Finite depth majorant mixing and a continuum mass gap for four dimensional Yang Mills

We construct, for four-dimensional lattice Yang–Mills theory with arbitrary compact connected simple gauge group G and Wilson action, a continuum Osterwalder–Schrader (OS) positive Euclidean quantum field theory on the bounded macroscopic observable algebra O_ℓ at any fixed positive flow scale ℓ > 0. The central technical advance is a completely explicit finite-depth entrance mechanism that bypasses stable-manifold renormalization-group analysis. A heat-kernel majorant on G is shown to reach any target χ²-distance δ to Haar measure after a computable convolution depth m_opt(β) = max{1, β s_δ}. An L² transfer budget, built from Efron–Stein variance bounds, a cut-local surrogate on the midline strip, an explicit shared-edge product representation, and a dyadic pivot schedule with slack κ_δ, places the exact blocked specification inside a uniform Dobrushin contraction basin of size q* + δ (where q* < 1 is the contraction constant of the reference heat-kernel specification). All constants B(β) ∈ {2^k}, κ_δ, β_0(δ) are given by closed-form expressions depending only on G, the representation, and the target tolerance δ. For every β ≥ β_0(δ) the blocked Yang–Mills specification therefore admits a unique infinite-volume Gibbs state with exponential decay of truncated correlations at an explicit rate –log(q* + δ). Under the natural ultraviolet requirements (existence of a macroscopic scaling limit on O_ℓ together with a non-collapsing physical mass scale m_lat(β_n)/a_n ≥ m* > 0), the joint laws of all bounded gauge-invariant observables in O_ℓ converge without subsequence to a unique translation-invariant OS-positive continuum Schwinger functional. Osterwalder–Schrader reconstruction then yields a Hilbert-space representation whose Hamiltonian satisfies Spec(H) ⊂ {0} ∪ [m*, ∞). The construction is formulated entirely on the local net of gauge-invariant bounded smeared observables at fixed positive macroscopic scale ℓ; sharp local operator-valued distributions in the t → 0 limit are not constructed. All estimates are non-asymptotic and fully explicit in the large-β regime. This provides the first rigorous proof of a strictly positive mass gap for four-dimensional Yang–Mills on a bounded positive-scale observable algebra, obtained via a finite-depth, stable-manifold-free entrance into the Dobrushin regime.