Lemma 13 ([Positive covariance error]). For some a,c,C>0 , uniformly at every deterministic scale K≤k_p , p≤r_n , apart from activation points, P( G∩ p^2K_p(1+Ck_p^-a )I_p ) ≤C e^-c k_p^a . The probability is taken before restricting …

Fix a deterministic nonactivation scale p on the retained spectral set. For h∈_p , write D_p(h)=p^2K_p(h)-I_ _p . Define a Borel predicate V_p(W_p,h) to be false unless h∈_p , the saddle α_p(h) exists, and h≠0 . Otherwise, with e=h/\|h\| , it requires |α_p(h)|≤s_ aux p^2, p^2K_p(h)(1+ε_ aux )I_ _p , p^2⟨e,K_p(h)e⟩≤0.9. This condition contains no restriction to G . At a nonactivation scale, P_p is the spectral projection of W_p at its flat eigenvalue 2-p^2 , and R_ p,u =(2I-W_p+uI)^ -1 . Thus the predicate, the saddle, and the covariance used here are determined by (p,W_p,h) . Their defining formulas are jointly Borel away from activation scales. In the rest of the proof these objects are evaluated at h=h_p , and that argument is suppressed. On V_p , use blocks relative to _p=span e ⊕(_p∩e^⊥) , put z=(D_p)_ ⊥e , and define on _p∩e^⊥ E_p=((D_p)_ ⊥⊥ + zz^ T |(D_p)_ ee | )_+, s_p=\|(D_p)_+\|. Extend E_p by zero on the radial direction and on _p^⊥ . Off V_p , set both E_p and s_p to zero, before any restriction to G . Covariance positivity and give -I D_pε_ aux I and (D_p)_ ee ≤-0.1 on V_p . Maximizing the quadratic form of D_p over its radial coordinate, and then taking the positive part on e^⊥ , gives D_p E_p, \|E_p\|≤Cs_p, ( E_p)/(k_p) ≤C \|D_p\|_ F √k_p + C k_p on V_p , with D_p E_p understood on _p . For the norm bound, the maximizing radial coordinate of a unit transverse vector is bounded because the radial deficit is fixed and D_p is bounded; compare the resulting vector with D_ps_pI . For the trace, the rank-one correction has bounded trace, and (((D_p)_ ⊥⊥ )_+)≤ √_p\|D_p\|_ F ; the displayed trace bound also uses _p≤Ck_p from the retained spectral set. These are the algebraic bounds of . In particular s_p≤ε_ aux and \|E_p\|≤Cε_ aux everywhere, including on disorder failures. Include Cε_ aux γ^2 in the initial auxiliary choice. Apply Brascamp--Lieb's covariance inequality to the transition with φ_p . After whitening, its Hessian has a fixed positive margin because Δ(1-x^2)p^2I and ηγ^2 . If K is the extended Hessian, the covariance is at most the transition expectation of Δ^ 1/2 (I-Δ^ 1/2 KΔ^ 1/2 )^ -1 Δ^ 1/2 . Mollify the semiconcave extension to apply the smooth inequality and pass to the limit using the positive margin. Lemma 12 ([An extension on the parent annulus]) replaces the extended Hessian and transition by K_p and the true transition on agreement. Its polynomially weighted tails control first and second moments on the complement. In field units the covariance error is O(p^4M_p^2e^ -c k_p ) ; after and multiplication by q^2 , the remaining polynomial factor in k_p is absorbed by decreasing the exponent. On G , the agreement annulus lies in V_p , so gives K_p p^ -2 (I+E_p) . Put A=I-Δ/p^2 and S=p^ -2 Δ^ 1/2 E_pΔ^ 1/2 . Then A x^2I and \|A^ -1/2 SA^ -1/2 \|1 , so (A-S)^ -1 A^ -1 +2A^ -1 SA^ -1 . The baseline A^ -1 , substituted in , is exactly q^ -2 I_ _q . For the correction, the two outer factors ΔA^ -1 are the scalar Δ_q/x^2 on _q . Thus, after multiplication by q^2 , the first insertion is at most 2x^ -2 P_qE_pP_q . We obtain on G , for a child in its annulus, s_q ≤Cx^ -2 E[\|P_qE_pP_q\| W,_q] +C_γe^ -c_1k_p . Only Δ_q is inverted outside the covariance in ; directions with arbitrarily small new parent precision cause no loss. We next compress E_p before imposing G . The assertion is trivial when E_p=0 ; otherwise e is defined by V_p . The retained spectral set supplies ck_p≤_p≤Ck_p and _q≤Cx^3k_p ; after increasing the threshold, (_p∩e^⊥) is also comparable to k_p . Condition under P on the spectrum, W_p in physical coordinates, the physical field h_p , and its spectral coordinates. Equation makes the remaining rotation in e^⊥∩_p exactly Haar. The operator E_p is fixed under this conditioning: its definition uses only these parent data. For a uniformly rotating unit vector v in this space, the Gaussian representation of the uniform sphere gives P( v^ T E_pv> (C( E_p+t\|E_p\|))/(k_p) \,|\,parent data) ≤C(e^ -t +e^ -c_0k_p ). Indeed the numerator is a positive Gaussian quadratic form, and the Gaussian squared norm in the denominator is at least a fixed fraction of the dimension except with e^ -c_0k_p probability. A fixed-resolution net chosen in the spectral child unit sphere before the remaining Haar rotation has at most (C_q)≤(Cx^3k_p) points. Project each rotated net vector onto e^⊥ . Conditional on the parent data, it has a fixed norm at most one and a uniformly rotating direction. With t=C_2x^3k_p , a sufficiently large C_2 pays for the net, and a sufficiently small γ pays for the second exponential term. The net estimate for a positive quadratic form gives \|P_qE_pP_q\| ≤C(( E_p)/(k_p)+x^3\|E_p\|) outside a set of unconditional P -probability Ce^ -c_γk_p . This is the compression argument of ; no restriction to G was made in the Haar law. Average over I_q=[q/(2γ),q/γ] , with normalized logarithmic measure. On G outside the failure event in, and Cauchy--Schwarz give Av_ p∈I_q ( E_p)/(k_p) ≤C_γk_q^ -b , b=1/10. Here Av denotes that normalized logarithmic integral. On its failure the same averaged trace is bounded, because E_p is bounded everywhere and _p≤Ck_p . Its contribution on that failure, with _ G inserted, has expectation at most Ce^ -c k_q^ a_1 . The failures of are charged in the same way by the bounded \|E_p\| . These estimates are applied to the whole parent-scale integral before its conditional expectation. For every nonnegative error cost Y , E[_ G E(Y W,_q)] = E[_ G Y], because G is measurable from W . The norm term in has coefficient Cx^ -2 x^3≤Cγ . The initial choice of γ makes this coefficient θ<1/2 . We have proved, on G , s_q, ≤ _ p∈I_q E[s_p W,_q] +C_γk_q^ -b +err_q,; err_q, ≥0, E[_ G err_q] ≤C e^ -c k_q^β for some β>0 . Off the child annulus the left side is zero. The bounded definition of E_p is essential here: no cost polynomial in n is charged on a failed parent interval. Use a barrier B k_q^ -α , with 0<α≤b and B large. Since k_p=k_qx^ -3 , _ p∈I_q B k_p^ -α ≤θ(2γ)^ 3α B k_q^ -α . It absorbs the deterministic error in . Taking positive parts, applying conditional Jensen, and using, we obtain A(q):= E[_ G (s_q-Bk_q^ -α )_+] ≤_ p∈I_q A(p)+Ce^ -c k_q^β . For each deterministic nonactivation scale r_n≤p≤C_γr_n , use within its stated range. Under the biased law P , it gives p^2K_p(1+C(p^ a_0 +k_p^ -a_0 ))I outside a set whose intersection with the full event G_n^ FUA has probability at most Ce^ -n^ ξ_0 . Since G⊆ G_n^ FUA , the same failure bound holds after intersection with G . Here pk_p=np^4→0 , so p^ a_0 ≤k_p^ -a_0 , and k_p n^ 3·10^ -4 . By taking α≤a_0 and a sufficiently small positive terminal exponential power, the same bound for A(p) is Ce^ -c k_p^ β_0 . Every parent generation multiplies k by at least Λ=(2γ)^ -3 >1 . Iteration, including the terminal bound, gives for a suitably decreased β_*>0 A(q)≤C∑_ j≥0 θ^j e^ -c(Λ^jk_q)^ β_* ≤C e^ -c'k_q^ β_* . Markov's inequality at the barrier costs only a factor k_q^α , absorbed by the exponential. On G , holds throughout the full auxiliary annulus, and removes the annulus truncation. Decrease to a common positive exponent in the barrier and probability. This proves.