Studying Convergence of the
Connes–van Suijlekom Operator
BusyBots AI has built the first independent implementation of the CvS truncated Weil minimizer and conducted the first systematic convergence study across 15 prime cutoffs — spanning 113 orders of magnitude of convergence on an open question identified by Fields Medalist Alain Connes in his 2026 paper on the Riemann Hypothesis.
What is the Riemann Hypothesis?
Proposed in 1859 by Bernhard Riemann, it is the most important unsolved problem in mathematics. It governs the distribution of prime numbers — the atoms of arithmetic — and connects to cryptography, physics, and the deep structure of the number system.
The Zeta Function
Riemann's zeta function encodes information about every prime number. Its "zeros" — the points where it equals zero — control how primes are distributed. The hypothesis says all non-trivial zeros lie on a single line in the complex plane.
Millennium Prize
The Clay Mathematics Institute has offered $1 million for a proof or disproof. It is one of only seven Millennium Prize Problems. As of 2026, it remains unsolved after 167 years.
Connes' Approach
Fields Medalist Alain Connes published a framework in February 2026 showing that a specific mathematical object has zeros provably on the critical line. If these zeros converge to the Riemann zeros, the hypothesis follows. That convergence is the open question we are studying.
First Systematic CvS Convergence Study
We independently reimplemented the Connes–van Suijlekom operator framework from scratch, verified it against published results at c = 13, and extended the measurements to 15 prime cutoffs — producing the first systematic convergence data for this operator, spanning 113 orders of magnitude.
| Cutoff (c) | First Zero Error | Notes |
|---|---|---|
| 13 | 2.005 × 10−55 | Verified Matches Connes' ~2.6 × 10−55 (factor 1.3×) |
| 14 | 3.541 × 10−61 | Cross-validated CCM reports ~1.07 × 10−60 (our value ~30× smaller) |
| 17 | 1.634 × 10−76 | First measurement |
| 19 | 1.070 × 10−86 | First measurement |
| 23 | 5.520 × 10−103 | First measurement |
| 29 | 4.587 × 10−120 | First measurement |
| 31 | 1.141 × 10−124 | First measurement |
| 37 | 5.686 × 10−136 | First measurement |
| 41 | 2.760 × 10−142 | First measurement |
| 43 | 3.379 × 10−145 | First measurement |
| 47 | 4.270 × 10−150 | First measurement |
| 53 | 1.493 × 10−156 | First measurement |
| 59 | 3.911 × 10−162 | First measurement |
| 61 | 8.722 × 10−164 | First measurement |
| 67 | ~10−168 | Blind prediction passed |
Error is the absolute difference between the Galerkin-computed zero and the true first Riemann zero γ₁ = 14.134725… All measurements use N=100 basis functions and T=800 quadrature points. Cutoffs c ≤ 37 use 150-digit precision; cutoffs c ≥ 41 use 200-digit precision to avoid backward-error floor contamination. The data spans 113 orders of magnitude from c=13 (10−55) to c=67 (10−168). The c=67 row is a verified blind prediction — its value was predicted from the model before computation.
Empirical Scaling Observation
Across all 15 measured cutoffs, the first-zero error γ₁ is proportional to the smallest eigenvalue λmin of the Galerkin operator. This proportionality is consistent with standard predictions from Babuška–Osborn / Rayleigh–Ritz spectral approximation theory, and is consistent with the expectation that the CvS operator behaves as expected for a well-conditioned Galerkin truncation.
The proportionality constant C(c) grows slowly from ~7,000 at c=13 to ~18,489 at c=67, approximately linearly in log(c). The specific growth rate is an empirical observation whose theoretical explanation remains open.
What the Operator Reveals
Beyond convergence rates, the 15-cutoff dataset reveals structural properties of the CvS operator — observations that constrain future theoretical work on the convergence question.
The Full Story
Every iteration of this project is pre-registered before computation begins, adversarially reviewed after results land, and documented with full honesty — including failures.
How We Work
Every aspect of this project follows a discipline protocol designed to prevent the kind of motivated reasoning that plagues claims about famous open problems.
BusyBots AI
BusyBots is the AI research subsidiary of Click Fate Media. We build AI systems that do real intellectual work — not chatbots, not wrappers, but AI that reasons, computes, and self-corrects under human supervision.
The Riemann Project is our flagship demonstration: an AI system (Claude Opus 4.6) conducting genuine computational mathematics research under human supervision. The AI wrote the code, designed the experiments, conducted the adversarial reviews, characterized the eigenvalue-error scaling behavior, and drafted the paper — all with a human researcher providing strategic direction and quality control.
This is what AI-augmented research looks like. Not replacing mathematicians, but amplifying them — running computations that would take a human months in hours, exploring 20+ research angles simultaneously, and maintaining a level of documentation discipline that no solo researcher could sustain.
Read the Paper & Code
All computation is complete. The paper has been submitted and the code is open-source.