High-Precision Approximation of Riemann Zeros via the Truncated Weil Form
An independent, open-source study of the Connes–van Suijlekom truncated Weil form, carried out at extreme numerical precision. It includes what is, to our knowledge, the first independent out-of-sample test of Alain Connes' 2026 §6.4 prediction beyond the cutoffs reported so far — at c = 100.
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.
Connes' Approach
In February 2026, Alain Connes revisited a spectral route to the problem. A specific operator's ground state has zeros that provably lie on the critical line; whether they converge to the true Riemann zeros as a cutoff grows is the open question this project tests numerically.
An independent implementation, taken to c = 100
We built, to our knowledge, the first independent public implementation of the Connes–van Suijlekom (CvS) operator, reproduced the published c = 13 result, cross-validated it against the Connes–Consani–Moscovici (CCM) group at c = 14, measured the operator across fifteen prime cutoffs, and then pushed it far out of sample to c = 100 — where its predicted behaviour had never been checked.
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 eigenvalue should decay in the limit. We tested that formula where 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 move toward Connes' predicted value monotonically, and the deeper one lands about 3.3 away from Connes' predicted exponent of roughly 530 — an agreement of better than one percent. This is, to our knowledge, the first independent out-of-sample test of that prediction at a cutoff beyond those previously reported. The same computation recovers the first ten Riemann zeros to 307–329 matching digits — to our knowledge, the deepest such recovery from this construction in the public CvS/CCM literature.
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 prime 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. Rows c = 17–61 are, to our knowledge, the first public measurements at those cutoffs. The first-zero error is the absolute difference from the true value γ1 = 14.134725…
Two things we are careful about
The smallest-positive eigenvalue. At c = 100 the raw matrix carries a small, stable block of negative-sign eigenvalues, so we report the smallest positive one as the object of interest. The continuum positivity that would close the argument is equivalent to the Riemann Hypothesis itself, and 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.
What the operator reveals
Beyond raw convergence, the data shows clean structural regularities — reported here as observations, not explanations.
From the first line of code to publication
Every step was pre-registered before computation, reviewed afterward, and recorded in full — failures included.
Published, open, and reproducible
The full write-up is public on arXiv and Zenodo; the implementation is an open-source Python package; the data is downloadable, and the headline result reproduces in under a second.
- The c = 13 reproduction and the c = 14 cross-validation against CCM 2025.
- The fifteen-cutoff convergence sweep — 113 orders of magnitude.
- The c = 100 out-of-sample test of Connes' §6.4 prediction, and the 307–329-digit zero recovery.
- The finite-resolution reframing of the fit, and the candid smallest-positive-eigenvalue caveat.
Try it in two lines: pip install connes-cvs, then load the public data/ and reproduce the extrapolation with the bundled script.
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.
The mathematical foundation is entirely due to Connes, van Suijlekom, Consani, and Moscovici; the paper's 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.
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. No proof, theorem, or definition was originated by the model; every load-bearing mathematical claim was verified by the researcher, who conceived, directed, and verified all of the computational work and takes full responsibility for it.
Read the paper & run the code
The computation is complete, the paper is published, and everything is open-source and reproducible.