Conditional Continuum Limit of 4d SU(Nc) Yang-Mills Theory via Two-Layer Architecture, RG-Cauchy Uniqueness, and Step-Scaling Confinement

Building on the lattice results of Papers [E26I]-[E26IX] — which, per the series audit (all companions now at v2/v3), are WINDOWED and CONDITIONAL rather than unconditional — we give a conditional construction of a scaling-limit state for pure SU(N_c) lattice Yang-Mills theory in four Euclidean dimensions, along dyadic lattice spacings a_k = a_0 2^(-k). The construction proceeds via a two-layer architecture. Layer 1 (Local fields): for bounded gauge-invariant local observables, expectations converge — without extracting subsequences — to a unique limit; precompactness is trivial (| _{a,L}| <= 1), and uniqueness follows from a multiscale RG-Cauchy estimate (Assumption 3.5), the single hard analytic input of this layer: as already recorded in v2 (Remark 3.6, Appendix B), the naive asymptotic-freedom rate g_k^2 ~ c/k is NOT summable, so summability is a genuine hypothesis, not a consequence of the chain. Layer 2 (Confinement): the physical string tension sigma_phys > 0 is established through step-scaling of Creutz ratios at fixed physical loop size, conditionally on Assumptions 4.4, 4.7 and 4.9. The limiting state inherits Osterwalder-Schrader positivity and admits Hilbert-space reconstruction; the mass gap is conditional on a uniform physical transfer-matrix gap (Assumption A.2) and strong continuity (Assumption 5.5). Version 3 retags the input layer to the audited chain: the uniform-LSI inputs are conditional on (H-P0)+(H-YGZ)+(H-SFI)+(H-ABS), the DLR-LSI/mass-gap route on (H-DOB-blk), and all lattice statements hold in the volume window L_vol <= e^(C/g^2+O(1)). A new window-compatibility lemma (Lemma 1.4) shows this window is NOT an obstruction to the continuum limit: along the 2-loop trajectory the required lattice size L/a_k = 2^k L/a_0 satisfies ln(L/a_k) ~ 0.69 k while the audited window allows ln L_lat <= 32.3/g_k^2 ~ 3.12 k — a margin factor ~4.5 at the series' representative arithmetic. Assumption A.2 is now cross-referenced to its conditional lattice supplier (2602.0054 v2: transfer-matrix gap under (H-DOB-blk)+(H-P0), in kernel form). All quantitative claims, and exact validations of both layers in a solvable d = 2 toy, are adjudicated in a companion numerical suite.