The φ-Selection Conjecture: A Dynamical Selection Principle for Stationary-Action Configurations
Abstract
VERSION 2 (2026-09-26) — what changed since v1, reported honestly: P1 HAS RUN, AND F1 FIRED. The φ-selection conjecture is FALSIFIED in its tested form. In the two-harmonic standard map (a=0.5), φ⁻¹ ranked 8 of 17 rotation numbers by KAM-breakup threshold (Kc=0.3700); the winner was a near-rational, R1=[0;5,1,1,1,…], at Kc=0.6030 — a margin ~2400× the bisection error, stable under doubled resolution. Per the preregistered decision rule, this is a kill, not a null. The conjecture died on its first real test, one afternoon after it was posted. That is the system working as designed. WHY R1 WON (exploratory follow-up, not preregistered): R1 won by POSITION, not by irrationality. The second harmonic directly drives the 1/2 resonance, creating a giant island (measured half-width ≈0.176√K) that massacres everything near ω=0.5 — including φ⁻¹ at 0.618. Across all 17 rotation numbers, breakup threshold is predicted almost perfectly by distance from the two directly-driven giant resonances {0/1, 1/2}: Spearman ρ=+0.883, p<10⁻⁴. The Diophantine "irrationality selects" measure gives ρ=+0.069, p=0.80 — no correlation. (At a=0 the irrationality measure DID correlate, ρ=+0.556, p=0.025 — the second harmonic erased it.) R1's "weakness" (leading 5 in its continued fraction; Hurwitz 0.178 vs 0.382 for φ⁻¹) parked it at ω≈0.178, in the widest surviving gap. Every rank-faller sits in ω∈[0.36,0.62]; every rank-riser sits outside [0.30,0.73]. HYPOTHESIS FOR FUTURE WORK (not claimed): the survivor occupies the largest gap of the ACTUAL DRIVEN resonance web — which coincides with φ in the single-harmonic case (explaining why the original KAM anchor, Instance 0, found φ on top at 0.9717) and moves when the web is reshaped. Needs its own preregistered test. WHAT SURVIVES: the Instance-0 measurement (φ uniquely topped the single-harmonic map — untouched), the T1–T8 null battery on φ in masses and mixing angles, and layers 1–5 of the TOE architecture sketch. Files: Phi-Selection-Conjecture.md (updated: falsified status + P1 outcome + R1 analysis), TOE-Tuning-Program.md (the null battery, unchanged), R1-Upset-Analysis.md + r1_upset.png (exploratory report and figure). --- v1 description follows (2026-09-26, original posting): A falsifiable dynamical conjecture: of the stationary-action configurations available to a near-integrable Hamiltonian system, the ones that persist under perturbation are those whose characteristic frequency ratios are the most irrational (φ-closest). Mechanism: KAM/Diophantine theory — the golden mean is the hardest irrational to approximate (Hurwitz), so resonant configurations break first and the system is selected down to φ. Includes explicit domain restrictions, three falsifiers (F1–F3), one supporting instance (golden mean uniquely topping the KAM-breakup threshold at 0.9717 across 16 rotation numbers, apparatus validated against Greene's 0.97163540631), and a preregistered Test P1 (two-harmonic standard map, 17 rotation numbers, Greene residue criterion) — NOT YET RUN at v1 posting. Companion file records the preregistered null battery (T1–T8) that exhausted the spectral/flavor forms of φ-tuning in particle masses and mixing angles. Status at v1: conjecture, not result. Formalized with AI assistance (Melody, Muse Spark); all tests were preregistered before running and are reported as run.
