A Counterexample to a Linear-Junta Inverse Theorem for Spherical Triangles

We construct smooth, real, mean-zero, 1-bounded functions on high-dimensional spheres whose average product over zero-sum unit triangles tends to 7/16384, while their correlation with every function of m linear coordinates is O(sqrt(m/d)) when m=o(d). The bound is uniform over all choices of coordinates and all test functions of L2 norm at most one. The construction is a centered cosine of a balanced, rescaled quadratic form. A conditional coupling proves the projection estimate, and a multivariate Gaussian limit gives the positive triangle average and a nonzero limiting variance. This supplies a complete counterexample to the dimension-free quantitative interpretation of Dmitrii Zakharov's spherical-triangle inverse question, AIM Problem 2.4 / UnsolvedMath AIM-COMBINATORICS-0007. It does not settle the related zero-sum-free hemisphere conjecture or the planar and finite-field analogues. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.