Constraint-Driven Closure and the Yang–Mills Mass Gap

This capstone bridge note (Paper 6) closes the Route 1 series by supplying a minimal, model-independent bookkeeping layer that makes explicit the logical structure of any mass-gap statement. It formalizes the notion of an admissibility datum consisting of (i) a declared observable scope Σ (here the bounded macroscopic net O_ℓ at fixed positive resolution ℓ > 0) and (ii) a declared invariant package I (here the four macroscopic invariants (M1)–(M4): OS positivity, exponential Euclidean time-clustering, OS reconstruction, and a Hamiltonian spectral gap Spec(H) ⊂ {0} ∪ [m*, ∞)). The central technical observation is a scope-level closure lemma (Proposition 5.7): any demand that is not invariant under the equivalence relation induced by the declared invariants I cannot be imposed as a same-scope primitive without enlarging either the scope or the invariant package. Applied to the Route 1 construction at macroscopic scope Σ_ℓ, this immediately classifies: • stable-manifold RG structure (global trajectories, relevant/irrelevant decompositions) as a non-primitive upgrade beyond the finite-depth entrance witness that already supplies macroscopic clustering (Paper 2); • sharp t → 0 local operator-valued distributions (Wightman fields) as a strict ultraviolet scope enlargement requiring additional regularity inputs (Paper 5); • any interpretive narrative or renormalization-flow parameterization beyond OS semigroup time as likewise non-primitive (Paper 3). Under the natural admissibility datum that treats Σ_ℓ together with invariants (M1)–(M4) as primitive, Route 1 (Paper 1) plus the UV identification package (Paper 4) supplies a complete macroscopic invariant package. The resulting conditional-completion theorem (Theorem 6.1) then asserts that the Hamiltonian mass-gap statement is closed: no further same-scope requirement is needed. The note also supplies a clean status map of all nontrivial inputs (Table 1), a dictionary of interfaces isolated in Papers 2–5, and a neutral list of open upgrades that remain available once the macroscopic statement is secured. Written in standard mathematical-physics language and logically downstream of the technical work, this capstone can be cited independently whenever a discussion of “what counts as a solution” to the Yang–Mills mass-gap problem arises. Together with Papers 1–5 it provides a fully self-contained, scope-disciplined framework for a rigorous continuum mass-gap result at fixed positive macroscopic resolution.