Projected Heat Kernels from MTT Fixed-Point Data Conditional Sector Identities and the Physical-Replacement Boundary

This paper isolates the exact projected heat-kernel statement available from declared fixed-point data. On a specified compact Riemannian internal space, Euclidean problem, or spatial Cauchy-slice Hilbert space, a nonnegative self-adjoint elliptic operator A, a reducing orthogonal projector P, and a positive proper-time parameter determine a unique projected heat operator. We prove smoothness in the elliptic spectral setting, covariance under unitary equivalence, coherent-band error estimates, and the joint distributional limit to the full identity. If AP is zero, the projected heat operator equals P, so proper time creates no additional smoothing on the retained sector. Gap damping applies to the complementary evolution, while the projected operator annihilates that complement. These facts do not authorize a universal replacement of spacetime deltas, canonical commutators, gauge constraints, or local vertices. Such a replacement is a new physical model unless a selected realization supplies the operator domain, operator, projector, and scale, and separately proves locality or causal support, covariance, gauge or BRST compatibility, and the relevant algebraic identities. The result is therefore a conditional execution interface, not a derivation of a universal MTT regulator.