Compact Rigidity, Ordered KMS–Plancherel Transfer, and Reconstruction for Generalized von Mangoldt Heat Kernels

We study arithmetic heat transforms on normed free commutative monoids under Gaussian summability of their primitive lengths, without assuming an ordinary Euler-convergence half-plane. Divisor incidence determines the von Mangoldt selector even among entrywise matrix solutions. A semifinite KMS weight and a noncompact Plancherel carrier then give ordered heat evaluations with an exact absolute product-domain criterion: the unsliced object is integrable precisely where the ordinary Gibbs normalization is finite. The heat potential has an entire trace-class lift, and the corresponding connected sum and full-state Gaussian trace have the same finiteness domain. An explicit example satisfies all-time Gaussian summability but no positive ordinary Euler convergence. Compact recurrence and a represented Toeplitz-algebra obstruction explain why the selected heat operators require the stated ambient setting. A local extension to proper cancellative monoids relates positivity of the incidence selector to unique factorization, with an explicit two-torsion exception. For the inverse problem, classical Dirichlet-series uniqueness combined with dilation–Möbius inversion determines the normed free monoid from arbitrary exact samples approaching zero heat time, at known temperature. Unknown temperature has explicit countable ambiguity fibers; a second, occupation-sensitive channel gives joint recovery at fixed normalization and length units. Finite polynomial certificates recover selected integer counts with controlled errors, while global reconstruction remains unstable in total variation. The construction is unconditional and finite-place; its contour comparison retains an Euler-convergence hypothesis and makes no zero-side claim. The accompanying computational archive implements a reviewed, conditional three-sample certificate for selected primitive counts, with rigorous interval arithmetic, examples, reproducibility tests and explicitly inconclusive outcomes. The worked data are synthetic. Internal mathematical and software reviews are not external human peer review or a guarantee of universal software correctness.