Positivity of the 2I∗-projected Hessian of the doubled fuzzy sphere: coupling structure and Schur complements at one loop

We compute the 2I∗-invariant sector of the one-loop Hessian around the doubled fuzzysphere saddle point of the bosonic SO(3, 3) matrix model, by exact numerical diagonalisationat matrix sizes N = 7, 9, 13. Three results are established and two statements in the companionpapers are corrected. (i) The Hessian restricted to the 2I∗-invariant fluctuation sector ispositive semidefinite; at N = 13 its nonzero eigenvalues are exactly 23k(k+2) for k = 12, 20, 24, reproducing the Kaluza–Klein spectrum of P3 = S3/2I∗. No negative mode survives the projection.(ii) The invariant sector does not couple to any flat direction of the Hessian (coupling≤ 10−13), and the Schur complement obtained by integrating out all positive non-invariantmodes remains positive semidefinite. (iii) Under contour rotation of the negative directions— the Gaussian implementation of the Bromwich prescription — the fully reduced effectivequadratic form on the invariant sector is positive definite, and the resulting Gaussian process isreflection positive: the reconstructed one-dimensional theory is a unitary quantum mechanicswith self-adjoint H ≥ 0. Reflection positivity moreover forces the contour rotation: with anyunrotated negative mode the Osterwalder–Schrader inequality [2] fails. Corrections: the projectiondoes not commute with the Hessian, contrary to a companion claim; and the invariantsector couples linearly and strongly to the negative (“evanescent”) modes, so their decouplingis not structural but depends entirely on the contour prescription. As a by-product wefind that integrating out the complement strongly renormalises the Kaluza–Klein levels; thegolden-ratio mass relations quoted in the companion papers hold for the unmixed spectrumonly. All numbers are reproducible from the ancillary script.