0 over the entire segment. This round rigorously proves (not merely by sampling) the algebraic tail-coefficient bound for both fibers and the cubic-WKB envelope summation bound (<34, independent of viscosity), combining to give a source budget of <5.51×10⁴ν for n≥100 — but the "directional Green-factor cone" inequality needed to support this conclusion is itself not yet proved; the gap is STOP-C75, explicitly marked T_C→D=not yet reached. Section 16 of this round explicitly lists nine concrete to-do items for the next round (Directional Green Cone/Final Secant Closure) — including switching to a fast-Schur framework, proving the n⁻² algebraic factor, establishing post-transition exponential decay using Round 60's stable multiplier, certifying the first 99 layers, and eliminating the terminal secant pairing term — showing that, within the currently final round of the series, this route is still regarded as in progress, neither wrapped up nor abandoned, and is not a verdict of failure or a change of direction; this round includes numerical verification scripts and CSV data.">

← NS-X72 / X72-71 · Secant Riccati Linear-Viscosity Jost Bridge

NS-X72 · X72-71 Frontier (current) 2026-08

X72-71: Secant Riccati Linear-Viscosity Jost Bridge

Continuing from the difficulty left by Round 70, where the tangent kernel and the local source produce unnecessary cancellation at minimal ν (STOP-C74), this round instead compares the secant difference ΔR_n=R_n^ν-R_n^0 rather than the tangent derivative, proving the "Secant Riccati Volterra Identity" (an exact finite difference, with no Taylor remainder): the local source drops from O(ν) to O(ν²), and after summing over the transition layer of width ~ν^(-1/3), the central-graph displacement naturally yields R₁^ν-R₁^0=O(ν), consistent with the linear matching law observed numerically in Round 60; and, in the affine endpoint coordinates corrected in Round 69 (condition numbers of only 1.447 and 1.579), establishes the "Central Secant-Cone Bridge Lemma": if ‖R₁^ν-R₁^0‖≤8×10⁴ν holds for 0<ν≤10⁻⁶, then a₃,±(ν)>0 over the entire segment. This round rigorously proves (not merely by sampling) the algebraic tail-coefficient bound for both fibers and the cubic-WKB envelope summation bound (<34, independent of viscosity), combining to give a source budget of <5.51×10⁴ν for n≥100 — but the "directional Green-factor cone" inequality needed to support this conclusion is itself not yet proved; the gap is STOP-C75, explicitly marked T_C→D=not yet reached. Section 16 of this round explicitly lists nine concrete to-do items for the next round (Directional Green Cone/Final Secant Closure) — including switching to a fast-Schur framework, proving the n⁻² algebraic factor, establishing post-transition exponential decay using Round 60's stable multiplier, certifying the first 99 layers, and eliminating the terminal secant pairing term — showing that, within the currently final round of the series, this route is still regarded as in progress, neither wrapped up nor abandoned, and is not a verdict of failure or a change of direction; this round includes numerical verification scripts and CSV data.

This round's status: Frontier (current) — 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.

NS-X72 progress71 / 71

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