Haar Positivity: From Weil to Wightman

This paper identifies the positivity mechanism common to the arithmetic and quantum-field-theoretic sectors of the Shadow framework. The root object is a Haar-positive kernel: a function or distribution arising from a Haar convolution square P = Omega^v * Omega. Such kernels are of positive type and generate Hilbert spaces by the GNS construction. Four classical positivity conditions -- Weil's criterion for the Riemann Hypothesis, Wightman positivity for relativistic quantum fields, Osterwalder-Schrader reflection positivity in Euclidean field theory, and compact Haar projection onto gauge-singlet sectors -- are shown to be instances of this single construction, differing only in the ambient algebra and involution. The paper proves the abstract positivity theorems directly, states the Weil and reconstruction criteria in the form needed for comparison, and records how the Shadow framework uses each. The conclusion is precise: positivity is not imported into number theory from physics. Weil's criterion is already a Hilbert-space positivity condition. Quantum field theory recognizes the same structure under the names Wightman positivity and reflection positivity. Compact gauge theory recognizes it as Haar projection. One construction, four languages.