Curvature-Function Mode Spectrum, Hidden-Branch Opening, and a New Finite-Mode Candidate

Round 14's two-peak and chirality scans covered only a handful of directions. This round instead uses eight orthogonal Legendre modes over cumulative-density coordinates to build a complete perturbation basis for the curvature function, and computes the pressure-projected Hessian along the four-branch equal-height manifold: the result is 7 negative eigenvalues plus 1 small positive eigenvalue (1.33×10⁻⁶) — so Round 13's single-peak candidate is not a strict local maximum within this eight-mode space. Ascending along the corresponding Newton direction, the original four controlling branches keep rising, but after about m≈2.2 a ninth local minimum appears, and by about m≈3.228 this new branch begins to take over the global objective — the ascent cannot be extended indefinitely; it is confined to a narrow window bounded by a “branch opening.” Taking the candidate at m=3.2275 gives s₁₅=0.998914480716946, an improvement of about 1.37×10⁻⁷ over Round 13, confirmed as stable across 5 resolutions (6001 to 96001 points). A full-phase audit (96001×262144) finds the number of local minima has grown to 9, and the global minimum is controlled by an almost exact tie between the 120° branch and the p3|p3|p1 branch (in the former 270° neighborhood), with a gap of only 2.78×10⁻¹⁵. The series concludes here, ending on the honest, unfinished state of “a narrow ascent window just before the fifth branch takes over,” handing off to the five-branch event system of the next round.