moves the two halves oppositely, so the theorems cover every a. A finite modification Λ' adds or removes finitely many points.

E(Λ) = {e^{iλt} : λ in Λ} is a Riesz basis for L²(I) if it is complete and A Σ|c_λ|² ≤ ‖Σ c_λ e^{iλt}‖² ≤ B Σ|c_λ|² for some 0 < A ≤ B and all finite sums. Results of others used:

• Kadec (1964), sufficient only: sup_n |λ_n − n| < 1/4 gives a Riesz basis for L²(−π, π); 1/4 is sharp.
• Avdonin (1974), in Alemany and Nitzan's form: λ_n = n + δ_n give a Riesz basis for L²(−π, π) if (a) they are separated, (b) the δ_n are bounded, and (c) for some real c, block length N and d < 1/4, the mean of δ_n on each block of the fixed partition into blocks of N is within d of c; (c) for every N consecutive indices is the stronger sliding-window form.
• Pavlov (1979), in the form of Hruščev, Nikol'skii and Pavlov (1981), as Kozma and Lev state it: with n_Λ(b) − n_Λ(a) the number of points in [a, b), E(Λ) is a Riesz basis for L²(−π, π) if and only if (i) Λ is separated, (ii) n_Λ(x) − x is in BMO (its average distance from its own mean over bounded intervals is bounded), and (iii) a condition on its harmonic extension, not restated here, holds; only the necessity of (ii) is used.

Unconditional facts about S used: ∫_T^{2T} S(t)² dt = (T/2π²) log log T + O(T (log log T)^{1/2}) (Selberg); ∫_0^T S(t) dt = O(log T) (Littlewood); S is unbounded above and below (Selberg); S(t) = O(log t).

The results, as the paper states them

Theorem 3.1. If γ_n = γ has multiplicity m and n = N(γ⁻) + j with 1 ≤ j ≤ m, then x_n − n = −S(γ) + (m + 1)/2 − j; at a simple ordinate, x_n − n = −S(γ_n), and 3/2 is the only unfolding constant giving this.

Proposition 3.2 and Lemma 3.3. For non-ordinates 7 ≤ a < b, the sum of S(γ) over the ordinates in (a, b), with multiplicity, is (1/π) ∫_a^b θ'(t) S(t) dt + (S(b)² − S(a)²)/2, and their displacements sum to minus that. Beyond a cutoff, n_{Λ_a}(x) − x is S at the height unfolding to x, plus a constant.

Theorem 4.1. For every N ≥ 1 and real c, the block means (1/N) Σ_{n=k+1}^{k+N} d_n − c are unbounded above and below, over all k ≥ 0 and over the multiples of N. So Kadec's hypothesis fails for every centering, and Avdonin's (b) and (c) for every centering and block length, in both forms, also for Λ_{3/2} with 0 and for Λ_1. The theorem is unconditional, using only the monotonicity of θ and N and the unboundedness of S both ways; the failures occur at arbitrarily large heights.

Corollary 4.2. For non-ordinates 7 ≤ a < b the sum of S over the ordinates in (a, b) is O(log² b), or O(log² b/(log log b)²) under the Riemann hypothesis; so below height b every block of M > 4C log² b consecutive indices, C an absolute constant, has mean displacement under 1/4 in absolute value. This holds at a given height, while Avdonin's hypothesis asks for one block length at all heights, and none exists: the failure is slow.

Theorem 5.1. Let a be real and Λ' any finite modification of Λ_a, counted with multiplicity. For every bounded interval I and real c, the system {e^{iλt} : λ in Λ' + c} is not a Riesz basis for L²(I); in particular this holds for Λ_{3/2} with 0, for Λ_1, for Λ_{3/2}, and for the one-sided system {e^{ix_n t} : n ≥ 1}.

No hypothesis on the zeros is assumed; the theorem concerns the sequence counted with multiplicity. A Riesz basis has no repeated element, so its content is the distinct-frequency case, and a finite modification removes only finitely many repetitions: for the set of distinct points of Λ_a the theorem is proved only when all but finitely many ordinates are simple, and otherwise says nothing about that set.

The proof, in four steps: n_{Λ'}(x) − n_{Λ'}(0) = x + O(log x) gives Beurling densities D⁻ ≤ 1 ≤ D⁺ (whether they equal 1 is not known, or needed), so Landau's theorem forces |I| = 2π; on (−π, π) a Riesz basis would put n_{Λ'}(x) − x in BMO by (ii), bounding its mean-square oscillation by John-Nirenberg; by Lemma 3.3 that deviation is S in the unfolded variable plus a bounded term; and by Selberg and Littlewood its mean-square oscillation over the image of [T, 2T] is at least (1 + o(1)) log log T/(8π²) minus a constant, which tends to infinity.

Theorem 5.2 and Corollary 5.3. If Λ is locally finite and n_Λ(x) − dx has unbounded mean-square oscillation over bounded intervals, E(Λ) is no Riesz basis for L²(I) with |I| = 2πd, nor for any bounded I if also (n_Λ(x) − n_Λ(0))/x → d. Corollary 5.3 extends Theorem 5.1 to an L-function with N_F = θ_F/π + c_F + S_F, θ_F eventually increasing, θ_F' comparable to log t and S_F = O(log t), whenever the mean of S_F² over [T, 2T] tends to infinity and ∫_0^T S_F = O(T). The paper verifies these hypotheses for no family, nor for a fixed Dirichlet L-function or the Selberg class.

Remark 5.4. A sketch shows that n + c log(2 + |n|), n in ℤ, gives a Riesz basis of L²(−π, π) for |c| ≤ 1: Theorem 5.1 detects not that S is unbounded but that its local mean-square oscillation is.

Theorem 6.1. Unconditionally, Λ' counted with multiplicity has exterior Beurling-Malliavin density 1, and the radius of completeness of its distinct points lies between 2π/5 and π (part (a)); if all but finitely many ordinates are simple, it is π (part (b)). The lower bound rests on Conrey's theorem that at least two fifths of the zeros are simple and on the critical line, such zeros having distinct ordinates. It entered the paper on October 2, 2026: GPT-6 Astra's review of October 1 sketched it, a Claude Opus 5.5 session the same day found it independently, and Claude Fable 5.1 re-derived it.

Proposition 6.2. With 2πA_N ≤ 2πB_N the extreme eigenvalues of the Gram matrix of the first N unfolded exponentials on (−π, π), A_N is nonincreasing, B_N nondecreasing, A_N ≤ 1, and for N ≥ 2, with s_N the smallest gap, A_N ≤ 1 − sinc(s_N) ≤ π² s_N²/6 and B_N ≥ 1 + sinc(s_N). This upper bound is consistent with the observed decay of A_N, and the smallest gaps of order N^{−1/3} that the random-matrix model predicts would give A_N = O(N^{−2/3}). If inf_N s_N = 0 then A_N → 0; the converse is not proved.

What is left open

The paper asks whether the positive half of Λ_a is separated, that is, whether (γ_{n+1} − γ_n) log γ_n is bounded below by a positive constant (Question 6.3; the random-matrix model predicts not); whether E(Λ) is complete in L²(−π, π), and with what excess (Question 6.4); and whether E(Λ) is a frame for L²(I) when |I| < 2π (Question 6.5). Also open: the exact unconditional completeness radius, and whether B_N stays bounded. A comparison with random-matrix sections, proposed in comments of October 1, 2026, is not in the paper; it is recorded as the first item for a next revision.

Where the question came from

Of the four mathematics papers on this site, this is 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.

The chain grew inside Hypnos, the research harness described above (how it works, step by step), whose judge, Claude Opus 5, a frontier model made by Anthropic, writes each small-model line it keeps into the notebook as an entry.

• August 12, 2026 (a small model's line). Asked which known result an entry resembled, one of the three small models wrote: "The Kadets 1/4-theorem regarding the stability of bases of exponentials."
• August 12 (the judge's entry, Claude Opus 5). Kept, it became a question about twenty times as long, whether the unfolded ordinates meet Kadec's sufficient condition, with a test on the first 10,000 zeros and a prediction of early failure.
• August 14 and 20 (small models' lines). Two more recalls of the theorem, both kept.
• August 16 (a small model's line; the judge's entry). A small model claimed, falsely, that Kadec's condition holds for the first N zeros; the judge's entry recast it as a uniform bound on S at the ordinates (up to a shift of 1/2 from the line's unfolding θ/π + 1, which the paper's 3/2 removes), with Avdonin's averaged condition and Pavlov's characterization as the live question.
• From August 19 (programs written by the judge's model and graded by code). Kadec's bound first fails at n = 9; the largest displacement in the first 100,000 ordinates is 1.129.
• September 2 (a small model's line; the judge's entry). From a line about the first 50 rescaled zeros, the judge wrote that Kadec's condition, only sufficient, decides nothing by failing, and asked for measured lower Riesz constants against a random-jitter control. A recall of Pavlov's stability results, kept that day, is not among the 17 entries the paper cites.
• September 18 (a small model's line; the judge's entry; a program). A small model asked for the Riesz constants of the first 1,000 unfolded zeros; the judge's entry asked whether the lower constant stays away from zero or decays, the Riesz property then being lost; a program found it falling from 0.1115 at N = 100 to 0.0159 at N = 1,000.
• September 28 to 29 (Claude Fable 5.1, in the writing session). The owner asked a Claude Fable 5.1 session to read the harness's work and prove something it found there. It read a copy of the notebook, scored six chains, chose this one, and that night proved Theorem 5.1 by joining Pavlov's criterion with Selberg's mean-square theorem. The paper says Theorem 5.1 answers the chain's question "whether the Riesz property degrades or is merely uncertified (it is lost)", and Theorem 4.1 its fixed-block question.

Every question the writing session read in this chain was the judge's; the measurements were programs; the proof is Claude Fable 5.1's. Given the chain's own pairs of entries on September 29, the small models wrote no line containing the proof's mechanism, Pavlov's condition against Selberg's variance, and the harness had never drawn those pairs together. Whether Claude Fable 5.1 took its route from the notebook's mentions of Pavlov or from its own knowledge is not recorded. In the paper's words, the harness "supplied the motivating questions and the numerical observations, which are reported separately from the proofs." The notebook is private, so the entries and programs that the paper's Section 8 cites by number cannot be looked up.

How the paper was made

The paper's attribution divides the work: the model chose the problem from the harness's notebook, proved the theorems and wrote the programs and the text; David Ross set the task, ran the process and takes responsibility for the manuscript. He read every page only to check the account of the harness and the process against its private log, not as a review, and did not check the proofs. It is one of three manuscripts written from the harness's work on the night of September 28 to 29, 2026, and was written without reading the other two. Claude Fable 5.1 applied every review's edits, those of October 2 in a later session.

How it was reviewed

Four reviews were applied, all by AI models, each with a written answer to every finding. Each graded its findings blocking, major or minor; a blocking finding is one the reviewer judged would invalidate a statement or a proof as written.

1. September 29, 2026: Claude Opus 5.5 (Anthropic), a different model from the writer's company, in a fresh context. No blocking finding; five major, eight minor, the main one leading to the restatement for every unfolding constant and finite modification. Twelve items were applied that night; the owner settled the thirteenth, that most journals do not accept an AI author, with the attribution form at the top of this page. The report and its written answers.
2. September 29: GPT-6 Astra (OpenAI), the other company's model, kept independent of every other review. One blocking finding, since restated: Theorem 6.1 counted ordinates with multiplicity where completeness depends only on the distinct points. Five major, among them the two adjacent works, which it found. It ran both programs on the same table, recomputed all 100,000 displacements to every printed digit and checked the finite-section eigenvalues in double precision; all was applied that day. A consistency pass by a fresh Claude Opus 5.5 session on October 1 applied nine findings, none to a proof. The report and the written answers.
3. October 1: GPT-6 Astra again, fresh to the paper but with the earlier reviews in hand. No blocking finding; two major (a finite-height necessity claim the proof did not cover; the denial of the lower bound now in Theorem 6.1(a)) and four minor, all applied on October 2. The report and the written answers.
4. October 2: Claude Opus 5.5 (Anthropic), a different model from the writer's company, in a fresh session, the last review before publication. No blocking finding; one major, in the paper's README, which is not published here; nine minor, all applied; eight smaller suggestions not applied. It recomputed the first 1,000 ordinates independently, agreeing with the harness's table to 5·10⁻¹⁹, and every printed number. The report and the written answers.

No reviewer retrieved the originals of Selberg 1946, Avdonin 1974, Kadec 1964, Pavlov 1979, the cited page of Hruščev, Nikol'skii and Pavlov, or the book cited for the Beurling-Malliavin theorem; they were checked through secondary sources.

How to check it

A checker is a program that recomputes the paper's numbers, published beside the output it recorded. Two ship with the paper, in Python with mpmath.

verify_identity.py works at 40 digits on the first 10,000 ordinates: it checks Theorem 3.1 at sampled ordinates, computing S independently from ζ, checks Proposition 3.2 on random windows, and recomputes the Kadec and Avdonin statistics. Its outputs, verify_10000.json and verify_10000_40samples.json, give identity residuals below 10⁻¹⁹ and block residuals below 10⁻³³.

verify_kadec_1e5.py works at 30 digits on all 100,000 ordinates; its output gives the first displacement of size at least 1/4 (n = 9), the largest (1.129), the 43,588 (43.6 percent) of size at least 1/4, the block means D(M), below 1/4 at every tested length from 8 up, and the smallest unfolded gap (0.02186), matching the paper's Section 7 and Table 1 to the printed digits.

The paper's stated limits: the block check tests the algebra and its implementation, not a second continuation of the argument; the continuation check uses finite phase steps and does not certify against a missed winding; the finite-section constants and jitter control are quoted from the harness's earlier computations, the constants' eigenvalues confirmed in double precision only. No proof depends on the numbers. Both programs read the harness's table of the first 100,000 ordinates, which is not published, so they cannot be rerun from the published files alone.

What the paper does not claim

Beyond the limits stated with each result above, the paper claims nothing about the Riemann hypothesis itself, which appears only in conditional statements, and does not offer its numbers as evidence for its theorems. It claims no priority: its search was bounded, and Mikulik's full manuscript was not checked.

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.

This paper's programs read an unpublished table of zeros, so they cannot be rerun from the published files 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, with the PDF at the repository's root: the source, the two panels of Figure 1 (drawn by a program that is not published), the programs with their recorded outputs, and the reviews with their written answers, published as delivered, in the project's internal names. A SHA-256 is a fingerprint of a file's bytes: two files with the same fingerprint are the same file.

The PDF has 18 pages and SHA-256:

6de12107422fcf79de766da56461ccfa8618047f51c8073c56c0050b79d82c38