Collapse Kernel Tomography: Admissible Classicality, Convex No-Go Certificates, and a Screened Gravitational Test Kernel

Objective-collapse models are usually tested one model at a time: a spatial noise kernel is proposed, its parameters are constrained, and surviving variants are then modified by changing the kernel, the temporal spectrum, or the dissipation mechanism. We formulate a complementary spectral framework, collapse kernel tomography, in which stationary Gaussian mass-density collapse processes are represented by a non-negative spectral measure S(k, omega) and each experiment supplies a linear response functional. Null experiments define upper half-spaces; classicality requirements define lower half-spaces; and Farkas-type dual certificates can rule out entire spectral classes rather than individual parameter points. We show that lower requirements must first pass an admissibility test relative to the reference collapse law they are meant to constrain: a pointer criterion that is not satisfied by the maximal reference law cannot consistently be used as a ceiling on a bounded deformative subfamily. Applying this logic to a screened gravitational test kernel bounded above by UV-smeared Diosi-Penrose collapse removes the single-layer graphene-disk criterion as an admissible ceiling and yields a criterion-dependent silica-sphere frontier m_star = 1.683e-14 kg for l = 100 nm, Delta x = 2R, and Gamma_req = 100 per second. For the same kernel we derive a rigid-center-of-mass heating ceiling, reducing to 2 pi G hbar rho for a homogeneous bulk body; for silica this is 9.73e-41 W. A benchmark screened kernel with kappa = 2.1886e6 per meter and l = 100 nm predicts an order-one suppression relative to UV-smeared DP in the 1e-14 to 1e-13 kg probe range. Finally, we propose form-factor matched filtering using layered test masses, with the design rule p of at least 4l for usable and 8l for strong density-modulation contrast. The result is a convex, falsifiable geometry for collapse searches rather than a single-model parameter scan.

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