Companion I: Ultraviolet Gate and Route 1 Mass-Gap Chain

This document provides the complete proof home for the ultraviolet/public-scope gate and the Route 1 lattice-to-continuum mass-gap chain used in the main manuscript. It establishes the full lattice-side burden of the Yang–Mills mass-gap argument, including: the compact-simple ultraviolet gate (A1 input and group-scope closure), the one-shot entrance theorem into the KP/Dobrushin contraction regime, the same-scale Wilson-to-patched transport theorem, the finite-depth transport control and de-patching stability, the fixed-lattice vacuum spectral gap (QE2lat), and the continuum vacuum-gap transfer (QE3). The Route 1 chain is presented in fully explicit theorem-level form, with all load-bearing steps isolated into a finite set of closure nodes. Role in the Overall Proof This companion supplies the entire lattice-side and transport component of the proof. Within the global architecture: Companion I provides: the ultraviolet gate, the entrance mechanism, the fixed-lattice gap, and the transport of that gap to the continuum. Companion II supplies the constructive sharp-local field theory. Companion III supplies reconstruction and endpoint identification. The continuum mass gap is obtained by combining: Route 1 (this document) → Lane A construction (Companion II) → OS/Wightman reconstruction (Companion III) Main Technical Content The document proves a complete three-stage closure chain: 1. Ultraviolet Gate and Entrance Compact-simple A1 theorem One-shot entrance into the KP/Dobrushin basin Explicit control of contraction constants and physical scale 2. Fixed-Lattice Gap Exact conditional-expectation formulation of Wilson recursion Chart-truncated RG comparison with quantified error bounds Finite-depth stability of observables Establishment of a full fixed-lattice spectral gap 3. Continuum Gap Transfer Same-scale Wilson return (non-perturbative equivalence) Dyadic OS transport Non-collapse of the spectral gap Construction of a continuum vacuum gap Formal Verification Layer The results of this companion are integrated into a Lean4 verification framework that certifies: the Route 1 dependency structure, theorem-level closure ordering, separation of measure-theoretic, RG, and transport components, and non-circularity of the transport chain. The Lean verification operates relative to an explicitly imported Wilson lattice QFT substrate. Formalization of the Wilson measure and associated standard results is outside the scope of the verification layer. Scope and Boundaries This document: establishes the vacuum gap at the lattice and transport level, provides the only lattice-side input used in the full proof, and isolates all weak-window and RG-dependent assumptions into explicit nodes. This document does not: construct the continuum local field algebra (Companion II), perform OS/Wightman reconstruction (Companion III), or claim results for nonlocal operator sectors. Structure The proof is organized as a theorem-facing transport pipeline, with: a fixed measure-theoretic ledger (exact conditional expectations), a finite-depth transport architecture, and a single continuum transfer node. All load-bearing steps are localized and explicitly identified. Significance This companion isolates and resolves the most difficult component of the Yang–Mills mass-gap problem: the non-perturbative transport of a lattice spectral gap to the continuum theory. It does so with explicit control over: measure-level approximations, RG truncation error, and finite-depth stability.