Introduction: the construction and its dependencies

Introduction: the construction and its dependencies

The starting point is a finite probability law: group elements live on oriented lattice links, and the Wilson action measures each plaquette’s departure from the identity. The problem addressed by the supplied construction is to control one family of these laws while the mesh tends to zero and the physical period subsequently tends to infinity. Inserting local gauge-invariant functions into that same family is meant to produce a compatible hierarchy, rather than unrelated limits for different observables. The public theorem in Section 2 states the exact common-family and subsequence quantifiers.

The derivation proceeds through a finite linear model, an actual nonlinear integration, and two successive continuum limits. The quadratic model provides local inverse kernels and stationary response maps. Nonlinear charts compare group links with that model, while positive energy bounds pay for exceptional configurations. Marked local activities keep products of insertions attached to the same represented density. An exact reset identity and finite coupling shooting select calibrated histories on one calendar. Detector estimates then separate physical label classes and identify the equation image before the common continuum hierarchy is formed. These constructions supply the estimates and compatibility statements used in the limits and reconstruction.

Reflection positivity and slab bounds provide Hilbert-space control. The ground compression algebra supplies the phase space and its measure; tube continuation and tempered boundary limits supply fields on one common dense domain. The final part returns to actual finite current and stress functions. Their local equation estimates, coefficient calibration and Gaussian comparison determine a conserved physical stress and the centered trace. Positivity upgrades comparison along arbitrary retained scaling sequences to the full positive-distance trace limit. That limit supports the precise comparison exclusions stated in the public theorem.

Source and the two-paper release. The mathematical source is the expanded project extraction, Mathematical Translation [1], prepared with AI assistance. It presents the Yang–Mills construction associated with the Lean work credited to OpenAI. This paper develops that source as a complete mathematical exposition. Its definitions, hypotheses, estimates and derivations are included here, together with the formula amendments in Appendix A.

The accompanying flagship paper, Shadow Theory and the Wilson Yang–Mills Construction: A Source–Readout Reconstruction [2], applies Jeremy Rodgers’s Shadow Theory, source–readout and Source Lift programme to this construction [3]. Rodgers originated and directed that programme; AI assisted its execution and the preparation of the two expositions. The papers share the Zenodo release 10.5281/zenodo.24173681. Each contains its complete mathematical development: this paper presents the construction, and the flagship develops its source–readout realization and recovery. Their common subject does not require a reader of either paper to obtain an essential derivation from the other.

The given gauge group, faithful representation, four-dimensional lattice, Wilson action and calibration conventions are physical and mathematical inputs. The detailed construction supplies stress Ward identities, nested expansions and contact extensions on its common witnesses; the public theorem retains its original list of clauses.

How to read the argument. Read the finite setup and conventions first, especially the healthydomain derivative and support budgets. The quadratic section explains the aliases, metric amendment and filled stationary predictors. The nonlinear and marked-integration sections use those objects to prove an actual density identity. The long scale-history section then buys every fixed finite request in a specified order, and distinguishes a full Cauchy limit from an auxiliary extraction.

The label, shell-charge and quotient sections are needed for reconstruction and stress detection. The OPE, contact, Gaussian and trace sections retain their own finite horizons, comparison devices and domains. The closing assembly returns to the theorem and identifies all its witnesses on one family and one refinement.

Background. The finite law is Wilson’s lattice formulation [4]; compact group integration uses Haar measure [5]. Wick pairings [6], reflection positivity and Euclidean reconstruction [10, 11, 12], chessboard inequalities [13], positive-functional representations [7, 8], and the character-algebra approximation [9] form the mathematical background. Renormalization and cluster methods are described in [16, 17, 18, 14]; the field-theoretic framework is that of [15], and the conventional one-loop coefficient is associated with [19, 20]. These references credit the background; the actual hypotheses, constants and derivations used here are stated locally.

Accessible table transcription

Table transcription, source page 4

Notation	Role in this paper
G ⊆U(m)	Given closed group and faithful matrix representation; r = dim is
	R g
	the color dimension.
−1
h d, L	Physical fine mesh and physical period in the public family.
N =N p
	−1
ai, Li, ti	Intermediate spacing, integer block ratio, and ti = log(a).
	i
b
βi, βi	Prescribed reference clock and actual represented coupling; their com-
	parison is proved in the running argument.
θ, θ	Healthy-field thresholds; the local text specifies which schedule or fixed
	chart is being used.
Fd, Ed	Finite physical label filtration and coefficient spaces; the relevant equa-
	tion quotient is specified before use.
χ
χ, S, S	Character phase, its separated hierarchy, and the unselected barycenter
	hierarchy. A scalar cutoffalso denoted χ is always a function with an
	explicit position argument.
T, Θ	Conserved physical stress and its selected-vacuum centered trace. Finite
ab
	candidates are defined separately.