← NS-X72 / X72-68 · Central Sign-Cone Coarse Bridge Target

NS-X72 · X72-68 Coarsening 2026-08

X72-68: Central Sign-Cone Coarse Bridge Target

Continuing from Round 67's approach of compressing the final bridge into a total-variation estimate for the scattering derivative Σ(ν) (STOP-C71), this round uses the exact central identity at n=1 to convert the minimal quantity to be proved, u₃=-a₃, into a sign problem for two O(1) neighboring quantities e₁=u₂ and o₂=u₅/ν: the large fiber requires only e₁<0∧o₂>0, and the small fiber requires only the very loose conditions e₁≤-0.1, 0≤o₂≤100 to guarantee u₃<0. Rigorous endpoint intervals show both fibers sit deep inside this sign cone, so only the extremely crude derivative bounds |e₁'|<10⁵ and |o₂'|<10⁶ are needed to close the entire segment 0<ν≤10⁻⁶; the actual fixed-size tangent diagnostic values are only about O(1)–O(10²), leaving three to five orders of magnitude of margin. This round does not prove this uniform coarse derivative bound; the gap shifts to STOP-C72, left for the next round; this round includes numerical verification scripts and CSV data.

This round's status: Coarsening — 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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