Integrated Cross-Scale Derivative Bounds for Wilson Lattice Gauge Theory: Closing the Log-Sobolev Gap — a Conditional and Windowed Closure

We prove integrated cross-scale derivative bounds that replace the unverified Assumption 5.4 of the companion 2602.0041. Combined with two explicit large-field inputs (Hypotheses 3.2 and 4.2) and the conditional inequalities of 2602.0046, this yields — under the audited hypotheses listed below and within the stated volume window — the corresponding log-Sobolev assembly for the Wilson lattice gauge measure at sufficiently weak coupling, with constant independent of L_vol inside the window. The key decomposition into small-field and large-field contributions survives verbatim from v1, as do the sweeping-out modification (an L^1 bound in place of an essential supremum, and a shifted essential supremum over G_{k+1}), the Rothaus closure, the SU(2), d=2 toy-model analysis, and the correction lambda_1 >= alpha_* to Proposition 6.1(2) of 2602.0041. Version 2 corrects the logical status of the assembly after the quantitative audit of the series (ai.viXra:2602.0051-0056, all v2). (a) The Absorption step in the proof of Theorem 1.1 relied on the premises "p0(g) -> infinity as g -> 0 along the flow" and "if beta_k grows sufficiently with k": both use the inverted-sign flow of the series erratum and are withdrawn; with the correct asymptotic-freedom flow all statements hold in the window k <= k*(beta), i.e. L_vol <= e^(C/g^2+O(1)). (b) A new finding (F-ABS): the absorption condition (16) consumes Hypothesis 4.2 in the strong exponent form e^(-c beta_k eps_k^2), but the companion verification (2602.0056 v2) delivers only e^(-c_sf p0(g_k)) with c_sf = 2/(1+beta_0) — bounded along the window — so (16) fails at every scale k >= 2 with the Balaban-compatible thresholds (9), even under (H-P0). The strong form is therefore an additional explicit hypothesis (H-ABS), and its saturated variant eps_k = eps_* opens a second window k <= k_abs proportional to beta eps_^2. (c) The inputs are retagged per their v2 verifications: Hypothesis 3.2 is windowed and conditional on Balaban's small-field inputs (2602.0055 v2); Hypothesis 4.2 is form-level and conditional on (H-SFI)+(H-P0) (2602.0056 v2); the fiber LSI consumed by Corollary 1.2 inherits (H-YGZ) (2602.0053 v2). (d) Reference hygiene: v1's reference [8] was a self-citation of the present paper and is removed; the Wilson duplicate is removed; the audited chain 2602.0051-0056 (v2) is cited. What survives and is validated in the companion suite: the per-direction Wilson bound (Lemma 3.1), the energy-distance identity (13), the single-plaquette tail (Proposition 7.1, Table 1 reproduced digit-by-digit, with its caption/values normalization mismatch fixed), the impossibility Remark 7.2, the factorization step (21), the Rothaus closure, and Appendix A's lambda_1 >= alpha_.