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NS-INRS · DCRP52 Structural Characterization 2026-08

DCRP52: Null-Hessian Envelope Integrability and the X72 Lift

Building on the rank-one null-Hessian survivor cone DCRP51 obtained on the positive pseudo-eikonal sector (C>0), and drawing on X72 Round43's vorticity-stress nonlinear realizability cone (STOP-C47), this round asks whether Hessian integrability eliminates this null cone. It proves that it does not: every non-affine block is exactly a developable-envelope local normal form (its gradient curve lies on the pseudo-eikonal hyperboloid and moves along a wave-null direction), so unconditional local affine rigidity on the positive-C sector is false, and this envelope still satisfies X72 vorticity-stress realizability pointwise. But every non-affine realization necessarily produces a finite caustic/transition frontier. STOP-D52 hands the problem to DCRP53: can same-family continuation cross this finite transition set?

This round's status: Structural Characterization — Taken from the file's own Executive result / STOP node, meaning preserved, not a full verbatim translation. The original paper itself was authored directly in English; only this page's own commentary (title, status, and summary) is translated here.

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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