From Resonance to Rigor: A Longitudinal Case Study of Multi-Model LLM Hypothesis Generation in Number Theory

We present a longitudinal case study of a multi-model LLM orchestration system used as a hypothesis-generation engine for research-level number theory. The system — documented in three prior preprints (dimension-direct routing; rule-augmented skill orchestration achieving 202/202 on a 50-task agentic benchmark; and a dual-mode cognitive architecture) — separates a *Deliberation Mode*, which uses multi-model cross-validation to suppress hallucination, from an *Exploration Mode*, which channels multi-model divergence into structured hypothesis generation. Between April 9 and May 7, 2026, the system was deployed against the arithmetic of prime-degree isogenies of elliptic curves over $\mathbb{Q}$: 43 recorded multi-model runs (302 skill executions, 1,682 verification-tool calls) in the case-study corpus (BSD line plus resonance-methodology support), within a broader 95-run full corpus. The campaign contributed to three number-theory preprints with verifiable computational certificates, including a directional inequality on Birch–Swinnerton-Dyer (BSD) valuation flows ($a_\varphi S_\varphi \le 0$; 23,494 isogeny edges, zero violations), a Selmer-ratio mechanism ledger (80/80 local BSD ledger certifications; 62/62 exact-Selmer audits), and a visible residual transport identity (33/33 explicit targets matched; 18/19 height-diagnostic targets matched; 14/14 nonzero-residual core edges certified). We annotate 13 core hypotheses produced by the exploration corpus with resonance degree, downstream fate, and an echo-versus-extrapolation classification. Within the annotated registry, strong-resonance hypotheses survived at a higher descriptive rate (6/7) than the remaining hypotheses (2/6), and the surviving core is dominated by statements not located in prior literature (6 of 7 true-extrapolation items survived). We document the division of labor observed in practice — deterministic computation discovered the empirical pattern; exploration mode supplied the interpretive frameworks and mechanism conjectures that structured the follow-up papers — and we quantify, via this paper's own drafting process, the residual fabrication behavior of the deliberative mode when evidence is incomplete (Appendix B).