Sharp Local Fields as an Ultraviolet Scope Enlargement

This is a companion note (Paper 5) in the Route 1 series clarifies a central conceptual distinction in constructive approaches to four-dimensional Yang–Mills: the Hamiltonian spectral-gap statement proved in the macroscopic construction is logically independent of the existence of sharp local operator-valued distributions (Wightman fields) obtained by an ultraviolet limit ℓ ↓ 0 (or t → 0 in gradient-flow language). Route 1 (Paper 1) establishes Osterwalder–Schrader positivity and exponential clustering on a bounded macroscopic net O_ℓ of gauge-invariant observables at any fixed positive resolution scale ℓ > 0. Combined with the finite-depth entrance mechanism (Paper 2), the ultraviolet identification package (Paper 4), and the semigroup spectral-extraction theorem (Paper 3), this yields a strictly positive lower bound Spec(H) ⊂ {0} ∪ [m*, ∞) for the reconstructed Hamiltonian acting on the vacuum sector generated by O_ℓ. The present note shows that this spectral-gap conclusion is a property of the vacuum representation generated by the chosen observable algebra O_ℓ and is stable under restriction or extension of that algebra (Propositions 4.2 and 4.3). Constructing sharp pointlike fields is an ultraviolet scope enlargement requiring additional regularity inputs (uniform energy bounds, short-distance scaling, phase-space/nuclearity conditions) that are invisible at fixed positive resolution ℓ and are not forced by the existence of a macroscopic gap. The note makes explicit: • Two nested scopes: bounded local nets at fixed ℓ versus ultraviolet extensions to operator-valued distributions (Table 1 and Section 2). • A clean separation principle: the macroscopic vacuum-sector gap depends only on OS positivity and Euclidean time-clustering of bounded observables (Section 3). • Why the t → 0 demand in many Clay-style or Wightman formulations is a change of scope rather than a primitive requirement for the mass-gap statement on O_ℓ (Section 5). • Neutral listing of the extra structure needed for a sharp-field upgrade (Section 6). No new model-dependent estimates are introduced. The discussion is written in standard mathematical-physics language and provides a self-contained, citable clarification that the Route 1 mass-gap result on the bounded macroscopic algebra stands on its own, while the further ultraviolet completion to pointlike fields remains an open and logically separate problem. Together with Papers 1–4, this completes the logical and structural framework for a rigorous continuum mass-gap statement at fixed positive macroscopic scale.