← NS-X72 / X72-09 · Fourier-Triad Phase Coherence

NS-X72 · X72-09 Counterexample 2026-08

X72-09: Fourier-Triad Phase Coherence

Continuing from Round 08's STOP-C12, this round substitutes the abstract transfer rate ϑ back into the actual Navier–Stokes Fourier-triad convolution, establishing the triad transfer kernel T = A sin Φ (amplitude × phase coherence), and proves geometric constraints including the No-Free-Radial-Jump lemma and vanishing contribution from collinear triads. The core result is the phase-sign flexibility lemma: with the triad geometry and modal amplitudes fixed, flipping only the relative phase Φ can change the sign of the transfer, so the frequency geometry and modal amplitudes alone cannot determine the sign — a third single-observable rejection theorem, X_{Γ_triad,amp}, in a restricted context. ζ_{τ,s} is accordingly rewritten exactly as a signed phase-coherence triad integral, stopping at STOP-C13 (the triad-phase-coherence/commutator-sign gap), handing off to the next round's direct study of the dynamics of the phase Φ itself.

This round's status: Counterexample — 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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