From Coherence-Basin Statistics to a Local Quantum-Field Interface: Commutative Measures, the Selected Free CAR Net, and the Interacting Boundary

Abstract

This paper asks a narrower and more useful question than whether quantum field theory can be reconstructed from classical ensemble statistics. Let an admissible coherence-basin space carry a probability measure and bounded basin observables. We prove that these data have a canonical Hilbert-space representation by multiplication operators on an L^2 space. The representation is commutative. Convex mixtures, covariance kernels, Gaussian characteristic functionals, and classical coarse-graining follow within that layer, but canonical commutation or anticommutation relations, quantum interference, time ordering, a local net, and renormalized interactions do not. The correct connection to quantum field theory is an interface rather than an emergence-by-statistics theorem. A state on a quantum observable algebra restricts to a probability measure on every commutative record subalgebra; equivalently, a positive operator-valued measurement or instrument produces classical outcome statistics. Conversely, every state on a commutative subalgebra has extensions to the ambient C^*-algebra, but those extensions are generally nonunique. Identical basin statistics can therefore coexist with inequivalent quantum phases and correlations. This distinction can now be applied to the current Modal Triplet Theory (MTT) corpus. A companion paper owns the selected free-field construction: on the declared globally hyperbolic framed q79 representative, a twisted massless Dirac source composes with standard CAR/AQFT machinery to give an even local observable net with locality, covariance, the time-slice property, and a nonempty positive Hadamard state space. Basin and detector statistics may be read from commuting subalgebras or instruments of that net. The geometry-selected nonperturbative interacting gauge-BRST C^*-bridge, selected full-branch state, renormalization-group matching, and observable uncertainty packet remain open. The result is a precise classical-to-quantum interface and a closed selected free-QFT source, not a derivation of noncommutative QFT from probability alone.