AI-Guided Exploration of Arithmetic Dynamical Systems: Constraints, Failure Modes, and Search Strategies Toward Hilbert--Pólya Structures

Recent advances in AI-assisted mathematical discovery, program search, and formal reasoning motivate a practical design question: how can a long-horizon open problem be reorganized into a broad, falsifiable, and evidence-accountable human–AI search? We present a prospective candidate-engineering framework, grounded in a source-bound retrospective Phase-I design case on arithmetic dynamical structures relevant to the Hilbert–Pólya programme. The paper does not claim a proof of the Riemann Hypothesis, a completed Hilbert–Pólya realization, or a spectral identification of Riemann zeros. The framework has three layers. In Layer 1, a mathematician fixes a mathematical origin, permitted data, and an admissible transformation class; the present case uses a prime-symbolic origin, whereas a new mathematician-supplied basic idea begins a new Layer-1 record. In Layer 2, mathematician and AI jointly draft and version-freeze an AGENTS.md contract specifying candidate identity, lineage, gates, controls, portfolio policy, and stopping rules. In Layer 3, bounded AI work returns a source-linked evidence handoff for mathematician review. The accompanying roadmap requires a single candidate to establish an endogenous arithmetic source, orbit/repetition ownership, and a same-object zeta or determinant before stronger analytic and operator obligations are considered. The case comprises one early exploration entry and five system-family sessions. Its P1 Wiki retains catalogued provenance records rather than a common score of completed papers, theorems, or Route advances. No source-bound top-level direction documents an integrated same-object A0+A1+A2 chain in one natural candidate. We therefore motivate a broader but lineage-filtered candidate portfolio, screened early for arithmetic, clock, and determinant ownership rather than assembled retrospectively from partial successes. The associated Wiki is a provenance and navigation resource designed to preserve positive local results, controls, scoped failures, and reuse boundaries for subsequent researchers.