A guide to the research log
STATUS: RIEMANN HYPOTHESIS OPEN
The Riemann Hypothesis asks whether every nontrivial zero of the zeta function has real part \(1/2\). This log follows attempts to find arithmetic information strong enough to enforce that conclusion. It subsequently turns to B, investigating the provability and possible ZFC independence of fixed RH. B PROGRAM: CONTINUE. L7 complete. L8 not started.
Curated by @ykbballer91, with AI assistance. Research dates, public-update dates, and the release date are distinguished throughout. The current editorial state is maintained on the current-state page; earlier pages remain dated records.
Three ways to read
For the result and its limits: the latest L7 update explains the change of question, arithmetization, prime-support normalization and proof cost. Both B obligations I/II remain OPEN, 2 → 2; finite-Weil CAP/PAR belong to a separate HOLD route. Start with current state, then the roadmap. The update through Cycle 17 explains fixed-trial oscillation and its limits. CORE-S and the October 5 joint-transfer update remain dated history.
For the reasoning across attempts: read the timeline. Each track records its starting idea, tests, retained results, stopping point, and next question. Parallel investigations are not presented as a single serial proof.
For verification or reuse: complete new L0–L7 proofs, internal audits and code remain privately preserved. The public summaries alone do not provide full third-party reproduction evidence. Use the source map, references, and reproducibility notes. Original research reports, scoped internal audits, numerical outputs, and formal fragments have different evidential roles.
The recurring obstruction
Several natural structures already retain the zeta zeros. The difficult step is obtaining an independent arithmetic estimate that excludes growth away from the critical line. A positive metric, a bounded scaling action, a square-root Möbius estimate, or convergence of real-zero approximants may encode this requirement without proving it.
The earlier analytic route studied finite Weil forms and prolate proxies. Minimization within a restricted trial family does not identify the ground state of the whole finite system. Qualitative density and exact spanning of a fixed finite head do not supply uniform control as the parameters grow.
Reading the labels
PROVED refers to a particular auxiliary argument with its hypotheses. KNOWN is imported literature. CONDITIONAL retains unproved premises. NUMERICAL describes finite computations. FALSE and REJECTED preserve failed claims or routes. OPEN means the stated obligation is not discharged. See the full legend.
An internal AI audit is not external peer review. A successful script does not establish an untested asymptotic statement. A sufficient resolution condition is not asserted to be a sharp threshold.
Earlier auxiliary updates
The cyclicity result proves that the even derivatives of \(k\) are dense in even \(L^2\), without RH. The finite-head result gives exact rank statements, tail criteria, and two finite interval certificates. Neither closes the full-ground or complex-transform comparison.