Companion II: Lane A Sharp-Local Construction

Description

This document provides the complete proof home for the Lane A constructive chain, establishing the existence and structure of the sharp-local gauge-invariant Yang–Mills field algebra used in the main theorem.

It constructs, in fully explicit theorem-level form:

• the flowed continuum state,
• the bounded positive-time base state,
• the exact-dimension quotient formalism,
• the finite-truncation inverse-control package,
• the finite-cap mixed-correlator closure,
• the finite-cap sharp-local extension with OS structure,
• the inductive-union passage to the full local algebra,
• and bounded-base cyclicity in the reconstructed Hilbert space.

The result is a rigorously defined sharp-local Yang–Mills theory, expressed entirely in terms of gauge-invariant local observables generated by flowed curvature composites.

Role in the Overall Proof

This companion supplies the constructive field-theoretic core of the argument.

Within the global architecture:

• Companion I provides the lattice-side mass-gap and transport chain.
• Companion II constructs the continuum local theory on which that gap acts.
• Companion III performs reconstruction and endpoint identification.

The full proof chain is:

Main Technical Content

The Lane A chain is implemented as a strictly ordered constructive pipeline:

1. Flowed State and Positive-Time Base

• Construction of a unique flowed continuum state
• Dyadic convergence of Wilson expectations
• Definition of the bounded positive-time algebra

2. Exact-Dimension Quotient Structure

• Canonical quotient blocks for local operators
• Basis-independent coefficient extraction
• Elimination of representation-dependent ambiguities

3. Finite-Truncation Control

• One-shell transport relations
• Block-lower-triangular structure of coefficient maps
• Polylogarithmic inverse bounds on finite truncations

4. Finite-Cap Closure

• Mixed-correlator closure at fixed engineering cap
• Positive/unital functional construction
• Explicit reflection-positivity mechanism

5. Sharp-Local Extension

• Construction of a finite-cap sharp-local state
• Verification of Osterwalder–Schrader structure at finite cap
• Realization of renormalized local fields

6. Inductive-Union Completion

• Consistent gluing of finite-cap states
• Definition of the full sharp-local algebra
• Preservation of OS axioms on the inductive limit

7. Bounded-Base Cyclicity

• Density of the bounded positive-time algebra
• Explicit construction of approximating sequences
• Completion of the Hilbert-space structure

Formal Verification Layer

The Lane A construction is integrated into a Lean4 verification framework that certifies:

• the dependency structure of the constructive chain,
• the ordering of finite-cap → inductive-union → cyclicity steps,
• the separation of quotient, transport, and closure components,
• and the absence of circular dependencies.

The Lean layer operates relative to an explicitly imported Wilson QFT substrate, and verifies the theorem-level closure structure of the constructive pipeline.

Scope and Boundaries

This document:

• constructs the local gauge-invariant Yang–Mills field algebra,
• establishes its OS-consistent state structure,
• and provides the only constructive input used in the mass-gap theorem.

This document does not:

• prove the lattice spectral gap (Companion I),
• perform Minkowski reconstruction (Companion III),
• or address nonlocal operator sectors.

All statements are confined to the bounded-region local algebra generated by flowed curvature composites.

Structure

The proof is organized as a state-construction ledger, in which each step:

• introduces no hidden functional inputs,
• preserves explicit positivity control,
• and feeds directly into the next closure node.

No additional state-construction or positivity mechanisms are used outside this explicit chain.

Significance

This companion resolves the central constructive problem:

the explicit realization of a continuum Yang–Mills local field algebra with controlled positivity, locality, and reconstruction properties.

It provides a complete bridge from lattice observables to a rigorously defined local quantum field theory.