Positivity Before the Limit: Reflection Positive Neural Network Ensembles That Converge to P(φ)₂ in Finite Volume

Version 1 (3 October 2026). Comments and corrections are welcome at cody@frg.earth; corrected versions will be posted here. A neural network that depends continuously on finitely many random parameters, has real-analytic time lines and has a law invariant under time translations and time reflection is not reflection positive unless it is static, and a recent construction of the finite-volume φ⁴₂ measure from networks leaves open whether such a construction can be reflection positive at finite width or at finite cutoff. We show that it can, in two dimensions and in finite volume, in three senses. First, a regulator in the continuum with inverse propagator m² + w² + q² + a²w²q² gives Gaussian fields of finite variance that are reflection positive across both coordinate lines; the regulated P(φ)₂ measures in a box are reflection positive at every value of the regulator, and their densities converge in Lᵖ for every finite p, and their Schwinger functions and generating functionals converge, to those of the finite-volume P(φ)₂ measure, with errors of order a^(1/2). Second, an ensemble of resampling neurons, cosine features in space that are redrawn at random times, has the regulated two-point function at every width, and reweighted by the interaction it is reflection positive in time at every width and every regulator. We prove a rate for the width limit and a version of Nelson's bound at finite width, with constants that do not depend on the regulator once the width exceeds a power of log(1/a), and conclude that these ensembles converge to the finite-volume P(φ)₂ measure, in law and in all moments, along a width that is a power of log(1/a). Third, products of telegraph neurons and neurons built on the Mondrian process are reflection positive across both coordinate lines at every width. Without an interaction their two-point functions lie in a cone that stays at a positive distance from the free propagator in the supremum norm. With a mass counterterm in the interaction on an outer box, and a quartic term outside the box of the interaction, which keeps the weights uniformly integrable, they converge to the finite-volume P(φ)₂ measure in an iterated limit, of the width, of the outer quartic coupling, of the outer box and of the regulator in this order, with both positivities at every stage. For the quartic interaction, in the variant of this construction without outer box, the reweighted ensemble separates from its own width limit along every sequence of widths in an explicit range, from a multiple of a⁻² to a multiple of a⁻⁴. Finitely many frozen cosine features, as in the existing construction at a finite cutoff, are not reflection positive at any width, for almost every draw of the features. The interacting measures are in a fixed box, positivity with an interaction is proved for the reflections through its center, and the sufficient widths are not claimed to be sharp. Exact enumeration, Monte Carlo sampling and machine-checked scalar steps test the statements where a test is possible.