Lemma 18 ([Retention at order-n entropy]). Fix 0<λ<C_lev <∞ , C_E<∞ , and 0<ω<1 . On Gaussian disorder events of probability tending to one, every test with λn≤L≤C_lev n and (f)N≤C_E n¹-ω has an observation path to a product …

We use the path and the field tests in ; we verify that its entropy argument remains valid under. On source disorder events independent of the initial spin law, the calculation around Equation~(19.20) there, using its Lemma~17.4, Equation~(17.12), and Proposition~17.5, gives the following early field-test bound. On a fixed interval [t_*,t_b] of small positive scalar precision, the diagnostic and stability failures have integrated probability o(1) , uniformly under every initial spin law. This follows there by conditioning on the spin: the centered Gaussian-noise diagnostics have integrated failure o(1) , the additional spin shift has normalized norm O(t) , and the Gaussian low-band small-ball estimate is uniform in its center. No factor e^L is used for this interval. From t_b until the interaction coefficient reaches a fixed globally controlled range, the buffered stability test fails anywhere on the path with probability at most e^ -a n under the ordinary law Q_μ , for a fixed a>0 . This is the later field bound in . The word diagnostics and stability constants on the retained regions are those of Lemma 17 ([Square-root energy in a reweighted posterior mean]). Here is the order of the fixed choices in that construction. Choose first a large scalar endpoint t_ end and a small interaction coefficient below which covariance is globally bounded. Choose t_b>0 for the small-positive-field estimate, and obtain the buffer and rate a for the later test. The norm bound and the fixed total duration give constants C_ jump and C_ aft for the jump loss divided by t_*n and the loss after failure divided by n , uniformly when 1-β'≤t_*/3 . Choose a target drift budget η_1>0 so that 3η_1(1+C_ aft /a)<λ/4. Next choose 0<t_* t_b with C_ jump t_*≤η_1 , and finally 1-β'>0 sufficiently small relative to these choices, including 1-β'≤t_*/3 . Start with precision B_*=t_*I+(1-β')W, which is positive when 1-β' is small enough. Increase scalar precision to t_ end I+(1-β')W , and then traverse B_v=(t_ end +4v)I+(1-β'+v)W, 0≤v≤β'. The latter rate 4I+W is positive on the norm event. The endpoint is (t_ end +4β')I+W , so its posterior is a product. Before the globally controlled portion, the posterior interaction coefficient is either β'<1 or in a fixed compact subinterval of (0,1) . Reduce the available diagonal enlargement in to 0<η<(β')^ -1 -1 . Once these stability margins and C_ lead are fixed, choose the mean accuracy ε in Lemma 17 ([Square-root energy in a reweighted posterior mean]) so that its leading error, integrated over the bounded path and converted to a quadratic mean bound, is less than η_1 . Then fix its residual tolerance, selection depth, finite word diagnostics and constant C_ε , and finally take n large. On a valid posterior, apply to f and to 1 . With a_u=\|ν_u[X]-μ_u[X]\|/√n , this gives a_u≤2C_ lead ε+ (C_ε)/(√n) (2+√ _ μ_u (f) N_u ). By and Cauchy--Schwarz, the Q_ν -mean of the square root, also with any validity indicator, is at most √(f)/N . Under the resulting error is (C_ε)/(√n) (2+√(f)N) =O_ε(n^ -1/2 +n^ -ω/2 )=o(1). This bound is applied at deterministic observation times; inserting _ u<τ for a stopping time only decreases its Cauchy--Schwarz bound. Since a_u≤2 always, a_u^2≤2a_u . The early invalid paths cost at most four times their integrated probability, hence o(1) . Thus the normalized quadratic mean-shift integral, on valid paths and before any later stopping, is at most η_1+o(1) by the preceding choice of ε . Let u_g begin the globally controlled portion and let τ be the first failure of the later buffered test in [t_b,u_g] , with the immediate and no-failure conventions of the source. The initial jump leaks at most C_ jump t_*n entropy: compare to pure noise as above and use \|B_*\|≤Ct_* and \|X\|^2=n . The stopped form of, with the preceding integrated estimate, gives, with Q^τ denoting the law of the stopped observations, excluding the spin, D(Q_ν^τ Q_μ^τ) ≤(C_ jump t_*+η_1+o(1))n . For the event E of a failure before u_g , the ordinary probability is at most e^ -an . Data processing for its indicator then gives D(Q_ν^τ Q_μ^τ) ≥Q_ν(E)1 Q_μ(E) -2, Q_ν(E)≤ C_ jump t_*+η_1+o(1) a . These are exactly the deductions in , now with the enlarged energy range justified by Lemma 17 ([Square-root energy in a reweighted posterior mean]). After a failure the remaining precision rate and duration are bounded, and both spin means have norm at most √n ; the remaining loss per failed path is therefore at most C_ aft n . The total loss before u_g is bounded by (C_ jump t_*+η_1+o(1)) (1+C_ aft /a)n . The fixed choices above make this less than λn/4 . The initial entropy is at least L/2≥λn/2 , hence H(u_g)≥λn/4 . In the remaining portion the spin covariance is bounded for every field. The linear entropy test then gives - H≤CH ; over its bounded duration it retains a fixed fraction of H(u_g) . This proves the lemma.