The Riemann Project Three published papers on a spectral route to the Riemann Hypothesis
An independent, open-source study of one specific, recently proposed route to the most famous open problem in mathematics. Three papers measure, prove, and decode one finite construction at up to 1000 digits of precision. Every number has runnable code behind it, every correction is public, and no claim of proof is made.
What is the Riemann Hypothesis?
Proposed in 1859 by Bernhard Riemann, it is among the most important unsolved problems in mathematics. It governs how the prime numbers, the atoms of arithmetic, are distributed, and it reaches into cryptography, physics, and the deep structure of the number system.
The Zeta Function
Riemann's zeta function encodes information about every prime. Its non-trivial "zeros" control how the primes are spread out. The hypothesis says all of those zeros sit on a single vertical line in the complex plane: the critical line.
A Millennium Prize
The Clay Mathematics Institute lists it among the seven Millennium Prize Problems, with $1 million offered for a resolution. As of 2026 it has stood unsolved for 167 years.
The Spectral Dream
A century-old idea (Hilbert and Pólya) suggests the zeros might be the vibration frequencies of some natural operator: if such an operator exists with the right symmetry, the hypothesis follows. In February 2026, Alain Connes revisited a concrete candidate framework of this kind, and left one question open: does its finite version converge to the true Riemann zeros? If the finite versions home in on the true zeros, the route stays alive; if they drift, it is in trouble. Running that test is what this project does.
Three papers, one finite object
Connes' framework rests on an older equivalence. André Weil proved in 1952 that the Riemann Hypothesis is exactly the statement that a certain quadratic form, built from the primes together with an explicit archimedean term, never goes negative. The form itself is infinite. Connes and van Suijlekom showed how to truncate it, keeping only the prime powers up to a cutoff c; discretizing the result at a finite resolution N, the step carried out computationally by Connes, Consani, and Moscovici, turns it into an ordinary finite matrix that a computer can build, measure, and prove theorems about. That family of matrices, indexed by c and N, is the shared object of all three papers.
Imagine the infinite arithmetic of the primes compressed into a finite grid of numbers, small enough to hold in a computer, rich enough that the first Riemann zeros can be read out of it to hundreds of correct digits. The three papers ask three different questions about that grid: what does it do (measure it), why can it be trusted (prove exactly what it sees of the infinite object), and where are the primes inside it (find them, exactly).
Two words carry most of what follows. A matrix's eigenvalues are its short list of characteristic numbers; they are the same "vibration frequencies" as in the spectral dream above. And positive means none of those numbers dips below zero. Weil's theorem says the Riemann Hypothesis is exactly the statement that the infinite form never dips negative, which is why the smallest eigenvalue of the finite matrix is the needle every measurement below watches.
The three dials on the machine
Measure the matrix, then push it far beyond the published cutoffs
To our knowledge the first independent public implementation of the Connes–van Suijlekom operator: it reproduces the published c = 13 result, cross-validates against the Connes–Consani–Moscovici group at c = 14, measures fifteen cutoffs, and then jumps far out of sample to c = 100, where, to our knowledge, the framework's predicted behaviour had never been checked.
The framework's authors reported striking results at the first small cutoffs (up to c = 14). This paper rebuilt the whole machine from the published equations, sharing no code with the original, and confirmed those values. Then it kept going: fourteen additional cutoffs, and finally a leap to c = 100, far beyond the published range, to test the framework's predicted limiting behaviour where, to our knowledge, no one had looked. Along the way, the matrix gives back the first ten Riemann zeros to more than 300 correct digits each. For comparison: no physical measurement has ever reached even 15 significant digits.
The headline: an out-of-sample test of Connes' 2026 §6.4 prediction
Connes' 2026 letter gives a heuristic formula for how the operator's smallest-positive eigenvalue (see the caveats below) should decay in the limit. We tested that formula where, to our knowledge, no one had: at c = 100, on a sequence of increasingly fine approximations (N = 100, 150, 200, 250) carried to as many as 1000 digits of precision.
Two successive extrapolations both move toward Connes' predicted value, the deeper one landing closer: about 3.3 away from the predicted exponent of roughly 530, agreement at the under-one-percent level on the exponent. Four data points cannot rule out alternative convergence models; the paper's sensitivity discussion spells out what this does and does not establish. This is, to our knowledge, the first independent out-of-sample test of that prediction at a cutoff beyond those previously reported.
The c = 100 sequence
| N (resolution) | Precision | Smallest-positive eigenvalue λmin | Note |
|---|---|---|---|
| 100 | 500 digits | 1.22 × 10−191 | First point |
| 150 | 500 / 1000 digits | 6.42 × 10−248 | Cross-checked to 25 leading digits at 1000-digit precision |
| 200 | 500 digits | 4.87 × 10−295 | — |
| 250 | 500 digits | 2.08 × 10−334 | Deepest point; γ1–γ10 to 307–329 digits |
The extrapolations use two overlapping triples from this sequence; the underlying eigenvalues are stored as full-precision decimal strings in the public dataset.
The foundation: convergence across fifteen cutoffs
Before the c = 100 test, the operator was measured across fifteen cutoffs from c = 13 to 67. The error in the first Riemann zero shrinks smoothly and without interruption through 113 orders of magnitude.
Show all fifteen cutoffs (c = 13 → 67)
| Cutoff c | First-zero error |γ1 − γ1R| | λmin | Note |
|---|---|---|---|
| 13 | 2.005 × 10−55 | 2.865 × 10−59 | VerifiedReproduces Connes/CCM (~2.6×10−55) to a factor of ~1.3 |
| 14 | 3.541 × 10−61 | 4.835 × 10−65 | Cross-checkvs CCM 2025 (~1.07×10−60) |
| 17 | 1.634 × 10−76 | 2.030 × 10−80 | New cutoff |
| 19 | 1.070 × 10−86 | 1.265 × 10−90 | New cutoff |
| 23 | 5.520 × 10−103 | 5.959 × 10−107 | New cutoff |
| 29 | 4.587 × 10−120 | 4.366 × 10−124 | New cutoff |
| 31 | 1.141 × 10−124 | 1.045 × 10−128 | New cutoff |
| 37 | 5.686 × 10−136 | 4.670 × 10−140 | New cutoff |
| 41 | 2.760 × 10−142 | 2.122 × 10−146 | New cutoff |
| 43 | 3.379 × 10−145 | 2.519 × 10−149 | New cutoff |
| 47 | 4.270 × 10−150 | 2.994 × 10−154 | New cutoff |
| 53 | 1.493 × 10−156 | 9.615 × 10−161 | New cutoff |
| 59 | 3.911 × 10−162 | 2.328 × 10−166 | New cutoff |
| 61 | 8.722 × 10−164 | 5.063 × 10−168 | New cutoff |
| 67 | 1.478 × 10−168 | 7.993 × 10−173 | BlindDeepest in-sample; passed a pre-registered blind-prediction test |
All rows use a fixed resolution (N = 100) and 150- or 200-digit precision, chosen to stay clear of the arithmetic floor. Together with c = 100, these rows form, to our knowledge, the first independent multi-c survey of this spectrum beyond the framework authors' reported values at c ∈ {9, 12, 13, 14}. The first-zero error is the absolute difference from the true value γ1 = 14.134725… The two columns are two gauges of the same convergence: how far the recovered first zero sits from the true one, and how close the matrix's smallest characteristic number sits to zero.
What the operator reveals (for specialists)
Two things we are careful about
The smallest-positive eigenvalue. Some eigenvalues at c = 100 first came out negative. That turned out to be an artifact of the technical dial T, not of the object itself: a published correction (June 2026) traced the negative block to the finite archimedean cutoff, and a rigorous re-computation with guaranteed error bounds (an interval-arithmetic factorization, free of that cutoff) certifies the even sector non-negative at c = 100 for N = 100, 150, and 200. So we report the smallest positive eigenvalue as the object of interest; the full record lives in the repository's public ERRATA.
A note on scope: the matrix splits by symmetry into two independent halves, the even and odd sectors, and the headline measurements live in the even sector. The odd sector, where a small finite-cutoff negative has been reported through the public issue tracker, carries no cutoff-free certificate to date. And the continuum positivity that would close the whole argument is equivalent to the Riemann Hypothesis itself; we do not assume it.
The fit is finite-resolution, not the limit. A simple power law describes the data up to c = 67 at fixed resolution, but it is a finite-resolution rate, not the true limiting behaviour: the c = 100 measurement overturns its naive extrapolation by 49 orders of magnitude. We report what the numbers show, and what they do not.
BibTeX
The finite window is exact, and it comes with a price list
A finite computation can only ever see a window of an infinite object. This paper proves two exact finite theorems about that window: a dictionary that makes every even-sector value of the cutoff-free matrix an exact statement about the true zeros of zeta, and a sign theorem for the archimedean tail that any finite-T computation omits.
First half: the ideal window is not an approximation at all. Remove the technical dial T and you get the cutoff-free matrix Q∞, the ideal object the computed matrices approach. Every number that ideal matrix produces (in the even sector, where the headline results live) equals, exactly, a sum over the true zeros of the zeta function. The paper constructs an explicit change of language, a dictionary, between vectors of the matrix and the inputs of the 150-year-old formula that links primes to zeros. Nothing is lost in translation, and the translation is a closed formula.
Second half: what a real, finite-T computation still omits can only push one way. The omitted piece, the archimedean tail, is proved to have a definite sign, and its size has an exact budget. Together the two halves give a certification rule: certain finite computations genuinely certify facts about the ideal object, others certify nothing at all, and the rule tells you which is which before you compute.
The two theorems
The dictionary (Theorem 2.5). Every real even Galerkin coefficient vector v (Galerkin: the standard recipe for shrinking an infinite problem onto a finite frequency grid) determines, in closed form, a band-limited Guinand–Weil test function gv with
The construction factors through an exact source quotient of dimension 2N + 1 (membership characterized exactly, Corollary 2.4), and admits a non-collapsing pole-neutral subfamily.
The tail order (Theorem 3.2). Beyond the archimedean cutoff T, the omitted archimedean tail is a strictly positive definite, strictly totally positive Cauchy–Stieltjes increment. This yields a two-sided certification rule with an explicit budget:
The budget has teeth: resolving the 10−59 scale of the c = 100 negative block by brute archimedean cutoff would require T of order 1063. A cutoff-free interval LDLT factorization resolves it directly, which is exactly the certificate used in Paper 1's correction record.
Stated scope. The paper makes no claim regarding the Riemann Hypothesis, Weil positivity, or a prime-location bound. Its theorems are exact and finite-dimensional.
BibTeX
Where the primes live inside the matrix
Fix the resolution N and let the prime cutoff vary: the finite matrices trace a path. This paper studies the singular part of that path's second derivative and finds, sitting inside it, an exact finite copy of the prime side of the classical explicit formula.
Where are the primes inside this matrix? Not smeared across the spectrum: at exact addresses along the path. As the cutoff slides upward and crosses a prime power q, the matrix's rate of change jumps, the way a hiking trail's altitude profile kinks where the grade changes: the altitude itself never jumps, but its slope does, all at once. And the size of that kink is not approximately related to q. It carries exactly the classical von Mangoldt weight of q (a standard number-theory score that essentially records the prime's logarithm), up to one universal normalization shared by every prime power, in every one of the matrix's entries at once. Each prime arrives as a sharp, indivisible event with a universal shape, and the paper works out the exact finite geometry of those events.
The defining identity
The first-derivative jump of the matrix path u ↦ QN(u), u = log c, at a prime-power threshold u = log q:
This defines a negative semidefinite, rank-one matrix-valued von Mangoldt measure: a finite, cutoff-free realization of the prime side of the Weil–Guinand explicit formula on the operator path itself. The identity relating the prime side to the zeros is classical and is not reproved; the contribution is the isolation and naming of the finite matrix-valued measure and the exact finite geometry established around it.
Stated scope. The results are structural: the paper proves no positivity, no Riemann Hypothesis, and no prime-counting, next-prime, or factoring statement. The event algebra transfers to Selberg-class L-functions once a finite path with the same event structure is prescribed.
BibTeX
Checked by others, corrected in public
In open research, being checkable matters more than being right the first time. Everything here is built to be re-run by strangers, and it has been.
Try it: pip install connes-cvs, clone the repository for the public data/, and reproduce the headline extrapolation with the bundled script, in under a second.
From the first line of code to a three-paper series
Every major computation was pre-registered before running, reviewed afterward, and recorded. Failures included.
How the work is done
A famous open problem invites wishful thinking. This project runs on a discipline designed to prevent it.
In dialogue with the field
The work drew substantive correspondence through 2026, most extensively with Professor Alain Connes, whose program it builds on. That exchange confirmed a basis convention used throughout, clarified the qualitative motivation behind one of the constructions, and produced formulation comments that improved the manuscript. A number of other researchers across noncommutative geometry, spectral theory, and analytic number theory engaged with the work as well.
Through the summer of 2026 the project's public repository became a working surface of its own. Independent researchers reproduced the central spectra from scratch, in one case to more than 300 matching digits; probed the odd sector; and surfaced documentation defects through the public issue tracker. Every issue was answered and closed, each contribution is credited by name in the README, and the corrections are recorded in the public ERRATA.
The mathematical foundation is entirely due to Connes, van Suijlekom, Consani, and Moscovici; the papers' acknowledgments record individual contributions in full.
An independent research project
The Riemann Project is an independent research effort by Click Fate Media: an open, reproducible study of one specific, recently proposed route to the Riemann Hypothesis, carried out at extreme numerical precision and published in full as a three-paper series.
It follows a strict discipline: pre-registration, adversarial review, verification from multiple directions, open reporting of failures, and fully public code and data.
Modern AI tools were used as assistants for drafting implementation code and exposition, to specifications written by the researcher. Every mathematical claim was conceived, directed, and verified by the researcher, who takes full responsibility for the work; the papers each carry their own formal statement on the use of AI tools.
Read the papers & run the code
Three papers are published, the package is maintained, and everything is open-source and reproducible.