← NS-X72 / X72-70 · Dual Rank-One Riccati Kernel

NS-X72 · X72-70 Refinement 2026-08

X72-70: Dual Rank-One Riccati Kernel

Continuing from Round 69's conclusion compressing the first-order bridge into a single scalar scattering amplitude (STOP-C73), this round further proves, in parity-rescaled pointwise Riccati coordinates, that each local viscous tangent source is itself exactly rank one (nonzero only on the even layer, and exactly first order in ν), while the dual weight of the central scalar observable also exactly preserves rank one under pullback, yielding the "Dual Rank-One Scattering Kernel" identity, which splits the tangents of e₁', f', and o₂' exactly into a finite sum of scalar Jost kernels plus a single terminal pairing term. In numerical diagnostics over 10⁻⁸≤ν≤10⁻⁶, the absolute sum of the local kernel is only O(1)–O(10²), far below the sufficient bounds of 10⁵ and 10⁶ required by Round 68. This round does not prove the kernel's uniform summability or the decay of the terminal dual weight; the gap shifts to STOP-C74 (the dual-scalar-kernel-summability/terminal-weight gap), left for the next round to handle the majorant estimate; this round includes numerical verification scripts and CSV data.

This round's status: Refinement — 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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