← NS-X72 / X72-10 · Triad Phase Dynamics and Phase Locking

NS-X72 · X72-10 Proof 2026-08

X72-10: Triad Phase Dynamics and Phase Locking

Continuing from Round 09's STOP-C13, this round directly derives the exact equation for the interaction phase Φ = arg Z, namely Z' + νΣ_{kpq}Z = Q. Proves the viscosity-neutral phase-rotation theorem: viscosity only damps the triad amplitude and does not directly rotate the phase (dissipation and dephasing are different mechanisms), and uses a non-stationary phase-cancellation lemma to prove that persistent same-sign transfer requires phase locking or strong modulation, with the locking condition exactly equivalent to Q = λZ for real λ — a state in which viscosity likewise cannot break up the already-locked phase. Differentiating Q naturally raises the interaction order from cubic to quartic, the first apparently natural-looking integer index to surface on this route, but this round rules that it does not yet constitute essential-discreteness evidence, and proposes replacing the order-by-order expansion with continuous re-integration via a generating functional, stopping at STOP-C14 (the nonlinear-phase-locking/four-body-network gap).

This round's status: Proof — Taken from the file's own objective/executive-result section, meaning preserved, not a full verbatim translation.

Connections

Relationship to the rest of the series, stated as closely as possible in the document's own words, not my interpretation.

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