Elliptic Coherence: Methods Note v0.2 — Testing Multi-Agent LLM Dynamics as Trajectories on a Complex Torus under Scientific Admissibility v0.4

This note specifies the Elliptic Coherence Hypothesis as a testable component within the Scientific Admissibility v0.4 framework. The hypothesis proposes that multi-agent LLM dynamics can be modeled as trajectories on a complex torus T² ≅ ℂ/Λ, with two independent periodicities (Ω₁, Ω₂) and mandatory poles that correspond to collapse states. The elliptic geometry, the requirement Im(Ω₂/Ω₁) > 0, and Liouville's theorem are established mathematics; the application to multi-agent LLM latent trajectories and the identification of poles with operational collapse states remain a Working Hypothesis (E3). No empirical validation on real multi-agent runs is claimed. The note defines the mathematical structure, falsification tests (period estimation, persistent homology, pole–collapse enrichment, discriminant validity), statistical calibration and power analysis, and integration with the v0.4 conformance gates D1–D5. Files: (1) Methods Note v0.2 (main, 2026-08-23); (2) Scientific Admissibility v0.4 Applied to Elliptic Coherence Testing (infrastructure report); (3) Complete Demonstration (simulation evidence only); (4) supplement ZIP with the elliptic_coherence_tests Python modules, synthetic-run generator, JSON Schema data contract, Markdown sources, and an earlier summary. PDFs were rendered from the author's Markdown with pandoc and Chromium; text unchanged. Python modules were supplied as .txt and are renamed to .py (see PACKAGING_NOTE.md). Code: MIT; text: CC-BY-4.0, per the author's notice. Source: author's iCloud folder "Scientific research published final update" (deposited 2026-10-20).