Companion III: Reconstruction, Non-Triviality, and Faithful-Wilson Universality

This document provides the complete proof home for the endpoint packet of the Yang–Mills mass-gap argument. It establishes, in fully explicit theorem-level form: the Euclidean Osterwalder–Schrader (OS) dossier for the sharp-local theory, the OS/Wightman reconstruction and Haag–Kastler local net, the field correspondence between constructed observables and local quantum fields, the Minkowski vacuum spectral gap transfer, non-triviality of the theory (nonzero spectral weight), and the exact bounded-region local-net endpoint. It also proves qualitative faithful-Wilson universality: all faithful Wilson lattice regularizations of the same compact connected simple group GGG yield canonically isomorphic continuum local theories. Role in the Overall Proof This companion supplies the final closure layer of the argument. Within the global architecture: Companion I establishes the lattice spectral gap and transport. Companion II constructs the continuum sharp-local theory. Companion III reconstructs the physical quantum field theory and identifies the endpoint object. The full chain is: Route 1 gap (Companion I) → Sharp-local construction (Companion II) → Reconstruction and endpoint (this document) Main Technical Content The endpoint packet consists of four tightly coupled components: 1. Euclidean OS Dossier Verification of Euclidean invariance, symmetry, and reflection positivity Establishment of tempered Schwinger distributions Construction of the OS Hilbert space with dense positive-time algebra 2. OS/Wightman Reconstruction Analytic continuation to Minkowski space Construction of Wightman fields and vacuum representation Generation of the Haag–Kastler local net Uniqueness of the reconstructed theory up to unitary equivalence 3. Mass Gap Transfer and Non-Triviality Transfer of the spectral gap from the OS Hamiltonian to the Minkowski Hamiltonian Proof of vacuum uniqueness Explicit non-triviality witness (nonzero spectral measure) Exclusion of the trivial (identity) theory 4. Exact Local-Net Endpoint Identification of the continuum theory as a bounded-region local gauge-invariant net Demonstration that local observables are precisely gauge-invariant curvature polynomials Proof that extended-support line and surface operators lie outside the theorem-level algebra Faithful-Wilson Universality The document proves that: all faithful finite-dimensional Wilson regularizations of a fixed group GGG, within the admissible renormalization class and coherent normalization, produce canonically isomorphic continuum local theories. This universality is qualitative: the local field algebra and vacuum structure are invariant, explicit quantitative bounds remain indexed by the chosen regularization. Formal Verification Layer The endpoint packet is integrated into a Lean4 verification framework that certifies: consistency of the reconstruction pipeline, correct dependency of endpoint claims on Lane A and Route 1 inputs, non-circularity of the reconstruction and gap-transfer chain, alignment between manuscript statements and theorem-level outputs. The Lean layer operates relative to an explicitly imported Wilson QFT substrate, and verifies the closure structure of the endpoint assembly. Scope and Boundaries This document: completes the construction of the physical Yang–Mills quantum field theory, establishes its local net structure and vacuum sector properties, and provides the final theorem-level identification of the model. This document does not: prove the lattice spectral gap (Companion I), construct the sharp-local algebra (Companion II), or include nonlocal operator sectors in the theorem claim. All results are confined to the bounded-region local gauge-invariant algebra. Structure The proof is organized as a final theorem packet, where: all inputs are explicitly traced to earlier companions, all reconstruction steps are standard and isolated, and the endpoint object is uniquely identified. No additional assumptions or hidden constructions are introduced at this stage. Significance This companion completes the program by showing that the constructed object is: a non-trivial, local, relativistic quantum field theory with a strictly positive vacuum mass gap. It provides the final bridge from constructive analysis to the standard axiomatic framework of quantum field theory.