Ultraviolet Trajectories and Conditional Completion

This companion note (Paper 4 in the series) isolates and makes fully explicit the ultraviolet (UV) identification inputs required to convert the finite-depth entrance mechanism and Euclidean clustering results of the Route 1 construction (Paper 1) into a rigorous continuum statement on a bounded macroscopic observable net O_ℓ at any fixed positive physical scale ℓ > 0. Route 1 proves, for four-dimensional lattice Yang–Mills with compact connected simple gauge group and Wilson action, that after an explicit blocking depth the exact blocked specification enters a uniform Dobrushin contraction basin. This immediately yields uniqueness of the infinite-volume Gibbs state and exponential Euclidean clustering of bounded gauge-invariant observables, together with Osterwalder–Schrader (OS) positivity. To pass from these lattice/block-scale Euclidean statements to a continuum theory on O_ℓ one must still specify: • a tuned UV trajectory β_n → ∞ with lattice spacings a_n ↓ 0 selected by a renormalization prescription at scale ℓ, • an identification of macroscopic observables along the trajectory, • existence of the macroscopic scaling limit on O_ℓ, and • a non-collapse condition ensuring that the physical (continuum) decay rate remains bounded away from zero. The note formulates these requirements cleanly as a UV package (Assumption 3.1, items (U1)–(U5)) and proves the conditional completion theorem (Theorem 4.4): once the Route 1 entrance/clustering outputs are available and the UV package holds, the induced functionals converge pointwise to a translation-invariant OS-positive Schwinger functional S on O_ℓ; the standard OS reconstruction applies; and the reconstructed Hamiltonian satisfies Spec(H) ⊂ {0} ∪ [m*, ∞) with the same m* appearing in the non-collapse condition. All statements are made on bounded macroscopic nets (no sharp local operator-valued distributions or t → 0 limits are constructed). The note also supplies: • precise definitions of lattice approximants, trajectory identifications, and induced functionals (Section 2); • a convenient lattice-time reformulation of the non-collapse condition (Appendix A); • a clean separation of what is internal to the entrance mechanism versus what remains external (the general UV-completion problem for 4D Yang–Mills). This note can be cited independently as the precise interface between the non-perturbative lattice control of Route 1 and any continuum conclusion on bounded positive-scale observables. Together with the companion notes on finite-depth entrance (Paper 2) and semigroup spectral extraction (Paper 3), it completes the logical skeleton for the macroscopic mass-gap result.