AI refines math bound on Riemann hypothesis zeros to 0.673

The work begins from Anthropic's Theorem D, which establishes that the proportion of simple (non-repeated) zeros of the Riemann zeta function on the critical line satisfies a lower bound of H_MT = (3/2) − (1/√2)cot(1/√2) ≈ 0.672500703679. An AI-generated research draft then proposed a strengthening using a seven-point refinement argument. This refinement proves that a specific function F₆ is bounded below by 191/50000 for all nonnegative gaps, and from this, with block size m=267, a final calculation yields the improved bound of 0.6730213619501665335.

The authors provide complete reproducibility artifacts. The key computational step uses interval arithmetic to exhaustively verify that the proposed bound holds. A clean run on CPython 3.12.3 with the python-flint library (version 0.8.0) confirms the result by searching a grid of 4000 cells with precision of 128 bits, exploring 786,215 nodes and pruning 393,472 of them across a maximum tree depth of 43. The exact verification output, including cryptographic hashes of the kernel and second-derivative tables used in the computation, is committed to the repository as certificates/seven-point.expected.json. The authors note that continuous integration reruns the exhaustive verifier from a clean checkout and compares every deterministic field to ensure reproducibility.

Critically, the authors emphasize that this is an unreviewed candidate refinement. They do not claim it proves the Riemann hypothesis or independently replaces Anthropic's analytic results; instead, the matrix inequality, kernel normalization, seven-point combinatorics, shifted-block pinching, and final arithmetic are all covered in a technical audit, while the analytic trace estimates, tail bounds, and optimized test family remain dependencies of Anthropic's original paper and its Lean 4 artifact. The work is released under the MIT license and preserves the upstream provenance chain. The authors welcome attempts to falsify the argument or reproduce the certificate on other architectures once the repository becomes public, signaling openness to verification and error-correction.