Riemann Hypothesis Program
PROGRAM / RIEMANN HYPOTHESIS
The SEERAVERSE RH Program is a long-term program that simultaneously studies the proof path from finite structure to analytic rigidity, using the Riemann Hypothesis as its subject, and the auditability of AI-assisted research.
Program Positioning
This program does not rely on making the entire SRI dependent on a single mathematical problem. It is an example of research capability demonstrating how a research system, including AI, can manage hypotheses, calculations, proof candidates, counterexamples, and revisions for high-difficulty, long-term, interdisciplinary unknown problems.
Current Research Path
- Defining structure and positivity at finite depth
- Identifying constraints preserved and propagated across depths
- Investigating connection conditions with structures in complex analysis
- Explicitly stating missing conditions required to reach rigidity on the critical line
- Separating computational evidence from mathematical proof records
Boundary of Claims
SRI does not treat preprints, finite range computational results, or working hypotheses as proofs of the Riemann Hypothesis. Public deliverables explicitly state the version, publication date, DOI or repository URL, verification status, and unresolved gaps.
Published Results
Papers, preprints, technical notes, diagrams, and computational Artifacts are managed in the Publication Registry. Critiques, recalculations, counterexamples, and alternative paths from external researchers are welcome.
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Mathematical Background
The Riemann Hypothesis is a mathematical unsolved problem stating that the non-trivial zeros of the zeta function lie on the critical line in the complex plane. While deeply related to prime number distribution, emphasizing only its importance does not constitute a research path. In this Program, we clearly distinguish between structures verifiable by finite objects and proofs required for infinite/analytic objects.
Canonical Proof Chain
The current research path connects Pick positivity at finite depth, constraint propagation across depths, analytic continuation, and rigidity on the critical line. For each link, we prepare definitions, lemmas, dependencies, candidate counterexamples, and computable checks, ensuring that progress to the next stage is not based merely on intuitive similarity.
Scope Guaranteed by Finite Computation
Finite matrices, interval arithmetic, numerical searches, and certificates can guarantee properties within specified ranges and precision. However, success in a finite range does not constitute a proof for the infinite general case. We record numerical errors, search ranges, implementations, and dependent libraries, assigning a Claim Level separate from analytic claims.
Publication and Artifact
We manage research artifacts not only through paper bodies but also through definition tables, dependency graphs, computational runners, inputs, outputs, SHA256 hashes, audit notes, and known failure cases. If a new version is released, the old version is not deleted; instead, it indicates which claims were added, weakened, or retracted.
Points for External Verification
In particular, we focus review efforts on the transition from finite to analytic domains, conditions under which positivity is preserved, and additional assumptions necessary for rigidity on the critical line. We welcome not only favorable evaluations of SRI but also contradictions with existing theorems, circular reasoning, implicit assumptions, and counterexamples.
How to Read Current Results
Do not assume that a proof of the Riemann Hypothesis has been completed merely by reading the title or abstract of any Preprint. It is necessary to check the Claim Boundary, premises, distinction between finite and infinite domains, peer review status, and subsequent versions in the main body. SRI News serves as an entry point for conveying published facts; the basis for mathematical validity resides in Publications and Artifacts.
Role of AI-Assisted Research
AI assists in organizing existing literature, decomposing proof candidates, visualizing lemma dependencies, searching for counterexamples, and generating code. However, the local consistency of a long derivation and its overall validity are separate issues. We do not equate AI confidence with proof strength; this requires combining different models, formal checks, and review by human mathematicians.
Auditable Research Chain
We aim for a structure where every claim can trace back to referenced definitions, used lemmas, computational artifacts, and unresolved dependencies. If a paper is revised, the changes are shown as a difference (diff) link. This allows verification of specific parts rather than accepting or rejecting the entire body of research.
Next Milestone
The immediate tasks include refining conditions for extending properties from finite to analytic domains, establishing reproducible procedures for computational certificates, and preparing dependency diagrams for external review. Milestones are set as next verifiable units of work, not as announcements of proof completion.