Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three

Proof. The variables \lambda V_x have the iid law \mu_\lambda(dv)=(2\lambda)^{-1}\mathbf1_{[-\lambda,\lambda]}(v)\,dv. Its concentration function is \sup_a\mu_\lambda([a,a+t])=\min\{1,t/(2\lambda)\}, so it satisfies the bounded-support Hölder hypothesis of the fixed-disorder edge localization theorem of Elgart and Klein. Their author-preprint Theorem 2.2 and Corollary 1.8(i)–(ii) [9] give a deterministic A>0 and almost-sure pure-point spectrum with exponentially decaying eigenfunctions on [-2d-\lambda,-2d-\lambda+A). Take \delta_{d,\lambda}=\min\{A/2,\lambda/2\}. The almost-sure spectrum is [-2d-\lambda,2d+\lambda] [9, author-preprint equation (1.17)], hence its projection on the nonempty open interval J_{d,\lambda} cannot vanish. This interval lies below -2d, while I_d lies above -2d. Intersect the localization and spectrum events with the probability-one event of Theorem 1.1, at the same fixed coupling.□

Proof. Condition on the history before a query. Its source index is then fixed. Coordinates whose values are already recorded contribute no new information. If a query includes both known and unqueried digits, apply the following factorization to its unqueried part. For a sign query, the product law factors into the queried coordinate and the other unqueried coordinates. Adjoining the answer leaves the latter factors unchanged. For a whole true suffix, apply the same argument to its block of digits: test against finite cylinder functions of the other coordinates and then use the monotone class theorem. This is a statement about conditional expectations, so it also covers a continuous residual answer without conditioning on an individual value of probability zero. The same factorization applies to unused auxiliary signs.

For a uniform order with r elements still unexposed, each possible next element has exactly (r-1)! completions. Its remaining order is therefore uniform after that entry is revealed. An answer subsequently queried at the selected index uses no unexposed order entry. Choices and failure decisions made from the enlarged history add no information. Induction proves the assertions after every fixed finite number of operations.

For precision at a random operation time, let \tau be a stopping time of (\mathcal F_q^{\mathrm{ret}}) with \PP(\tau<\infty)=1, and let \mathcal F_\tau^{\mathrm{ret}} be its stopped sigma-field. For D\in\mathcal F_\tau^{\mathrm{ret}}, apply each finite-operation conditional-expectation identity against D\cap\{\tau=q\}\in\mathcal F_q^{\mathrm{ret}} and sum over q. These countably many events exhaust D up to a null set and give the same identity conditional on \mathcal F_\tau^{\mathrm{ret}}. A whole-suffix query remains one operation in this argument.□

Proof. Because \mathcal C carries constants to constants, W is unchanged by addition of a constant to f. The minimization is a finite-dimensional least-squares problem, so its minimum is attained and is a quadratic form in f. Zero energy forces \alpha=0 and v constant, so the induced form on the quotient is positive definite.

For a neutral y, completion of squares gives

Neutrality makes \mathcal C^*y orthogonal to constants. The inverse matrix of a positive quadratic form is its dual, which proves both the identity and the comparison assertion on the quotient.□

Proof. Counting grid frequencies in dyadic annuli and using (9) gives

For completeness, a ball of radius r\ge2\pi/L contains at most C_d(1+Lr)^d\le C_dNr^d grid points. On an annulus r/2<|k|\le r, the summand is at most C r^{-2}. Decomposing the ball of radius a into dyadic annuli, discarding empty annuli below the first mesh frequency, gives C_d\sum_{j\ge0}(2^{-j}a)^{d-2}\le C_da^{d-2}. If a is smaller than the first nonzero frequency, the sum is empty. The weaker last bound deliberately retains the exponents needed later; it is valid because a\le1 and d-2\ge1.

At each frequency |\widehat{y_\Sigma}(k)|\le\|y\|_1, while at most two copies per site gives \|y_\Sigma\|_2^2\le2\|y\|_2^2. The low frequencies in the quadratic form are therefore bounded by Ca\|y\|_1^2, and Parseval bounds the high frequencies by Ca^{-2}\|y\|_2^2. For y\ne0, choose a=(\|y\|_2/\|y\|_1)^{2/3}\le1. This proves the claimed inequality. For y=0 there is nothing to prove.□

Proof. The case l=1 is immediate. Translate Q to \{0,\ldots,l-1\}^d. Define

This gives a centered function on \mathbb T_{2l}. Each original site has 2^d preimages. Every internal cube edge is copied 2^d times, and differences on the folding edges vanish. Consequently

The smallest positive eigenvalue of \Lambda_{2l} is at least c l^{-2}, so Parseval proves the first assertion.

For the weighted assertion put M=\max_Q w and S=\sum_Qw. If S=0 there is nothing to prove. Otherwise a=(M/S)^{1/3}\in(0,1]. Split V into frequencies 0<|k|\le a and |k|>a; its zero Fourier coefficient vanishes. With normalized inverse Fourier transformation, Cauchy–Schwarz gives

Thus (10) and Parseval imply

These inequalities include the case of an empty low-frequency range. Restricting to Q, using |z_1+z_2|^2\le2|z_1|^2+2|z_2|^2, and applying the exact reflection identities above, we obtain

The chosen value of a gives Sa=Ma^{-2}=M^{1/3}S^{2/3}, proving the claim.□

Proof. Represent each displacement by coordinate differences of least absolute value. A distance ball is contained in the coordinate box [-R,R]^d, proving the upper bound. For R\le L/4, a coordinate box of radius \lfloor R/d\rfloor embeds without identifications and is contained in the distance ball, giving the lower bound after adjusting the constant for bounded R. The number of representatives at distance r is at most C_d(1+r)^{d-1}, by counting nonnegative integer d-tuples of sum r and their signs. Thus the left side of (12) is bounded by C_d\sum_{r\ge0}(1+r)^{d-1+a}e^{-ur}, which is at most C_du^{-d-a} by comparison with its integral.□

Proof. On (-\pi,\pi)^d make the one-to-one change of coordinates x_j=2\sin(k_j/2). Then a(k)+2d=|x|^2 and dk=\prod_j(1-x_j^2/4)^{-1/2}dx. For 0<s<0.04 the entire sphere of radius \sqrt s lies in the coordinate domain, so polar integration shows that the normalized density of a+2d equals

This density is positive and smooth on (0,0.04). On every compact subinterval its derivatives are bounded and the density has a positive lower bound, with constants allowed to depend on d.

Choose a smooth cutoff \chi supported in (a_d/5,2a_d) and equal to one on a neighborhood of \mathcal J_d. Split the Stieltjes integral into this localized density and the complementary part. In the latter part the real denominator stays a fixed distance from zero for energies in -2d+\mathcal J_d, so it is continuous and bounded up to the real axis, as is its derivative. In the former part, subtract \rho_d(\alpha+2d) locally from the density. The difference vanishes linearly at the real energy; its Cauchy integral has a continuous upper boundary value. The constant part is an explicit logarithmic integral. This proves upper-half-plane continuity and gives the boundary imaginary part \pi\rho_d(\alpha+2d).

For the first inequality in (13), integrate the bounded localized density against ((s-(\alpha+2d))^2+v^2)^{-1/2}; its integral over the fixed support is at most C\log(2/v). For the second, integration by parts gives

There is no boundary term. The same absolute-integral estimate bounds this expression, while the complementary contribution stays bounded.

For the perturbation assertion, first translate all phases in the continuum torus integral. Put x_j=2\sqrt{c_j}\sin(k_j/2), so the new minimum is -2\sum_jc_j and the excess energy is |x|^2. The Jacobian is

Choose \varepsilon_d so small that c_j\ge1/2 and 2d\varepsilon_d<a_d/10. Energies in -2d+\mathcal J_d then have excess above the shifted minimum in a fixed compact subinterval of (a_d/5,2a_d). The corresponding polar density and its derivatives are uniformly bounded there. A cutoff equal to one on that compact interval gives the same integration-by-parts estimate, uniformly in the coefficients and phases.□

Proof. We first solve the scalar equation, then prove the two smallness bounds, and finally identify the inverse form of D.

The scalar fixed point. By Lemma 3.5, the boundary values G_0(\alpha+i0) for \alpha\in-2d+\mathcal J_d form a compact subset of the upper half-plane. Choose a_0>0 and B<\infty so that they lie strictly inside the rectangle

The closure of I_d lies in the interior of -2d+\mathcal J_d. Hence, when t is small, E+t\Re r\in-2d+\mathcal J_d for every E\in I_d and r\in\mathcal B. Uniform boundary continuity on this compact interval shows that r\mapsto G_0(E+tr) maps \mathcal B into its interior with fixed slack.

For a periodic C^1 function, comparing each grid cell with its corner gives an integral-to-grid error at most CL^{-1}\|\nabla f\|_\infty. On \mathcal B, set z=E+tr and R(k)=(a(k)-z)^{-1}. Since \Im z\asymp t,

The finite-grid map \Phi(r)=N^{-1}\sum_kR(k) consequently preserves \mathcal B, provided C/(Lt^2) is sufficiently small. Here t\asymp4^{-n_0} and L\ge4^{20n_0}, so

The second estimate of Lemma 3.5 holds for every r\in\mathcal B, without a fixed-point hypothesis. Differentiating the finite sum and using (16) gives

It is less than 1/2 for small t. Integrating the complex derivative along any segment in the convex rectangle makes \Phi a contraction. Iteration from one fixed starting point in \mathcal B constructs its unique fixed point. The iterates are continuous in E and converge uniformly, so the fixed point is continuous in E.

At this solution, translation invariance gives g_{ii}=r. Taking imaginary parts of the scalar equation, and cancelling p>0, yields the exact finite-grid normalization

The Fourier symbols and the remaining bounds. For each grid displacement \zeta\in\mathcal K_L, Fourier transformation gives

The first symbol is real, either because F is real and even or by the change k\mapsto-k-\zeta and R(-k)=R(k). Its zero mode is 1 by (18), and Cauchy–Schwarz gives |\widehat F(\zeta)|\le1. Both convolution operators are unitarily diagonalized by Fourier transformation, and Parseval gives the row identity:

Thus the two smallness assertions require a uniform bound for \widehat K and an averaged square bound for \widehat F. At the fixed point q=p is scalar, so D=p^2(\Id-F). The inverse-form comparisons will follow from a two-sided estimate of 1-\widehat F(\zeta) at every nonzero grid frequency. At higher frequencies we will also use that \widehat F is small to retain the sharp diagonal coefficient.

An absolute-product bound, including tangential intersections. We first pass to the continuum frequency torus. For every nonzero displacement \zeta\in[-\pi,\pi)^d, we claim

For quadratic dispersion, Erdős, Salmhofer and Yau obtain a related inverse-displacement two-resolvent bound by radial and angular integration [12, author-preprint Appendix A, Lemma A.1, equations (A.2) and (A.4)]. The proof below supplies the required near-bottom lattice estimate by convex slicing, including tangency.

In this estimate the displacement need not lie on the finite grid. For small t the real excess energy \Re z+2d lies in \mathcal J_d. If either dispersion has excess at least 0.04, the corresponding denominator is separated from zero by a fixed positive constant. Lemma 3.5 then bounds this part of the integral, including the factor t, by C_dt\log(1/t). This is absorbed by the right side of (20), since |\zeta|\le\pi\sqrt d.

In the joint region where both excess energies are below 0.04, the frequency vectors k and k+\zeta have unique small unwrapped representatives: every coordinate has absolute value at most 2\arcsin(0.1)<0.201. Their difference is the representative \zeta, so |\zeta_j|<0.402. Set x_j=\sin(k_j+\zeta_j/2). The joint region is contained in |x_j|<1/4, where the change of variables and its inverse have bounded Jacobians. The two dispersions become

Since \cos(\zeta_j/2)\ge\cos(0.201)>0, their Hessians are uniformly positive and bounded on that box. Their difference is linear:

Suppose that the two energy constraints

hold simultaneously, where s_1,s_2>0. Their difference gives |b_\zeta\cdot x|\le s_1+s_2. Thus x lies in a slab normal to b_\zeta of width at most C\max(s_1,s_2)/|\zeta|. On each slice of this slab, the remaining task is to bound the narrower energy shell, including when the slice is tangent to an energy surface.

For either sign, every affine hyperplane H meeting the box, every E'\in\RR, and every s>0, we claim

where \operatorname{vol}_{H} is (d-1)-dimensional volume on H. Fix the sign and extend each one-variable summand outside its coordinate interval by its second-order Taylor quadratic at the endpoint. This gives a globally C^2 function f with c\Id\le\nabla^2f\le C\Id, uniformly in \zeta. On a hyperplane H that meets the box choose x_0\in H in the box. The restriction f|_H is coercive and uniformly convex, and therefore has a unique minimizer x_*. If P_H is projection to the direction space of H, strong convexity and P_H\nabla f(x_*)=0 give

Consequently H intersected with the box lies in a ball of a fixed radius R_d about x_*. In polar coordinates in H, of dimension q=d-1\ge2, one has

Along each ray the energy is increasing. Changing from r to the energy and integrating an interval of length 2s proves the asserted shell bound, also when the interval contains the minimum. This explicitly includes tangency of the hyperplane to an energy surface; no nonzero-gradient assumption at the constrained minimum is used.

On each slice of the slab above, apply this shell bound to the narrower of the two energy constraints. Fubini’s theorem and the bounded coordinate Jacobian give joint volume at most

Partition each denominator into dyadic sizes s_i\ge ct, with the innermost bin of size comparable to t and the outermost of fixed size. On a pair of bins the product inverse is at most C/(s_1s_2), while the containing shells have the volume just bounded. There are O(\log(1/t)) bins for each denominator. Multiplication by t and summation prove (20).

Uniformly in \zeta,

Thus the error in either t-weighted product average is at most \varepsilon_L from (17). These absolute quadrature errors will be used only for product estimates, not to prove the gap at very small nonzero frequencies.

Smallness of K and the fourth-power rows. The absolute-product estimate is small for |\zeta|>\sqrt t. To control \widehat K also at smaller grid frequencies, use cancellation in the signed product. For \zeta\in\mathcal K_L with |\zeta|\le\sqrt t, use the elementary identity

The two denominators have the same strictly negative imaginary part, so none of the interpolated denominators vanishes. The interpolated dispersion (1-u)a(k)+ua(k+\zeta) is a cosine sum with shifted phases and amplitudes satisfying

Lemma 3.5 bounds the integral of its inverse square by C\log(1/t), uniformly in u and \zeta. Consequently

For larger \zeta, (20) bounds it by C\sqrt t\log^2(1/t)+\varepsilon_L. These estimates tend to zero uniformly, giving the asserted bound on \|K\|.

For the fourth-power row sums, the Parseval identity above and the absolute-product bound give the following estimate. All frequency sums in this calculation are over \mathcal K_L:

The frequency sum is bounded uniformly by the d-dimensional shell count used earlier. Here N^{-1}\le C_dt^{20d} and \varepsilon_L^2\le Ct^{36}; hence the last expression is o(t^{1+\delta}), providing the stated fourth-row slack.

We have obtained a well-defined initial diagonal and the two smallness bounds. It remains to prove the gap of F and identify the resulting inverse form. The gap at the smallest frequencies requires a direct estimate of the finite sum, which we give next.

The gap at every grid frequency. In this part \zeta\in\mathcal K_L. The exact normalization and (19) imply

The dispersion is Lipschitz, so the resolvent identity and |R|\le C/t give

Bound one factor by C/t and use (18) for the other. Since also |\widehat F|\le1, this proves

For the lower bound choose a sufficiently small fixed number \zeta_0>0. Suppose 0<|\zeta|\le\zeta_0, and put e=\zeta/|\zeta| and \alpha=\Re z+2d\in\mathcal J_d. In the coordinates x_j=2\sin(k_j/2), choose fixed 0<\theta_1<\theta_0 and a fixed small c_0>0, and define

All these points are in the bottom coordinate neighborhood. The radii are bounded above by a small constant and below by a positive constant. Writing x=\rho\omega, the derivative identity

uses |\sqrt{1-u}-1|\le u and \|(x_j^3)_j\|_2\le\rho^3. Taking \cos\theta_0\ge3/4, and using \rho^2<0.02, proves \nabla a(k)\cdot e\ge\rho\ge c_1>0 on P_e. The constants are uniform in e,\alpha and t. Shrink \zeta_0 so that the segment from k to k+\zeta remains in this small neighborhood. Its Hessian 2\diag(\cos k_j) is positive there, so the directional derivative along the segment stays at least c_1.

We next count grid points directly. Polar integration in the x coordinates, with its bounded positive Jacobian, gives

The angular margin from \theta_1 to \theta_0 is fixed, while the radial energy margin is c_0t/2. Both the coordinate map and its inverse have uniformly bounded derivatives. It follows that P_e^- lies at Euclidean distance at least c_dt from the complement of P_e: a displacement of size c_dt changes the energy by at most c_0t/2 and the angular coordinate by less than its fixed margin. The mesh-cell diameter is 2\pi\sqrt d/L=o(t). Associate to each point of P_e^- the corner grid point of its mesh cell. For small t that grid point belongs to P_e. Each cell has normalized volume N^{-1}, and hence

This argument is uniform in the direction of \zeta, including directions that vary with the grid size.

At each of these points,

Their nonnegative contribution to (22) therefore gives

This argument is valid even for |\zeta| of order L^{-1}: no additive quadrature error has been subtracted from the gap.

For |\zeta|\ge\zeta_0, the absolute-product estimate and the grid error give

when t is small. Combining this with (23) and (25) proves

The three inverse-form bounds. Equation (26) and D=p^2(\Id-F) now show that D has precisely the constants as kernel. Thus, for \zeta\in\mathcal K_L\setminus\{0\},

The same gap comparison gives

Together with (9), this proves the two rough comparisons and the lower bound by c\Id.

For the sharp upper bound split at |\zeta|=\sqrt t. Above this scale,

By making the disorder threshold smaller, we obtain \widehat X(\zeta)\le(1+t^\kappa)p^{-2} on this range. Below it, t/|\zeta|^2\ge1 and the rough upper bound is at most Ct/|\zeta|^2. Adding the nonnegative high-frequency term and using (9) proves the last line of (15) with its stated coefficient 1+t^\kappa.

All smallness conditions used above hold once t_{n_0} is sufficiently small. Since t_{n_0}\asymp4^{-n_0} uniformly in m\ge4n_0, one choice of \lambda_0(d) ensures them all. This completes the construction of the initial state.□

Proof. Let V_{xi}=k_x\mathbf1_{\{x\text{ above }i\}} on the remaining ports. The physical imaginary damping at i is t p_i^{\rm phys}\sum_{x\text{ above }i}k_x^4\ge ct. Moreover,

Applying the physical Ward identity to g=Vg^{\rm phys}V^\top proves the virtual one. Since p\ge c, c\|h\|^2\le\|\Im h\|/\sqrt t\le\|h\|/\sqrt t, proving the norm bound.

At the initial duplication, the matrix Q_{xi}=k_x^2\mathbf1_{\{x\text{ above }i\}} satisfies Q^*Q=\Id, and K_0=QK^{\rm phys}Q^\top. Also

because the fourth powers of the two weights over a site sum to one. The stated bounds follow from \max(w,\theta)\le3/4; restricting the columns to reserves introduces a further factor \theta.

At a stage start, q_i^{\rm phys}=p_i^{\rm phys}. Expanding the two terms in D_0 gives (29). Its dual formula follows by maximizing 2\Re\langle y,f\rangle-D_0(f) separately over the averages \bar f and the weighted-mean-zero fluctuations above each site. The broad comparison with W^*, and its dual with W, follow from the physical comparisons and the fact that the two weights are bounded away from zero.

For the sharp bound, retain the exact diagonal cancellation in (30). Weighted Cauchy–Schwarz gives

The coefficient of the diagonal form is therefore exactly 1+t^\kappa, as claimed.□

Proof. Sherman–Morrison on the physical inverse, followed by (28), gives (34). For the moment calculation write \varepsilon=\sqrt t and \xi_0=\varepsilon\xi. The denominator is 1+O(\varepsilon) uniformly, and

Now use \EE\xi=0, \EE\xi^2=1 and boundedness of \xi. No vanishing third moment is needed.

Entrywise squaring (34) and using the scalar pivot a_x in \Id-K gives (36)–(38). For D, the old edge conductance is p_ip_jF_{ij} and its pivot is p_x(q_x-p_xF_{xx}). Eliminating x adds conductance p_ip_jp_xF_{ix}F_{xj}/(q_x-p_xF_{xx}). Subtracting this from the conductance obtained by expanding |h_{ij}-\gamma h_{ix}h_{xj}|^2 proves (39).□

Proof. Since z=(\Id-K)U and Fp=q,

Taking imaginary parts first bounds q_i for small t; substituting back proves (43).

For (44), Ward gives

Apply the weighted Schur test to the matrix on the right using weights |h_{ij}|, omitting its zero rows and columns. Its weighted absolute row quotient is at most

This proves the operator estimate.

To prove (45), put M=\diag(pq), fix i, and define

The charge y is neutral. The variational formula for X gives

On the other hand, (31), Schur duality, and the lower half of (41) imply

Here we used p_j^{-2}=(q_j/p_j)M_{jj}^{-1} and q_j/p_j=1+O(t^\beta); both \beta and .10\delta exceed \kappa. Since

comparison yields

The row bounds and the active-site count give

For example, the numerator in the last expression is (\sum_jq_jF_{ij})^2\le C, whereas \Tr M\ge cN. Lemma 3.2 now gives

The exponent difference \frac13-\frac{2\delta}{3}-\kappa is positive, and N^{-1}=4^{-20dm}\le C_dt^{1+\delta+\kappa} for n\le m. This proves (45).□

Proof. Extend \mu_x=\EE\gamma_x by zero outside \mathcal B. The moment and lag bounds give \|\mu\|_\infty\le C\sqrt t\,t^\beta. If the omitted pivot rows and columns are restored temporarily, the sum of the linear means in (38) is -2h\circ(h\diag(\mu)h). Its absolute row sum is at most

Terms incorrectly restored with x=i or x=j have magnitude at most CtF_{ij} and hence total row and column sums O(t). For the quadratic mean, |\EE\gamma_x^2-a_x^{-1}|\le C\sqrt t; its absolute row sum is bounded by C\sqrt t\sum_xF_{ix}\sum_jF_{xj}\le C\sqrt t. Symmetry and the Schur test prove (46).

For the linear square bound, fix an output i and put v_j=h_{ij}u_j. Restoring the omitted inner index j=x gives

The restored term equals h_{xx}h_{ix}^2u_x; its squared sum over i,x is at most CtH_*\|u\|^2. Deleting the output i=x can only decrease these estimates. Sum over i and use the bounded column sums of F to obtain C(T_*^2+tH_*)\|u\|^2. The realized scalar \gamma_x is bounded, so the same estimate applies to the linear part of J_x.

For its quadratic part,

Summation over x gives CH_*\|u\|^2 because \|K\| is bounded. The quadratic coefficient is uniformly bounded. Combining these estimates proves the first stack in (47). Every J_x is complex symmetric, so applying the same bound to \bar u proves the adjoint stack.□

Proof. All matrices and coefficients below are evaluated at the fixed prestate. In particular, the values of f do not depend on the hypothetical pivot. The zero-padded operator L_x uses f|_{V\setminus\{x\}} and has zero output at x. Whenever we split a difference through x, we use the value f_x of this same common potential.

Write d_{ij}f=f_i-f_j. The uniform bounds on p give

For brevity put

The prestate bounds and Lemma 4.2 imply

Indeed, q_x=p_x+O(t^\beta), F_{xx}=O(t), and hence c_x=1+O(t^\beta+t). The scalar moment bounds give \EE|\gamma_x|^2=1+O(\sqrt t) and \mu_x=-\sqrt t(g_{xx}-r_x)+O(t). Since \beta<1/2, these imply (52).

Decompose L_x=L_x^{(1)}+L_x^{(2)} according to its survivor edge weights

Both parts are real symmetric Laplacians; their edge weights may have either sign.

The sum of the means. Let

and retain the endpoint weights by introducing

First include all intermediate indices x\in S, also when x=i or x=j, in the linear mean. Its quadratic form becomes

Writing M_f=\diag(f) and using \langle A,B\rangle_{\mathrm{HS}}=\Tr(A^*B), this is exactly

The diagonal matrix M_f commutes with C_\mu=\diag(\mu_x\mathbf1_{\{x\in S\}}/p_x). Therefore

and the ideal property of the Hilbert–Schmidt norm gives

Since p is bounded above and below, \|H_p\|\le Ct^{-1/2} and \|C_\mu\|\le C\sqrt t\,t^\beta. Cauchy–Schwarz in the exact pairing thus bounds its absolute value by Ct^\beta D(f). In particular this is a signed cancellation with the actual p_i p_j weights retained; no regularity or constancy of p has been used. For an edge ij, removing the intermediate index x=i, if i\in S, costs at most

The removal of x=j has the same bound. Summing these edgewise errors costs at most Ct^{1+\beta}D(f).

The quadratic mean has absolute form bounded by

Use

and the bounded row and column sums of the symmetric matrix F. After extending the x-sum to V, the last expression is at most Ct^\beta\mathcal E(f). This proves (48) for every subset S.

The linear output stacks. For each i\in V, define the complex vector

The reality of f allows the real part in (53) to be taken after summing over j. Therefore, for i\ne x,

Moreover,

Temporarily include j=x in (54). The bound of Lemma 4.3,

gives

For the squared \ell^1 norms, first apply Cauchy–Schwarz in i:

It remains to remove the artificially included j=x terms. Their absolute values are at most

Their squared \ell^2 stack is bounded by

and their squared \ell^1 stack is bounded by

These are absorbed in the preceding bounds, since \max F\le\sqrt{H_*}. Dropping the output i=x only reduces the majorants. This proves both required stacks for L_x^{(1)}.

The quadratic output stacks. Put \varepsilon=\max_{i,j}F_{ij}\le\sqrt{H_*}. By (52), the absolute output is bounded by a constant times

Here we have included all omitted indices, which only increases the nonnegative majorant. Cauchy–Schwarz in j, followed by splitting differences at x, gives

In the last step we used \sum_iF_{ix}^2\le\varepsilon\sum_iF_{ix}\le C\varepsilon. For the other stack, the total mass \sum_{i,j}F_{ix}F_{xj} is bounded uniformly in x, so

Both estimates remain true with the x-sum restricted to S. Since \sqrt{H_*}\le t^{\delta/2}, combining the linear and quadratic bounds proves (49). No factor depending on the number of candidate pivots has been used.

The dual norm stack. The Laplacian property implies that every y_x=L_xf is neutral. Lemma 3.2, including its version for at most two ports per site, gives

Hölder’s inequality over x, followed by (49) and b\le t, yields

The last inequality uses 1/3+\delta/6>\delta/2. This proves (50).□

Proof. All matrices without a prime are fixed at the prestate. Expectations in the proof are conditional on that prestate. Set

Thus R_F\le C. For a hypothetical deletion x, write

The diagonal bound gives a_x=1+O(t), so a_x is bounded away from zero. Lemma 4.2 gives

The exact equation. Let B_x=\Schur_{\{x\}}(\Id-K). In the block decomposition with deleted coordinate x, the off-diagonal column of \Id-K is -k^x. Eliminating U_x from (\Id-K)U=z therefore gives

On the other hand, the update gives

Subtracting the two equations yields

Since B_x^{-1}=P_{R_xR_x},

The inverse in this formula will also follow from the convergent series below.

Moments and the forcing array. The identity g_{ii}-r_i=[(\Id-K)u]_i and the row-sum bound imply |g_{ii}-r_i|\le Ct^\beta. Expanding the denominator of \gamma_x using |\xi_{0,x}+tr_x|\le C\sqrt t, centering, and variance t, gives

In particular,

For the middle estimate, the extra term is z_x(a_x^{-1}-1)=O(t^{3/2+\beta}).

Put Y^x=Z^x/\sqrt t. Its direct forcing satisfies

The linear part of J_xu_{R_x}, apart from its bounded coefficient, is

For fixed i, restore the term j=x and set D_i=\diag(h_{ij})_j. The resulting vector indexed by x is D_i hD_i u, of norm at most

The restored term h_{ix}^2h_{xx}u_x has \ell^2_x-norm at most C\sqrt t\sqrt{H_*}\,t^\beta. Removing the output x=i only reduces this norm. The quadratic part is bounded by

and hence has A_2-norm at most Ct^\beta\sqrt{H_*}. Consequently,

because \beta+\delta/4=.67\delta>.50\delta.

For the summed mean, the direct forcing costs at most C\sqrt t\sum_xF_{ix}\le C\sqrt t. Extend \mu_x by zero off S. Before the survivor omissions, the summed linear part of \EE J_x is -2h\circ(h\diag(\mu)h). Its absolute row sum is bounded by

Contracting against u contributes another factor t^\beta. Each omitted set x=i or j=x costs at most

and their intersection satisfies the same bound. For the quadratic part, |\nu_x|\le C\sqrt t, while

Its survivor omissions decrease this absolute bound. We have proved

This estimate controls the signed sum of the conditional means; it does not assert an A_1 bound on the forcing itself.

The restricted inverse. Let M_x=\Id-e_xe_x^*, and define Q^x=M_xPM_x. Since Y_x^x=0, the padded vector P_{R_xR_x}Y^x equals Q^xY^x=M_xPY^x. For an array W, write (QW)^x=Q^xW^x. Minkowski’s inequality gives

For means, the output mask depends on the deletion index. Precisely,

For every hypothetical outcome, (59) gives A_2(Y)\le Ct^{\delta/2}. Jensen’s inequality therefore bounds the second term by CL_Pt^{\delta/2}. Thus

Two array bounds for the correction. We next control repeated applications of the random matrices J_x. The bounds below are pathwise and apply to any array W, including one depending on the same outcomes as J_x. Write A_2(W)=A_W. Up to uniformly bounded coefficients, the linear and quadratic outputs are

Both are set to zero when i=x. Cauchy–Schwarz in j gives

Also |W_j^x|\le A_W, so

Writing (JW)^x=J_xW^x, we obtain

All omissions only reduce the nonnegative upper bounds above.

For the A_1 estimate, apply Cauchy–Schwarz over (x,j):

The quadratic part satisfies

Therefore

The remainder and the final mean. Define the columnwise operator \mathcal C by (\mathcal CW)^x=Q^xJ_xW^x, and put

Equations (61) and (63) give A_2(\mathcal CW)\le q_*A_2(W). For small t, q_*<1/2. Equation (57) thus has the convergent series solution

Apply (64) to the outermost correction in every term of positive degree. It gives

Both J_x and Y^x depend on \xi_{0,x}; their expectations need not factor. The remainder estimate is valid for every collection of hypothetical outcomes, so averaging it bounds the sum of the absolute conditional mean remainders. Combining it with (60) and (62) yields

as required. No independence between a correction and its forcing was used.□

Proof. The linear part of R_i^x, for x\ne i, is

Cauchy–Schwarz gives

For the mean, extend \mu_x=\EE\gamma_x by zero outside \mathcal B\setminus\{i\} and put B=h\diag(\mu)h. The Ward row bounds and \|\mu\|_\infty\le C\sqrt t\,t^\beta imply

Therefore the absolute linear mean, summed over x, is bounded by

The error term accounts for restoring j=x: its individual value is \mu_xh_{xx}F_{ix}^2 and its summed absolute value is at most CtH_*.

For complex a,b, the elementary fourth-power expansion gives

Apply this to a=h_{ij} and b=\gamma_xh_{ix}h_{xj}. The sum over x,j of the absolute nonlinear errors is at most

After division by H_*, the total bias is thus bounded by

since \beta-\delta/4=.17\delta>\kappa. The square stack of the nonlinear errors is bounded by the square of their absolute sum, and is smaller than Ct^\kappa H_*^2. Combining with the linear square stack proves Lemma 4.7. All bounds remain valid after restricting either of the two index sets.□

Proof. Only the pullback and centering assertions remain to explain. The Schur comparison following (41) gives D_{\rm cur}(Bv|_{\rm cur})\le CW(Bv)\le C\|v\|_{\mathcal H}^2. The form mean and dual square bound of Lemma 4.5 consequently imply

The latter follows by applying the assumed dual bound to L_x(Bv|_{\rm cur}) and summing over x. The remaining rows of the table are Lemmas 4.4, 4.6, and 4.7.

If there are M remaining slots in a uniformly random ordering, each conditional mean and second moment is the corresponding sum divided by M; skipped slots contribute zero. Center the joint random slot-and-scalar increment. For a self-adjoint increment Y,

The centered jump increases by at most a factor two. Apply this to the Hermitian dilation for a complex or rectangular increment. Predictable stopping preserves all these bounds, including the prospective first breaking step because its prestate still satisfies the hypotheses.□

Proof. Write the physical active inverse as g=M^{-1}, where the real part of M is the terminal Schur matrix. Its dissipative part gives \|M^{-1}\|\le C/t. Every off-diagonal entry created by eliminating a connected terminal component joins sites at collapsed distance zero. Consequently conjugation by Q(i)=\exp\{atd_T(z,i)\} leaves all these entries unchanged. Only direct active-active hopping entries change. There are at most 2d in each row and column, and their relative change is O(at). Thus \|QMQ^{-1}-M\|\le C_d at. For a sufficiently small fixed a, a Neumann series gives \|QM^{-1}Q^{-1}\|\le C/t. Set z=j and multiply by \sqrt t to obtain (68). The bounded port weights preserve this estimate.

For completeness, the additive error in (67) cannot simply be absorbed into a polynomial prefactor after exponentiation. Split instead at ordinary distance t^{-2}. Below this distance, the crude bound Ct^{-1/2} suffices, since u t^{-2}=t. Above it, d_T(i,j)\ge\dist(i,j)/4 for small t, and (68) is stronger than (69).□

Proof. The Ward identity and bounded positive p give \sup_i\sum_jF_{ij}\le C. Since F is symmetric, the column sums obey the same bound. Tilt K by Q(i)=e^{u\dist(z,i)}. Each row and column sum of QKQ^{-1}-K is bounded by

The contribution from distances at most t^{-2} is O(t). For greater distances, the stronger estimate |h_{ij}|\le Ct^{-1/2}e^{-ct\dist(i,j)} from the proof of Lemma 5.1 makes the sum exponentially small. The row-column norm bound therefore gives \|QKQ^{-1}\|\le\rho<1, uniformly over z. Hence

Summing in k proves the entrywise assertion.

To improve the absolute row sum, use a different estimate for short powers. Entrywise |K^k|\le F^k, and Fp\le(1+Ct^\beta)p, so

These sums are uniformly bounded for k\le k_0=\lceil C_1\log(1/t)\rceil. For the other powers, use

which remains valid with two ports per site. Indeed, choosing minimal torus displacement representatives embeds each distance shell into \mathbb Z^d, where the shell of radius r has at most C_d(1+r)^{d-1} sites. Summing the geometric shell series gives the displayed bound uniformly in the torus size. Choose C_1=C_1(d) so that t^{-3d}\rho^{k_0}\le1. The first k_0 powers then cost Ck_0, and all remaining powers together cost C.□

Proof. Reserve ports keep the physical damping at least ct, so the updated inverse exists. By the single-update formula,

Since \delta/2>\beta, this proves the new diagonal estimate. The formula h'_{ij}=h_{ij}-\gamma h_{ix}h_{xj}, with |\gamma|\le C, also gives

The inherited stage-start data and the current cap conditions are the prestate hypotheses of Proposition 4.8; Lemma 5.2 supplies its inverse-row input. That proposition gives \|J_x\|\le Ct^{\delta/4}. Write I-\widehat K=\Schur_{\{x\}}(I-K). The Schur contraction proved after (41) gives \|\widehat K\|\le\|K\|\le\rho_0, while the exact next-step identity is

Thus \|K'\|\le\rho_0+Ct^{\delta/4}\le\rho_1<1 for small t. The terminal set has not changed, so its metric bound has not changed either. Lemmas 5.1 and 5.2 now apply.

The explicit formulas for J_x,L_x contain a factor joining each surviving endpoint to x. Lemma 5.1 therefore gives (72). In the exact lag formula \Delta\widetilde U=P'Z^x/\sqrt t, the vector Z^x has the same single-center decay by its definition and the prestate bound on U. The entrywise bound on P' preserves that decay under convolution. The fourth-power proxy formulas are finite sums of the same factors. Each convolution loses at most a power of u^{-1}, because \sum_j e^{-cu\dist(i,j)}\le C_d u^{-d}; decreasing the exponent constant absorbs the convolution. This proves (71) with a polynomial prefactor whose degree may depend on d. Crucially, none of these deductions used a post-update fine cap.□

Proof. Use the displayed decomposition of B_Tf in the infimum defining W. Its \alpha part has no larger norm. The product rule and cube Poincare inequality give

This proves the first estimate, and duality gives the second.

For clarity, let \mathcal R be the Riesz map from the local space to its dual, and set A_x=\mathcal R^{-1}B_T^*L_xB_T. Every L_xy is neutral. Hence

Since each A_x is self-adjoint in this fixed inner product, this is the scalar identity bound

The mean estimate follows from the corresponding form bound. Both assertions hold for every subset of possible next ports; the square bound holds for arbitrary choices of their hypothetical outcomes. All maps here act on a fixed initial-port space; deleting a port merely pads the new matrix by zero there.□

Proof. For each included tile write J_T,L_T,\mathsf u_T,\mathsf R^\sigma_T for its accepted endpoint record or its current prefix record. Every included update has a valid prestate, so Lemma 5.3 gives the scalar and matrix tails through the last candidate update. Its proof does not require that candidate poststate to satisfy a fine cap.

For J, bounded overlap of the cubes gives

Indeed, test against two vectors and apply Cauchy–Schwarz to \sum_T\|1_{Q_T}f\|\|1_{Q_T}g\|. No sign assumption on J_T is needed. The discarded entries are negligible by (72) and distinctness of the pivots: an entry discarded for a tile has an endpoint at distance at least L_{\mathrm{tile}} from its pivot, and the remaining exponential kernel has uniformly bounded row and column sums up to a power of t^{-1}. Thus its total operator norm is at most t^{-C}e^{-cuL_{\mathrm{tile}}}. The analogous scalar argument uses bounded overlap at each monitored row and (71) elsewhere. It gives the stated bounds for the fixed records \mathsf u^{\rm sw} and \mathsf R^{\sigma,\rm sw}, up to the same negligible error.

For the energy form, test an arbitrary split f=\alpha+v on the initial ports, and write f_T for the restrictions of the pair (\alpha,v) to the local space \mathcal H_T. On Q_T\times Q_T, differences of B_Tf_T equal differences of f. The error is a tail. If x is in the tile and an endpoint lies outside Q_T, its distance from x is at least L_{\mathrm{tile}}. Extracting part of this distance from (72), then summing over distinct pivots, bounds the absolute error edge kernel by

Convolution sums only change the polynomial prefactor. For a minimal torus displacement representative a,

The shell count used in Lemma 5.2 gives, for s=0,2,

where for s=0 the zero displacement contributes one. Thus the gradient contribution costs at most C_du^{-d-2}, and the \alpha contribution at most C_du^{-d}. It follows that (77) contributes at most

For the pullback tail of one tile, apply this same kernel estimate to the zero extensions

Its error is bounded by the exponential prefactor times \|\alpha_T\|^2+b^{-1}\|\nabla v_T\|^2, which is at most C\| (\alpha,v)\|_{{\rm loc},T}^2 by the product and Poincare estimate in Lemma 5.4. Summing these local energies costs only bounded overlap. Since uL_{\mathrm{tile}}\asymp t^{-3}, all these errors are smaller than any power of t. The local form budgets give

Combining this with the two tail estimates and taking the infimum over splits yields

Here the last comparison is the initial comparison D_0\asymp W, not an assumed post-update comparison. The preceding overlap bounds similarly give \|J^{\rm sw}\|\le Ct^{.125\delta}\ll t^{.10\delta} and \|\mathsf u^{\rm sw}\|_\infty\le Ct^{.44\delta}\ll t^{.42\delta}. Restricting the latter record to survivors gives the actual transformed lag bound. The total normalized row-proxy increase is at most Ct^{.0005\delta}. Its upper-bound relation with actual row growth, together with the initial full-row bound .75H_* and the initial reserve-row bound \theta^2H_*, proves the two row conclusions. All the conclusions therefore hold through the candidate update.□

Proof. For u>0 with uJ\le1, functional calculus gives e^{uY_k}\le\Id+uY_k+Cu^2Y_k^2. Taking conditional expectations, using the martingale-difference property, and applying 1+x\le e^x yields \EE(e^{uY_k}\mid\mathcal F_{k-1})\le e^{Cu^2v_k}\Id. The Golden–Thompson inequality [16, 26], in the finite-Hermitian form stated in [13, Equation (1.1)], says \Tr e^{B+C}\le\Tr(e^Be^C) for finite-dimensional Hermitian B,C. Applied conditionally, and using that v_k is a predictable scalar, it shows that

is a nonnegative supermartingale with Z_0=D_s. At the first upper eigenvalue crossing of a, it is at least \exp(ua-Cu^2V). Stopping at that crossing and a finite horizon bounds its probability by D_s\exp(-ua+Cu^2V). Let the horizon increase, choose u to be a small constant times \min(a/V,1/J), and repeat with -Y_k. The union of the two crossings gives the result.□

Proof. Condition on the entire retained history at the trial start. There are M=L_{\mathrm{tile}}^d\asymp_d t^{-6d} permutation slots. While the trial is running, let \mathcal F_{k-1} be the full retained history immediately before slot k. It includes all earlier queries, their answers, exposed permutation slots and decisions, but not the next slot or its answer. Given \mathcal F_{k-1}, the next member is uniform among the r_k=M-k+1 unexposed physical sites. A terminal slot has zero increment. At each other possible slot, the scalar law conditional on \mathcal F_{k-1} and that newly exposed slot satisfies (33). The current numerical prestate is \mathcal F_{k-1}-measurable.

Let \tau be the first prefix violating a local budget or one of the four mutable caps, and put \tau=M+1 if there is none. The multiplier s_k=\mathbf1_{\{k\le\tau\}} is predictable: it asks only whether prefixes through k-1 satisfied the local budgets and validity. When s_k=1, evaluate the next raw increment at that valid prestate; when s_k=0, set the stopped increment to zero without another query. This defines the stopped increment as s_k times the raw increment where it is evaluated. Thus the first breaking update is included, and every retained update has a valid prestate. After stopping, the formal remaining increments are zero and \mathcal F_k stays equal to the history at stopping. The actual retained filtration continues through rollback, marking, and later trials. The poststate of the first breaking update need not satisfy the fine caps; Lemma 5.3 supplies all decay estimates needed through the breaking update. The next-slot law and its count r_k are used only when s_k=1; after stopping every variance envelope is zero.

We verify the scalar identity variance hypothesis of Lemma 5.6 separately for every monitored quantity. Put \ell=1+\log(1/t), let \EE_{k-1} denote conditional expectation given \mathcal F_{k-1}, and use the Hermitian dilation

All operators below act on the fixed spaces and scalar coordinate sets defined above.

For a monitored lag coordinate, take \mathsf H(\Delta\mathsf u_i^x) on \mathbb C^2. Its square is |\Delta\mathsf u_i^x|^2\Id_2, so the lag stack of Proposition 4.8 gives C_d\ell^2t^\delta\Id_2 after summing over all remaining possible bit ports. The record has zero increment at its own deletion and thereafter.

For J, write P_Q for the fixed Euclidean projection onto \mathcal P_0\cap Q and C_x=P_QJ_xP_Q on that space. Both square orders are controlled:

Consequently the sum of squares of \mathsf H(C_x) is bounded by C_dt^{\delta/2} times the identity on the doubled local port space. Zero padding makes these compressions meaningful even after ports are deleted.

For L, use the self-adjoint Riesz operator A_x=\mathcal R_T^{-1}B_T^*L_xB_T on \mathcal H_T. Lemma 5.4 gives directly \sum_x A_x^2\le C_dt^{\delta/2}\Id_{\mathcal H_T}. For either normalized row proxy, its increment \Delta\mathsf R_i^{\sigma,x} is real; Proposition 4.8 gives \sum_x|\Delta\mathsf R_i^{\sigma,x}|^2\le C_dt^\kappa. This is an identity bound on its fixed one-dimensional real space, again with zero increments from the row’s deletion onward.

Each of these pathwise stacks holds over any remaining subset and for every collection of allowed hypothetical scalar outcomes. They can therefore be averaged under any joint law with the required conditional marginals. Independence among hypothetical outcomes at different possible next ports is unnecessary. Averaging over the uniform next slot divides the relevant stack by r_k. For each of the four self-adjoint stopped raw increments Z_k^a, center the joint slot-and-scalar outcome by setting Y_k^a=Z_k^a-\EE_{k-1}Z_k^a. Then

with the explicit predictable scalar choices

Here \Id_a is the identity on the indicated fixed space. Centering increases the jump bound by at most a factor two. The harmonic sum obeys

Thus the deterministic total variance envelopes, centered jump bounds, and budgets, up to constants depending only on d, are

The same averaging of the arbitrary-subset mean estimates bounds the sum of norms of conditional means by

These are respectively o(t^{.44\delta}), o(t^{.125\delta}), and o(t^{.0005\delta}), so each is smaller than half its budget for sufficiently small t. Apply Lemma 5.6 to the centered sums at half-budget. The exponent margins for squared budget divided by V and for budget divided by the jump bound are, respectively, .25\delta and .125\delta for J,L, .12\delta and .06\delta for the transformed lag, and .004\delta and .002\delta for a normalized row proxy, using \kappa=.005\delta. For the lag, the two ratios are bounded below by

All these ratios tend to infinity by a positive power of t^{-1}, apart from logarithms.

The J dilation has dimension 2|\mathcal P_0\cap Q|\le C_dt^{-6d}, and \dim\mathcal H_T\le C_dt^{-6d}. Each lag test has dimension two and each row-proxy test dimension one; their total number is at most C_dt^{-6d}. The sum of all dimension prefactors in the concentration bound is consequently at most C_dt^{-6d}, independent of N. The union over these local tests has probability at most e^{-t^{-c}}, after decreasing c>0 and increasing the dimension-dependent starting scale. There is no union over the entire torus in this single-trial bound.

Suppose now that the trial fails at prefix \tau, but no local budget is violated there. Earlier accepted trials never crossed either test, and all earlier current prefixes satisfy validity. Thus every included update, including the one at \tau, has a valid prestate. The accepted earlier endpoint records satisfy the local budgets, and by hypothesis so does the current record at \tau. Their physical pivots are distinct by the record convention following (74). Lemma 5.5 therefore puts the poststate at \tau strictly inside all four mutable caps, contradicting the definition of \tau. Every trial failure thus includes a local budget crossing; a simultaneous violation of a local budget and a mutable cap is already counted in that event. The local crossing estimate proves the lemma.□

Proof. Lemma 5.7 and successive conditioning prove the stated conditional retry termination. Lemma 5.5 and the stopping rule maintain the mutable caps along every accepted sweep, using Lemma 5.3 through each candidate breaking update. The exact padded Schur identities then give the K estimate. The two-sided L cap keeps D_0+L^{\rm sw} comparable to D_0, with only the constant kernel, so Schur duality gives multiplicative transport of X on survivor-supported neutral charges. The only surviving ports are reserve ports, and on those ports

For a neutral physical charge y placed on the reserve ports, the exact initial identity (32) reads

Combining these two formulas with the multiplicative transport proves the first assertion. Applying each old comparison separately yields the three stated new coefficients. In particular the sharp coefficient follows from \theta(1+t^\kappa)+(1-\theta)=1+\theta t^\kappa.

Lemma 4.3 gives the actual virtual diagonal difference g^{\rm cur}_{ii}-r_i=O(t^\beta); division by the reserve factor \sqrt\theta preserves that bound physically. The reserve-row conclusion of Lemma 5.5 is (\theta^2+Ct^{.0005\delta})H_*: its fixed proxy bounds the possible increase after discarding favorable removed-column losses. At the endpoint the virtual h is already the new physical h. Finally, \sup_{.25\le\theta\le.625}\theta^{1-\delta}<.7, so the strict last inequality holds for small t.□

Proof. Let T_0 denote the existing terminal set. The true principal matrices on T_0 and on T_0\cup B are invertible. Eliminating the latter set therefore produces a real symmetric survivor matrix with damping t\diag(p_R), whose inverse exists by dissipativity. Reversing these Schur eliminations shows that the full active matrix, with the new diagonal on B, is invertible as well.

Let V be its diagonal change on B; its entries include the removal of -tr_i. In particular \Im V_{ii}=tp_i>0 and V is invertible. The finite-dimensional resolvent identity gives

To see that the middle inverse exists, write \mathcal M=g^{-1} for the old active matrix and E_B:\CC^B\to\CC^{B\cup R} for coordinate inclusion. The finite-rank determinant identity gives

and both full determinants are nonzero. Thus no estimate on the norm of this middle inverse is assumed.

The old scattering matrix S is symmetric unitary. Its R block S_{RR} is a contraction, whereas the new survivor scattering matrix S' is symmetric unitary. Hence \Delta S=S'-S_{RR} satisfies \|\Delta S\|\le2. Since g_{BR}=g_{RB}^\top, (83) yields

where the coefficient C_* incorporates the invertible diagonal p_B factors. Its size is irrelevant. Left range inclusion gives \Pi\Delta S=\Delta S. Transposing this equality and using \Delta S^\top=\Delta S gives \Delta S\Pi^\top=\Delta S. An orthogonal projection is Hermitian, so \Pi^\top=\overline\Pi; thus

The conjugation on the right is necessary for a generally complex range. The two projections have the same real diagonal. Cauchy–Schwarz and the operator norm bound imply

Similarly, \Delta S\Delta S^*\le4\Pi gives \sum_j|\Delta S_{ij}|^2\le4\Pi_{ii}\le8l_*. The factorization has rank at most |B|, whence \|\Delta S\|_{\mathrm{HS}}^2\le4|B|.

Finally,

Bounded positive p gives respectively \max|\Delta h_{ij}|\le Cl_*/\sqrt t=C\sqrt t\,t^{e_0}, \max_i\sum_j|\Delta h_{ij}|^2\le Cl_*/t=Ct^{e_0}, and \sum_{ij}|\Delta h_{ij}|^2\le C|B|/t. The fourth-power row sum is at most the largest squared entry times the row square sum, namely Ct^{1+2e_0}t^{e_0}. This proves (81)–(82).□

Proof. Write \mathcal M=M-t\diag(r), so g=\mathcal M^{-1}. The exterior block is dissipative: for every z\in\CC^R,

Cauchy–Schwarz implies \|\mathcal M_{RR}z\|\ge ct\|z\|, and hence \mathcal M_{RR} is invertible with \|\mathcal M_{RR}^{-1}\|\le C/t. For any coefficient vector z\in\CC^B, the vector w=g_{RB}z obeys

Every nonzero off-diagonal entry of M is either a direct lattice hop or comes from the terminal Schur correction. The latter correction is block supported on active neighbors of a connected terminal component: the terminal matrix and its inverse are block diagonal by those components. Any two such neighbors have collapsed distance zero. Consequently the source f is supported on \{i:d_T(i,B)\le1\}.

Let \rho_i=(d_T(i,B)-1)_+ and let \mathcal E be multiplication by e^{at\rho_i}. Then \mathcal E f=f. The function \rho is one-Lipschitz in the collapsed pseudometric. The diagonal and every terminal Schur entry commute with \mathcal E, including arbitrarily large Schur entries. Only direct hopping entries can contribute to

For each such entry, |\rho_i-\rho_j|\le1, so its change is at most e^{at}-1\le C at for t\le1. There are at most 2d direct neighbors in each row and column. The absolute row and column sum bounds therefore give \|E_a\|\le C_d at. Using (86) now gives the exact identity

Choose a>0 so small that \|\mathcal M_{RR}^{-1}E_a\|\le1/2. Absorption proves

All matrices are finite, so no domain issue arises from the exponential weight. Crucially, the right side is the norm of w itself, not the norm of a vector used to represent it.

Because \mathcal C=2it\sqrt{p_R}\,g_{RB}\sqrt{p_B}, the matrix \sqrt{p_B} is invertible, and \sqrt{p_R} commutes with \mathcal E, bounded positive p transfers (87) to \|\mathcal Ev\|\le C\|v\| for all v\in\Ran \mathcal C. Thus |v_i|\le C e^{-at\rho_i}\|v\| on this entire range. The squared norm of coordinate evaluation on \Ran \mathcal C is \Pi_{ii}; equivalently, \Pi_{ii}=\sup_{v\in\Ran \mathcal C,\,\|v\|=1}|v_i|^2. This proves (85). When \mathcal C=0 the projection is zero and the same conclusion holds. Rank deficiency and arbitrarily small positive singular values never entered the estimate.□

Proof. In the support identity (84), use Lemma 6.2 at both endpoints in place of the uniform bound \Pi_{ii}\le2l_*. The estimate |\Delta S_{ij}|\le2\sqrt{\Pi_{ii}\Pi_{jj}} and \Delta S=2i\sqrt t\,\sqrt{p_R}\,\Delta h\,\sqrt{p_R} give the result.□

Proof. All expectations in the proof are conditional on the prestate. Write m=m_*=m_x. The Sherman–Morrison identity gives, on surviving indices,

The denominator is bounded away from zero because g_{xx} is bounded and |\xi+tr_x|\le C\sqrt t. Expansion to second order gives

Here the remainder follows from \EE|\xi+tr_x|^3\le Ct^{3/2}. Since \beta_w<1/2, it follows that

For i\in R', let

The Ward identity and the diagonal lag bound show that

Each omitted index in B\cup\{x\} contributes at most C\sqrt{H_*}. Thus

The cross block in the old normalization is, exactly,

There is no complex conjugate on u_x in this formula, because the resolvent update is symmetric.

The unitary cancellation. Define b_x=U_{R'}^*v. The estimate needed for b_x is stronger than the direct bound \|U_{R'}\|\|v\|. Orthogonality of the B columns of S to its x-th column gives

Since S_B is symmetric, \overline\Gamma=\Id-S_BS_B^*, and \Gamma S_B^*=S_B^*\overline\Gamma. Functional calculus therefore gives the intertwining identity

Also S_{Bx}=\mathcal C_x^\top, so symmetry gives

As \|S_B\|\le1, we obtain \|U^*S_{Rx}\|\le\sqrt m. For i\in R', S_{ix}=2i\sqrt t\,p_xv_i. Restoring and then subtracting the x-th term in U^*S_{Rx} yields

The Gram increment. Set

Multiplication of (99) gives

Let \|\cdot\|_1 denote the trace norm. Equations (97)–(100) imply

For example, the conditional mean of the two mixed terms is bounded by 2|\EE c_x|\|b_x\|\sqrt m\le Cmt^{\beta_w}. The final bound follows from \Tr D_*^2\le\|D_*\|_1^2. Since m\le4t^{1+e_0}, we have \|D_*\|=o(1), pathwise.

The new Gram matrix satisfies the exact congruence

Indeed, W^*W=Q+D_* and \tau G_\tau^{-1}=\Id-Q. Inverting this congruence shows that the new leverages are

No commutation of D_* and G_\tau is used here.

Total leverage. By cyclicity of the trace and W^*W=Q+D_*=\Id+D_*-V_\tau,

The identity

and 0\le V_\tau\le\Id imply

The quadratic remainder is controlled by \Tr D_*^2, without a factor depending on |B|.

Squared leverage. It remains to show that the loss of the pivot row dominates the redistribution of leverage among the survivors. Put

Then

Since \|U\|\le1,

Cauchy–Schwarz and (98) therefore give

To handle normalization, expand

The first-order term with the old rows has conditional mean bounded by

because \|\sum_i m_iu_i^*u_i\|\le m. Replacing the old rows by the updated rows costs at most

For completeness, if M_R=\diag((m_i)_{i\in R'}), then

Its trace norm is at most Cm\sqrt m+Cm^2\le Cm\sqrt m, using \|U_{R'}\|\le1, \|M_R\|\le m, and \|v\|\le C. Taking its trace against D_* proves (108). Moreover, \|W\|\le2 and \max_i\|w_i\|^2\le Cm. The second-order normalization term consequently costs at most Cm\Tr D_*^2 after summing with weights m_i.

The square terms in (106) are small as well. Pointwise Cauchy–Schwarz gives

Write R_D=(\Id+D_*)^{-1}-\Id. Since R_D is Hermitian,

Summing this inequality, and using \|R_D\|_{\mathrm{HS}}^2\le C\Tr D_*^2, gives

All the errors just obtained are o(m^2). More explicitly, after division by m^2, their bounds are constant multiples of

Every exponent is positive under (95). This proves the claimed drift of \mathcal P.□

Proof. At this prefix, \mathcal P\le4l_*\mathcal A, so \Phi\le0. At every later endpoint, including one reached by a failing heat,

The latter bound needs neither a leverage cap nor a diagonal cap: it follows from the definition of ridge leverage and \mathcal A\le|B|. The same statement holds at the projection limit. Sum the conditional drift (112) over the heats, including the last heat, and include the nonpositive ridge changes from (111). Stopping first at H\wedge k gives

Letting k increase proves the expected-count estimate.

This estimate holds conditionally after every later reached prefix before stopping, with the same constant. Take a block length k_{\mathrm{block}}=\lceil 2Cr_*/l_*\rceil, enlarging C if necessary. Conditional Markov inequality bounds the probability of a further block of k_{\mathrm{block}} heats by 1/2. On a prefix at which a stop has already occurred the probability is zero. Iterating over blocks therefore gives \PP(H\ge k k_{\mathrm{block}})\le2^{-(k-1)}, which implies the stated exponential bound after adjusting its constants. Finally,

for sufficiently small t, proving (113).□

Proof. A path minimizing the new cost can be chosen simple by deleting cycles. Suppose it meets k\ge1 different groups, and let A be its old cost. Each site of their union is incident to at most two edges of the simple path. Its cost therefore decreases by at most 2Mk, even if the path returns to a group after leaving it. In the chronological sequence of group visits there are at least k-1 transitions between different groups. To obtain disjoint segments, list the groups visited along the path, suppress consecutive repetitions of the same group, and use the segment from the last visit to one group before the first visit to the next. There are at least k-1 such transitions. Each has old cost at least the old collapsed distance between its endpoints, hence at least \mathcal L. This argument already includes all shortcuts through old terminals. Repeated visits can increase the number of transitions but do not increase the count of distinct groups in the loss 2Mk. Hence A\ge(k-1)\mathcal L and

If the path meets no new group, the old lower bound applies directly.□

Proof. We first prove the metric assertion under the stated location hypothesis, and then use it to continue that hypothesis to the next pivot. Distinct snapshot components are separated by more than R_j. Their new terminal sets, including the current provisional additions, are therefore separated by at least R_j-2\ell_j. For an actual success, the number of new terminals is less than n_h+|B|\le n_h+r_*\le Cn_h for small t_j, since \nu<1+e_h. A provisional group, including its one prospective pivot, obeys the same bound from the heat limit. Apply Lemma 7.1 with M=Cn_h to all additions of the stage at once. Starting with the ordinary metric before any clearing, the multiplicative losses are bounded by a constant times

This sum can be made arbitrarily small. The additive defects, through stage j, are at most

where e_h=.0046<.03. In these estimates t_i\asymp4^{-i} uniformly in the final resolution. They also imply the inequality \mathcal L>2M required in the metric lemma. This proves (128).

At the current running heat prestate, the metric just proved supplies (114) with \ell=\ell_j\asymp t_j^{-4}. The selection implication (115) therefore places the next proposed pivot within \ell_j/2 of B. Its underlying full-range estimate uses only survivor damping and terminal Schur structure, so this application needs neither a lower singular-value bound for the cross block nor a provisional bound on K or D. Induction over the proposed pivots proves that there is no first violation of the location hypothesis. In particular the argument applies before a prospective first budget-breaking update; it does not assume that update will be accepted. Once that pivot is localized, applying the same separated-group bound with it included also gives the metric estimate at its prospective poststate. A subsequent rollback removes its numerical metric cost.

One anchor from each snapshot component forms an R_j-separated set. Packing disjoint balls of radius a fixed fraction of R_j bounds the number of components by CN/R_j^d. Each success, or the current provisional trial, removes at most Cn_h sites. The total fraction removed through all stages is bounded by

for large n_0. Failed trials are rolled back and have no terminal cost. This proves the active-count assertion, including provisional prefixes and their prospective outputs. Moreover, r_*=t_j^{-\nu}\le C4^{\nu m/2}\ll N=4^{20dm} for j\le m/2. Removing the at most r_* core sites from an active set of size at least N/2 leaves a nonempty exterior. At DONE, the precompletion terminal and active sets are those of the committed state at trial start if no heat occurred, and those of the last heat output otherwise. The same metric and active-count bounds therefore hold there.

A snapshot component of size at most r_* has diameter at most (r_*-1)R_j. Adding the pivot neighborhood gives the stated D_s. Assign exactly one cube to each snapshot component, centered at its anchor. For all cubes containing a given point, the disjoint balls of radius R_j/3 about their anchors fit in a ball of radius CD_s. The d-dimensional volume bounds therefore give overlap at most C_d(D_s/R_j)^d=C_dt_j^{-d\nu}. There is no factor for the number of sites in a component: all its edits use this single enclosing cube. All these cubes fit on the torus: j\le m/2 implies D_s\le C4^{(8+\nu)m/2}\ll4^{20m}.□

Proof of Proposition 7.3. Put \epsilon=t^{.10\delta}<1/2. We prove the form bound and the existence of its Schur complements by induction on the prefix length k. In the step for k, suppose the form bound and the identity below hold at every shorter prefix. For each a\le k, let S_{a-1}=\sum_{s<a}L_s, padded to A_0, and E_{a-1}=A_0\setminus A_{a-1}. The induction hypothesis is |S_{a-1}|\le\epsilon D_0 together with

For the empty prefix, S_0=0 and E_0=\varnothing, so the identity is immediate. The initial lower comparison with W makes the kernel of D_0 precisely the constants. At a later prefix, the form bound gives the same kernel for D_0+S_{a-1}. The Schur fact from Section 2 then gives the constant kernel of D_{a-1} and, since A_a is nonempty, the existence of its pivot on O_a. Thus L_a is defined. Minimization over eliminated coordinates also gives, for every extension F to A_0,

The two energy contributions. On surviving pairs the changes of K and F are exactly

where old matrices are restricted to A_a\times A_a and \circ denotes entrywise multiplication. Let D^{\rm edge}_{A_a} be the Laplacian made from the old edges with both endpoints in A_a. For a potential f on A_a, the fact that p is frozen gives

The first term uses the single endpoint difference a^{(a)}. The form B_a is nonnegative because the Schur complement minimizes the old energy over O_a, while the surviving-edge energy does not depend on those coordinates. It carries old energy through the removed sites, so its far part will require the old resolvent decay separately from the decay of a^{(a)}.

Fix a real potential f on A_0 and one split f=\alpha+v|_{A_0} for the whole prefix. Put

The last norm sums nearest-neighbor edges of the cube; harmless enlargement of Q'_a includes its boundary edges. We use this same split for every a and take the infimum defining W(f) only after summing the local estimates.

Uniform tails for the survivor changes. We first record the tail estimate for a^{(a)} and the two survivor changes it produces. Put \rho_i=\dist(i,z_a) and \sigma_i=(\dist(i,O_a)-Cm')_+. Since O_a lies within distance D_s of z_a and Cm'=o(D_s),

If (i,j)\notin Q_a\times Q_a, then \rho_i+\rho_j\ge100D_s, and hence

It follows from (131), after decreasing c and increasing P, that

The inverse of a real symmetric active matrix with damping -it\diag(p) has norm at most C/t. Thus |h^{(a-1)}_{ij}|\le Ct^{-1/2}, and (140) also holds for the absolute values of the tail entries of \Delta K^{(a)} and \Delta F^{(a)}, after increasing P.

On the d-dimensional torus, uniformly in its side length, for every fixed q\ge0 and 0<t\le1,

For q=0 take 0^0=1. Indeed, counting coordinate representatives of minimal length gives at most C_d(1+s)^{d-1} sites in the shell s\le\dist(i,z)<s+1. The sum is bounded by C_{d,q}\sum_{s\ge0}e^{-cts}(1+s)^{d+q-1}, which is at most the right side by comparison with its integral. In particular the zeroth and second moments cost Ct^{-d} and Ct^{-d-2}. Distinctness of the anchors and the triangle inequality consequently give

Therefore the sum of all tail kernels has row and column sums at most t^{-P}e^{-ctD_s}. The same conclusion holds after arbitrary restrictions to survivors or to a committed prefix.

To use the same tail in the energy form, extend the fixed \alpha by zero to the torus. For every lattice displacement u, a shortest nearest-neighbor path and Cauchy–Schwarz give

Apply the row-sum bound to the \alpha differences and (142) to the v differences. Equations (140)–(141) then imply

We henceforth denote such bounds by \varepsilon_t times the indicated norm, where \varepsilon_t=t^{-P}e^{-ctD_s} with P allowed to increase. Because tD_s\asymp t^{-7-\nu}, \varepsilon_t is smaller than every fixed positive power of t. Since b\le1, the right side of (143) is at most \varepsilon_t(\|\alpha\|_2^2+b^{-1}\|\nabla v\|_2^2).

The diagonal, fourth powers, and K. For a site surviving a prefix, its diagonal increments telescope. There are at most Ct^{-d\nu} halos Q_a containing that site. Their near increments obey |\Delta g^{(a)}_{ii}|\le Ct^{.42\delta}; all other increments sum to at most \varepsilon_t, by (140) and \Delta g=a^{(a)}/\sqrt t. Together with (129), this proves (133). In particular, at every committed prestate the Ward identity gives

The first two bounds in (130) imply

For the near pieces, Minkowski’s inequality in \ell^4, followed by halo overlap, costs at most

per row. The sum of the \ell^4 norms of the tail rows is at most their summed \ell^1 norm, which is bounded by \varepsilon_t. Telescoping on the final surviving columns, and observing that discarding columns decreases the initial row norm, proves (134). The strict final inequality holds because .0225\delta-d\nu>0.

For K, at a committed prestate (144) and Cauchy–Schwarz yield

The matrices are symmetric, so the same bound holds for column sums. The near pieces have at most Ct^{-d\nu} overlapping row supports, and the tails have norm at most \varepsilon_t. On A_k the exact telescoping identity is

The row and column sum bound therefore gives \|K^{(k)}\|\le\|K^{(0)}\|+Ct^{.125\delta-d\nu} +\varepsilon_t, which implies (132).

The local energy of a survivor change. We now estimate the near part of the first term in (139) using the common split fixed above. For nonnegative symmetric weights w_{ij} supported in Q_a^2, let l_i=\sum_jw_{ij}, L_\infty=\max_i l_i and L_1=\sum_i l_i. The difference inequality and Lemma 3.3 give

Apply this with w_{ij}=|a^{(a)}_{ij}|^2 on surviving pairs. By (130) and b=t_{n_0}t\le t,

Since 1/3-.23\delta>.25\delta for \delta=.01, we obtain

For the old factor in the cross term, apply the preceding-prefix bound (138) with a local extension. Choose a cutoff \chi_a equal to one on Q_a, supported in Q'_a, and with |\nabla\chi_a|\le C/D_s. Set F=\chi_a\alpha+\chi_a(v-c_a)|_{A_0}. Cube Poincare gives

and hence W(F)\le CW_a. Differences of F agree with those of f on Q_a^2. The old Laplacian edge weights p_ip_j|h_{ij}|^2 are nonnegative; therefore (138) bounds their near commutator energy by CW_a. Weighted Cauchy–Schwarz with (146) proves that the near survivor-edge change has absolute form at most

Indeed its cross term costs the square root of CW_a\cdot Ct^{.25\delta}W_a, and its quadratic term costs Ct^{.25\delta}W_a. The bounded factors p_ip_j only alter the constants.

The old energy carried through O_a. We next estimate B_a, the second term in (139). Test the old Schur minimum with the constant choice f_x=c_a for all x\in O_a. This gives

All old edges with both endpoints in O_a vanish for this trial.

For i\in Q_a, put l_i=\sum_{x\in O_a}F^{(a-1)}_{ix}. The row estimate above gives fourth-power rows below .8H_* at the old state a-1. Cauchy–Schwarz and |O_a|\le Cn_h imply

By symmetry and (144), \sum_i l_i\le Cn_h\le Ct^{-1-.46\delta}. Thus

Apply Lemma 3.3 to the v-c_a term in (148), and the supremum bound to its \alpha term. The resulting near contribution is at most Ct^{.27\delta}W_a.

The far part of this constant trial uses the old terminal geometry. If i\notin Q_a and x\in O_a, then

The old Combes–Thomas bound of Lemma 5.1 therefore gives, after summing the at most Cn_h removed sites,

In (148) use

After summing distinct anchors, the first two terms are bounded by (141) and (142), respectively. For the last term, cube Poincare gives the sufficient pointwise estimate

Summing the far weights costs Ct^{-d}, and summing these local gradient energies costs the halo overlap Ct^{-d\nu}. All remaining factors are powers of t^{-1}, absorbed into \varepsilon_t. Hence the total far contribution from the Schur trials is at most

Closing the comparison induction. Use the decomposition (139). Equations (147), (143), and (149), followed by halo overlap, imply for the current prefix and the one fixed split of f that

Take the infimum over all splits and use W\le C D_0. Since

the resulting estimate is at most \tfrac12t^{.10\delta}D_0(f) for sufficiently small t. This is a strict improvement of the induction bound. It also makes the kernel of D_0+S_k precisely the constants. Since A_k is nonempty, its Schur pivot on E_k exists. The zero extension of L_k and associativity of Schur complements now give

This continues (137) and gives the constant kernel of D_k. Thus the form induction closes. The diagonal and row estimates proved above also preserve the trial-start margins. The application uses no comparison for D or cap on K at an intermediate heat of a provisional trial.

Finally (137) holds also at prefix k. For a neutral charge supported in A_k, inversion of a Schur complement gives the inverse quadratic form of D_0+S_k on its zero extension. The inequalities (1-\epsilon)D_0\le D_0+S_k\le(1+\epsilon)D_0 therefore imply (136) by inversion on the space modulo constants.□

Proof. The dissipative inverse bound gives \|g\|\le C/t, and Ward gives

For a vector z in a sufficiently small fixed \ell^\infty ball, put Z=\diag z and g_z=(M-t\diag(r+z))^{-1}. The Neumann series converges because \|tZg\|\le C\|z\|_\infty. Its first diagonal derivative is t\sum_jg_{ij}^2z_j=(Kz)_i. Thus

For completeness, every diagonal term of order k\ge2 is bounded using the two endpoint rows and k-1 intervening operator norms:

The same expansion, replacing one factor of z by z-w, proves

There is no volume or negative power of t in either constant.

Set P=(\Id-K)^{-1}. The reset equation is

Take a=t^{.35\delta}. The image of the radius-a ball has radius at most

and its Lipschitz constant is at most C\log(1/t)t^{.35\delta}=o(1). It is therefore a contraction. Its unique fixed point in that ball is \zeta. Iteration from zero proves measurability in the initial data. For small t, its imaginary part cannot move p out of a fixed positive bounded range.

The exact resolvent identity, in the h normalization, is

Both old and new inverse operator norms are O(t^{-1}); their endpoint row norms in the h normalization are bounded. For the new rows this follows also by multiplying an old row by the convergent Neumann factor. The identity gives \|\Delta h_i\|_2\le C\rho, and Cauchy–Schwarz using its two endpoint rows gives |\Delta h_{ij}|\le C\sqrt t\,\rho. Interpolating these two estimates proves \|\Delta h_i\|_4\le Ct^{1/4}\rho. Since

symmetry and the row-sum bound prove the norm assertion for K.

For a real diagonal potential f, put B=\sqrt t\,\diag(\zeta). Then h'=(h^{-1}-B)^{-1} and [f,B]=0, so [f,h']=h'h^{-1}[f,h]h^{-1}h'. Factoring the outside matrices gives the exact identity

Write the two outside factors as L and R. Each, and each inverse, differs from identity by O(\rho) in operator norm, since \|h\|=O(t^{-1/2}). Keep the nonconstant p factors explicitly: if P_p=\diag(\sqrt p) and C_f=P_p[f,h]P_p, then

The outside factors and their inverses are \Id+O(\rho), because P_p and P_p^{-1} have bounded norm. Their multiplication changes the Hilbert–Schmidt norm by two-sided factors 1+O(\rho). Finally p'_i/p_i=1+O(\rho) uniformly, so replacing P_p by P_{p'} has the same relative cost. Because

we obtain (151). These two-sided inequalities preserve the constant kernel, and inversion on the neutral subspace proves the final assertion.□

Proof. Initialization supplies the assertions at the first stage, with \|K\|=o(1). Enlarge the permitted bounds on r, p, and the low comparison constants by fixed factors, retaining positive lower bounds. We show that the estimates obtained under these enlarged ranges remain inside them. The form equivalence D\asymp W used below follows from the upper rough comparison for X, and from the two lower bounds X\ge cb\Lambda^+ and X\ge c\Id.

Apply Proposition 5.8 at the old scale t_{\rm old}, and write t=\theta t_{\rm old}. On neutral physical-site charges define the comparison form

The exact reserve-port identity in Lemma 4.1 identifies \mathcal X with \theta X_0 restricted to reserve charges, where X_0 is the duplicated stage-start inverse form. Proposition 5.8 places the actual post-sweep inverse form between (1-\epsilon)\mathcal X and (1+\epsilon)\mathcal X, with \epsilon=Ct_{\rm old}^{.10\delta}. The low scale changes from b_{\rm old} to \theta b_{\rm old}. The physical lag after the sweep is O(t^{.42\delta}), its fourth-row sums are below .7t^{1+\delta}, and its K norm has increased by at most Ct_{\rm old}^{.10\delta}.

If this stage is within the clearing cutoff, consider its snapshot components in their prescribed order. The post-sweep state satisfies (129): its lag and row margins were just obtained, and the form equivalence supplies its comparison. At the first trial, and then at any trial whose preceding prefix has the bounds of Proposition 7.3, the lag and row hypotheses of Proposition 6.6 hold. The location induction of Lemma 7.2 applies throughout the stopped trial. On DONE, the heat and core calculation following Proposition 7.3 supplies its two operation inputs for the new numerical endpoint. Applying that proposition to the enlarged prefix gives a total relative form error O(t^{.10\delta}). Its proof bounds the total K increment by Ct^{.125\delta-d\nu} and the final lag by Ct^{.42\delta-d\nu}=o(t^{.36\delta}). In fourth-row norm the cumulative change, divided by t^{(1+\delta)/4}, is O(t^{.0225\delta-d\nu})=o(1). For large n_0 it moves the sweep’s .7 bound to a bound strictly below .8, uniformly at every committed prefix. These bounds supply the starting margins for the next component. Failed trials restore the previous analytic state; their new pending marks do not alter any of these bounds. If no clearing is scheduled, the sweep endpoint itself has the stronger required estimates.

The committed endpoint has \|K\|<1/2 once the total summable increments are chosen small. Lemma 5.2, applied with its lag and the terminal metric, gives \|(\Id-K)^{-1}\|_{\infty\to\infty}\le C\log(1/t). Proposition 7.4 now restores g_{ii}=r_i. Its fourth-row norm cost relative to t^{(1+\delta)/4} is

The full fourth-row bound is therefore restored. Its form error is O(t^{.35\delta}), and the change of r and the additional K cost are bounded by the same power.

It remains to check the three individual comparison constants. For the sharp upper bound, (154) transforms the diagonal coefficient 1+t_{\rm old}^{\kappa} into 1+\theta t_{\rm old}^{\kappa}. Clearing, sweep, and reset change this by O(t_{\rm old}^{.10\delta}); the change from p to p' at reset has the smaller error O(t^{.35\delta}). For \theta\in[.25,.625], continuity and \kappa<1 give

Thus, for small t_{\rm old},

because .10\delta>\kappa=.005\delta. For the rough upper bound the high coefficient is at most

Choosing C_0 large first and n_0 large second keeps this below C_0, since \theta\le.625. Both low comparison coefficients change only by factors 1+O(t_{\rm old}^{.10\delta}); the lower factor is interpreted as 1-O(t_{\rm old}^{.10\delta}).

All these errors are bounded by a constant times 4^{-c n} for a fixed c>0. Their sum from n_0 onward can be made arbitrarily small, and the corresponding positive products stay bounded above and away from zero. This closes the enlarged ranges for the low constants, r, and p. It also keeps \|K\|\le1/3, using the initialization margin and the summable norm increments. Bounded p,q give \|D\|\le C, hence X\ge c\Id on neutral charges. Relative form comparisons and Schur complementation preserve the one-dimensional constant kernel. The geometry and active-count bounds are supplied independently by Lemma 7.2. After the clearing cutoff the terminal set no longer changes, and the permissible metric defect t^{-1.03} only increases as t decreases, so the same metric bound continues to hold.

Finally, if a_\lambda=(\lambda^2/3)4^{n_0}, then a_\lambda\in(1/4,1] and t_n=a_\lambda(4^{-n}+4^{-m}). The geometric sums and smallness choices above are therefore uniform in m and in the energy after the initialization window has been fixed. One fixed dimension-dependent lower threshold for n_0 suffices simultaneously for all m\ge4n_0, a_\lambda\in(1/4,1], and E\in I_d. For d\ge4, the intrinsic choice of the final disorder threshold will be made in Definition 8.2 after the probability estimates. The argument proves preservation along every finite history produced by the rules; almost-sure termination of ordinary retries is a separate consequence of their conditional failure bound, used in the next section.□

Proof. Put

The radius W_j will contain both the possible reach of a stage-j failure and the connected exploration of a huge component near the root. Passing to the much earlier scale r makes a ball of radius 3W_r small compared with that component. We will retain only k_j of the separated older sites obtained this way. The estimates below verify these three uses of the choices.

We define a deterministic family of finite rooted trees, called witnesses. A leaf of a scale-j witness rooted at x specifies one birth label of index s with n_0<s\le j and s>r, whose anchor is within distance W_j/2 of x. Alternatively, when r\ge n_0, an internal vertex specifies k_j child centers within distance W_j/2 of x, separated from each other by more than 3W_r, and a scale-r witness at each child. The family at scale n_0 is empty, since there are initially no pending sites. The scales strictly decrease down a branch, so every tree is finite.

Extracting a witness and separating its leaf labels. We first show that a pending site yields a witness all of whose leaf births occurred. A site newly marked by a clearing failure has a stage-j birth nearby, giving a leaf by (158). Otherwise it lies in a huge snapshot component. Explore a connected set of

distinct vertices starting at x. Each is within distance M_jR_j\le C4^{(8+\nu)j} of x. If one has latest birth index s>r, its birth anchor is within C4^{(8+\nu)j}+C4^{(8+\nu)s}<W_j/2 of x, once n_0 is large. This again gives a leaf.

If no such latest birth exists, all explored sites were pending at end r, by persistence. Necessarily r\ge n_0. A ball of radius 3W_r contains at most

sites. Greedy packing of the explored set therefore supplies at least c_2 4^{0.9\nu j} centers with the required separation, more than k_j for sufficiently large n_0. Choose k_j and recursively extract their witnesses. Their centers lie within W_j/2 of x.

Every birth anchor in a scale-j witness is within W_j of its root. For a leaf this is immediate. For an extracted branch it follows from

For the entire deterministic witness family the same conclusion follows from W_j/2+W_r<W_j. Thus different child witnesses have disjoint sets of possible physical birth anchors: their support balls have radius W_r and centers more than 3W_r apart. Induction shows that all leaf birth labels in a witness are distinct. This is the reason for spatial separation; no spatial independence is needed.

Summing deterministic witness weights. First fix a deterministic possible witness tree, including all its centers and leaf labels. Its leaves have distinct labels, so (157) bounds the probability that all its leaves occur by the product of their p_s’s. Assign this product as the tree’s weight. Let Z_j be the supremum over roots of the sum of these weights over all allowed scale-j witnesses, and put Z_{n_0}=0. The deterministic family is finite at every scale. A union bound now bounds the pending probability by Z_j. The adaptive extraction above proves only existence of a member of this family; we do not condition on the tree extracted from a realized component. A related expansion into deterministic trees of separated lower-scale events appears in Sznitman’s treatment of cascading events [24, Definition 3.1 and proof of Theorem 3.4]. Here the chronological conditional bound (157) supplies the product weight for distinct birth labels.

There are at most CjW_j^d possible leaf labels, since there are at most j scales, two types, and CW_j^d anchor positions. In counting branches, order the child centers and discard their separation constraint to obtain an upper bound. Hence

The second term is absent when r<n_0, and vanishes at r=n_0.

Closing the numerical recursion. Choose

Then c_{\rm h}<c\alpha_{\rm h} and \beta_{\rm h}+c_{\rm h}\alpha_{\rm h}-c_{\rm h}\ge.15\nu>0. We prove Z_j\le e^{-4^{c_{\rm h}j}} by induction. The leaf term in (159) requires care when \alpha_{\rm h} is tiny. If r<n_0, then

Its logarithm is at most

after one fixed increase of n_0: the entropy is O_d(n_0/\alpha_{\rm h}), whereas 4^{cn_0} dominates both that quantity and 4^{c(n_0+1)/4}. This treats the entire, possibly very long, nonrecursive range. If r\ge n_0, use j<(r+1)/\alpha_{\rm h} and c_{\rm h}j\le c(r+1)/4 instead. Now 4^{cr} dominates the entropy O_d((r+1)/\alpha_{\rm h}) and the target exponent 4^{c(r+1)/4}, uniformly for r\ge n_0. The same half-bound for the leaf term follows. These thresholds depend on d but not on m.

For the second term, assume r\ge n_0 and Z_r>0; otherwise it vanishes. A single fixed increase of n_0 ensures, for every integer u\ge n_0,

Such a choice is possible because exponential growth dominates the fixed linear function on the right. Since j<(r+1)/\alpha_{\rm h}, the induction hypothesis now gives

Using k_j\ge4^{\beta_{\rm h} j}/2 and r\ge\alpha_{\rm h} j-1, we obtain

Together with the first half-bound this closes the induction. Condition (161) explicitly includes the first recursive scales, when the parent index can be nearly n_0/\alpha_{\rm h}; it requires no dependence of n_0 on m.

Finally, these geometric counts are valid on the torus. Since j\le\lfloor m/2\rfloor, every support radius is at most W_j\le4^{5m}, whereas the side is 4^{20m}. All exploration and separation estimates therefore use the same uniform d-dimensional ball bounds as on the lattice.□

Completion of the proof of Proposition 2.1. Initialization and Proposition 7.5 provide the committed-state invariants throughout the construction. Lemma 2.2, together with the ordinary and clearing query rules verified in Sections 5 and 7, supplies all conditional random-input hypotheses, including after failed trials. Preliminary auxiliary retries terminate almost surely. By (162) and (163), with probability 1-o_m(1) the cutoff finds no pending site and every subsequent sweep, including the whole-tail sweep, succeeds.

On that event all terminal sites have their exact true potentials, and the last sweep makes every remaining active potential true as well. The active set A has cardinality at least N/2 by the deterministic induction. Its final frozen coefficients satisfy |r_i|\le C and c\le\Im r_i\le C; the endpoint diagonal-lag bound of the last ordinary sweep gives an active diagonal imaginary part at least a fixed positive constant. The final active inverse is precisely the Schur inverse of

The true terminal principal block is invertible on the probability-one event established in Section 2, and the active Schur block has strictly negative definite imaginary part before inversion. Thus the full inverse exists and has the asserted active diagonal entries.

The analytic interpolation exponents remain 4/3,2/3 in all dimensions: Section 3 proves a deliberately weaker Fourier bound requiring only d\ge3. The common choice of \nu leaves d\nu=10^{-5}\delta, below every analytic gain consumed by packing. The additional factor d in the history scale ratio is needed for the count of spatial witnesses in Section 8.

All numerical lower bounds on n_0, including (161), are fixed independently of m, a_\lambda\in(1/4,1], and E\in I_d. For every starting index above this dimension-dependent cutoff, the certificate probability tends to one as m\to\infty and is at least 1/2 uniformly for all m\ge4n_0 after a further fixed increase of the cutoff. This proves the certificate with its stated uniform constants.□

Proof. First fix k,n_0,m,u,E and A. Write

The product of the potential cube [-1,1]^N and

is compact. For each positive integer q, impose on this product the additional conditions

The inverse is continuous on the determinant-bounded set, so the resulting set is compact. Its projection onto the potential coordinates is compact. Taking the countable union over q and the finite union over A with |A|\ge N/2 gives exactly the feasibility event, hence a Borel set. This argument uses the full finite matrix, without any extra condition on a Schur denominator.

For existence, all constants and starting-index requirements in Proposition 7.5 and the witness proof depend only on d and on the fixed window. They are uniform in u\in(1/4,1] because t_n=u(4^{-n}+4^{-m})\asymp4^{-n}. Equation (164) makes the total failure probability at most 1/2 simultaneously for every m\ge4n_0 once n_0 exceeds one fixed cutoff. On success the certificate has uniform bounds |r_i|\le C_r, \Im r_i\ge c_p, and \Im G_{ii}^{\rm cert}\ge c_g, with all three constants positive where appropriate. Choose an integer k\ge10 above that cutoff and above C_r,c_p^{-1},c_g^{-1}. Then the constructed success event implies (165) for every allowed set of parameters. Although the construction also uses auxiliary randomness, its success implies the Borel feasibility event in the true variables alone. The latter event therefore has probability at least the joint success probability, which is at least 1/2. Thus such k exist.

Finally, if 0<\lambda<\sqrt3\,2^{-n_*(d)}, set

Then n_0\ge n_*(d), 1/4<u\le1, \sqrt{3u}\,2^{-n_0}=\lambda, and s_m=u4^{-m}. Definition 8.2 applies directly. Universal quantification over u and E refers to their individual finite-box probabilities; it does not require a common probability-one event over those parameters.□

Proof. For \Im z>0 define

Then \Im M(z)>0, and the Schur identity gives

For any a\in\RR and v>0,

Hence \int\rho(s)\Im g_s(E+i\eta)\,ds \le\pi\|\rho\|_\infty. Multiply by a nonnegative continuous compactly supported function \varphi(E), integrate in E, and divide by \pi. The spectral theorem and Fubini identify the inner energy integral as the integral of the Poisson convolution of \varphi against \mu_s. That convolution converges uniformly to \varphi as \eta\downarrow0. Since \mu_s(\RR)=1, dominated convergence gives

Regularity of the two Borel measures proves (167). For a fixed value of the conditioning variables the same calculation applies verbatim. No cyclicity assumption on (1,0) is needed.□

Proof. Write H=H_{\lambda,\omega}. For \Im z>0, write

and put

When a diagonal resolvent entry has a finite complex boundary value, write q_x(E+i0)=\Im G_{xx}(E+i0). Our first objective is to prove that, for Lebesgue-almost every fixed E\in I,

The probability-one event here may depend on E. Once this assertion is proved, the last part of the proof will use Fubini and own-site averaging to obtain the almost-sure spectral statement on the whole interval.

A true finite-volume resolvent bound. Fix E\in I and suppress the subscripts on \eta_m,N_m. We seek a lower bound for a bounded function of q_i^{(m)}(E+i\eta) which is valid for the true operator even though the certificate is chosen adaptively.

Let \mathcal S_{m,E} be the event in the true potential variables that a certificate exists. This event is Borel. For each of the finitely many possible A, constrain the true potential vector and r to their compact ranges and impose |\det(H_m-E-\eta\diag_A r)|\ge1/j, j\in\NN. The inverse inequalities define a closed set there. Its projection onto the potential coordinates is compact; taking the union over j and A gives \mathcal S_{m,E}. An auxiliary construction with success probability at least 1/2 therefore implies \PP(\mathcal S_{m,E})\ge1/2 for the true product law.

Almost surely, every true principal matrix of H_m-E is invertible. Indeed, each of its determinants is a nonzero polynomial in the independent continuously distributed diagonal variables, and there are finitely many subsets: the product of all diagonal variables in a nonempty block has coefficient one in its determinant. This event applies to an adaptively selected subset as well. For any qualifying pair (A,r), let S_A be the real symmetric Schur complement of H_m-E after eliminating A^c. The active block of the certified full inverse is

Define G_A=(S_A-i\eta\Id)^{-1} on the same active space. Here \widehat G_A acts on \ell^2(A), and dissipativity on that space gives \|\widehat G_A\|\le(\eta p_-)^{-1}. The inverse identity and the Ward identity give, respectively,

Set C_R=1+(B+1)/p_-. For i\in A, these identities imply

Thus, with c_*=q_*/(p_+C_R^2)>0,

Put T=(H_m-E)^{-1}, and regard B_A=P_ATP_A as an operator on \ell^2(A), where P_A is coordinate projection. Block inversion gives S_A^{-1}=B_A, and hence

Thus G_A is an active-space intermediary: it is the compression of an inverse damped on A. To compare it with the full scalar resolvent, we separate the true spectral modes very close to E before forming a second compression. Those modes may make T large, but the difference they create after compression has rank at most their number. Away from them, the inverse-square weight is bounded by a multiple of the scalar resolvent’s Poisson weight.

For 0<\epsilon\le1, define

The identity

shows that \operatorname{rank}(B_A-B_c)\le r_Q, for every set A. For self-adjoint matrices B,C,

Thus D=G_A-G_c has

Put M_\epsilon=1+\epsilon^{-2}. Since Q commutes with T, the matrix T_c is self-adjoint, and so is its compression B_c on \ell^2(A). Its own functional calculus gives

Together with contraction of the coordinate projection this implies

For the last inequality, outside the cut interval one has \eta/(u-E)^2\le M_\epsilon\eta/((u-E)^2+\eta^2). The argument neither commutes P_A past Q nor uses an operator-monotonicity assertion about f_\eta.

Define the bounded continuous function \Phi(u)=\min\{2M_\epsilon u,c_*\} for u\ge0. Equations (170) and (172), together with the squared triangle inequality, give

Summing and using (171) yields

Every qualifying certificate implies this same inequality for the true scalar resolvent and the true rank r_Q. Thus it holds on the existential event \mathcal S_{m,E} without choosing a certificate measurably. Outside that event the left side is nonnegative, so pointwise it is at least \frac{c_*}{2}\mathbf1_{\mathcal S_{m,E}}-8r_Q/(\eta N). Taking expectations under the unconditioned true product law gives

Under this law, at each site the true diagonal value is uniform on [-\lambda,\lambda] and independent of the other site values. Conditioning on those other values, applying Lemma 9.1, and summing gives the finite-volume Wegner estimate [28]

The rank estimate is under the unconditioned true law; the success event enters only through its probability in (173). Choose 0<\epsilon\le\min\{1,c_*\lambda/64\}, fixed as m varies. Using \PP(\mathcal S_{m,E})\ge1/2, for all sufficiently large m Equation (173) therefore gives

The expression being averaged now depends only on the true potentials. Translation invariance of that true product law on the torus yields

The construction of A need not be translation equivariant.

Infinite volume and the first boundary conclusion. We next transfer (174) to the infinite lattice. Couple the potentials on a root cube of radius comparable to L_m, less than half the torus side, to the infinite iid model. Cutting the boundary bonds of this cube gives the same root component in the two models. Exponential conjugation of a scalar-damped nearest-neighbor resolvent gives

Indeed, conjugation by e^{a\eta\dist(x,\cdot)} changes the hopping in operator norm by at most Ca\eta; a Neumann series and \|G\|\le\eta^{-1} prove the estimate for sufficiently small fixed a. Truncated weights give the same conclusion on the infinite lattice. The bound also holds after boundary bonds are removed. A cube of side comparable to L_m has 2d faces, each containing O_d(L_m^{d-1}) crossing bonds. Resolvent identity across these bonds, with the root a distance \asymp L_m from every face, gives

Indeed, \eta_mL_m\asymp4^{19m}, whereas the logarithm of the prefactor is O_d(m). This estimate is uniform in the potential. Since \Phi is bounded and Lipschitz, it follows that

Each diagonal resolvent entry is the Borel transform of a finite measure, and thus has a finite complex boundary value for Lebesgue-almost every energy; see [25, Corollary 3.25]. Fubini gives a deterministic full-measure subset of I on which these limits exist almost surely. Bounded convergence in (176) shows that, for almost every fixed E\in I,

All subsequent energy limits concern the true resolvent. In particular, no joint choice of the auxiliary construction over uncountably many energies is required.

Boundary nonvanishing and propagation of positivity. The shortest-path observation and rank-one comparison below have precedents in Jakšić and Last [17, author-preprint Section 1.1 and proof of Theorem 2.4]. We give the boundary-value argument used here. We first claim that, for every fixed realization and pair x\ne y, G_{xy}(E+i0) exists finitely and is nonzero for Lebesgue-almost every E. This form will let us apply Fubini before returning to a fixed energy.

To prove the claim, real symmetry gives

Both quadratic forms are Herglotz functions, so G_{xy} has finite boundary values a.e. To prove nonvanishing, let \ell=|x-y|_1. The first nonzero coefficient of the convergent expansion

in the (x,y) entry occurs at n=\ell. A product of fewer than \ell nearest-neighbor steps and diagonal factors cannot join x to y; at length \ell every factor must be a hopping step on a shortest path. Thus (H^\ell)_{xy} equals (-1)^\ell times the positive number of shortest lattice paths and G_{xy} is not identically zero in any dimension considered.

Write each quadratic Herglotz function m_\pm as m_\pm=i(1+b_\pm)/(1-b_\pm) with b_\pm bounded analytic and |b_\pm|<1. Their difference is a quotient of bounded analytic functions with nonzero numerator. A nonzero bounded analytic function f on the upper half-plane cannot have zero vertical boundary values on a positive-measure set. For completeness, normalize |f|\le1 and choose z_0 with f(z_0)\ne0. Jensen’s inequality after mapping \{\Im z>\eta\} to the disk gives

for 0<\eta<\Im z_0. To obtain this boundary inequality from Jensen on smaller circles, use Fatou’s lemma and the continuity of f across the line \Im z=\eta; the single boundary point corresponding to infinity has zero harmonic measure. Another application of Fatou’s lemma as \eta\downarrow0 rules out a positive-measure zero-boundary set. If the finite boundary value of G_{xy} were zero, the bounded-analytic numerator would tend to zero too, since the denominator factors 1-b_\pm are bounded. Applying the uniqueness statement to that numerator proves the claimed nonvanishing.

For each realization, intersect the boundary statements over the countable set of sites and pairs, and then apply Fubini. Together with (177), this gives a deterministic full-measure set I_0\subset I such that for every fixed E\in I_0, (177) holds and, almost surely, all diagonal and off-diagonal boundary values are finite and every off-diagonal value for distinct sites is nonzero. Fix such an E for the rest of this propagation argument. The event that some site has q_x(E+i0)>0 is translation invariant and has positive probability. Ergodicity of the iid translation action makes its probability one. One elementary proof of the required ergodicity approximates an invariant event by a finite-coordinate cylinder event, translates the latter to make the two cylinder events independent, and lets the approximation error tend to zero.

Fix an ordered pair x\ne y and condition on all potentials except the true diagonal value s_x. For almost every such complement, the finite-boundary and nonzero-off-diagonal properties hold for almost every s_x\in[-\lambda,\lambda]. Call these own-site values admissible. This full-measure set may depend on the complement and on E; the argument requires admissibility only for almost every own-site value at this fixed energy. There cannot be two distinct admissible values s_1,s_2 for which q_x>0 and q_y=0. To see this, evaluate at s_1 and set

The boundary imaginary-part matrix on \{x,y\} is positive semidefinite, being a limit of such matrices. Its y-diagonal is zero, so its (x,y) entry vanishes. Hence a is real; it is nonzero by the preceding argument. Moreover, \Im g>0, so 1+hg\ne0 for every real h. The upper-half-plane rank-one identity therefore passes to the boundary and gives

a contradiction. The set of admissible s_x giving q_x>0,q_y=0 therefore contains at most one point and has zero conditional probability under the uniform own-site law. The inadmissible values already form a conditional null set. Thus

Countably many pairs, combined with almost-sure existence of a positive site, prove (169) for every E\in I_0. This completes the fixed-energy part of the proof.

Exclusion of singular mass. We now use the fixed-energy assertion (169) to prove an almost-sure statement about the spectral measure on I. Positivity a.e. in energy is not by itself sufficient: a spectral measure could also have singular mass on an exceptional null set. After fixing the other site potentials, we construct an energy set on which every value of the remaining site potential has a finite diagonal boundary value. Its complement can then be used in conditional spectral averaging. This conditional spectral-averaging step is the purity principle of Aizenman and Warzel [4, Proposition 2.2], credited there to Mira Shamis via Simon–Wolff’s arguments; compare also Jakšić and Last [17, author-preprint Section 4].

Fix a site x, and write \overline\omega for the potentials at the other sites. Let H^c_{\overline\omega} be the compression with site x removed, and let b be the hopping vector from x to the remaining sites. Both are independent of the true diagonal value s at x. For a real value s at that site, the cavity identity (168) becomes

The function M_{\overline\omega} has finite boundary values for almost every energy, since its second term is a Borel transform of a finite positive measure. Its finite boundary values have nonnegative imaginary part. By (169) and Fubini, for almost every \overline\omega and almost every E\in I, the diagonal boundary value exists finitely and has positive imaginary part for almost every s\in[-\lambda,\lambda]. If the finite value M_{\overline\omega}(E+i0) were real, the cavity formula would give zero imaginary boundary part for every s\ne M_{\overline\omega}(E+i0), a contradiction. Thus the set

has full Lebesgue measure in I for almost every complement. The boundary condition defining S_{\overline\omega} is Borel and jointly measurable in (\overline\omega,E): at positive height the resolvents are jointly measurable and continuous in height, so boundary existence and its bounds can be expressed using rational heights. The set depends only on the complementary configuration, and hence is independent of s. At each point of S_{\overline\omega}, the positive imaginary part of M_{\overline\omega}(E+i0) keeps the cavity denominator nonzero for every real s. Thus the diagonal resolvent has a finite boundary value there for every s.

We spell out the measure-theoretic consequence. Let \mu be a finite positive measure with Borel transform g, and let S be a Borel set on which \limsup_{\eta\downarrow0}\Im g(E+i\eta)<\infty. Put

These sets are Borel, since continuity at positive height allows the supremum to be taken over rational heights. Then S=\bigcup_k S_k. For E\in S_k and 0<r<1/k,

Let B\subset S_k be a Lebesgue-null Borel set. For each \varepsilon>0, cover B by open intervals J_j with \sum_j|J_j|<\varepsilon and |J_j|<1/k. Discard intervals missing B and choose E_j\in B\cap J_j. The intervals (E_j-|J_j|,E_j+|J_j|) still cover B, so Equation (178) gives

Letting \varepsilon\downarrow0 proves \mu|_{S_k}\ll dE, and hence \mu|_S\ll dE. This proves directly the bounded-boundary criterion in the Borel-set form we need; compare [25, Theorem 3.23 and Corollary 3.24].

Apply this deterministic argument, for each s, to its site measure \mu_s on S_{\overline\omega}. The bounded-Poisson subsets S_k may depend on s, but the larger set S_{\overline\omega} does not. Therefore

For its complementary Borel null set B_{\overline\omega}=I\setminus S_{\overline\omega}, conditional spectral averaging gives

For almost every complement, this proves \mu_s(B_{\overline\omega})=0 for almost every own-site value s\in[-\lambda,\lambda]. Together with absolute continuity on S_{\overline\omega} for every s, it follows that \mu_x|_I\ll dE almost surely under the true product law. Countably intersecting over x gives simultaneous absolute continuity of all basis measures on I.

The vectors \chi_I(H)e_x consequently belong to \mathcal H_{\mathrm{ac}}(H). Their linear span is dense in \chi_I(H)\ell^2(\ZZ^d), and the absolutely continuous subspace is closed. This proves the spectral-subspace inclusion. Finally, (169) and Fubini give q_0(E+i0)>0 for a.e. E\in I on a probability-one event. If some point of I lay outside the closed spectrum, it would have an open neighborhood in I disjoint from the spectrum. The resolvent is real there, so its diagonal imaginary boundary parts vanish throughout that neighborhood, a contradiction. Thus I\subset\sigma(H) as well. Moreover \mu_0(I)>0: if \mu_0(I)=0, the Poisson integral of \mu_0 tends to zero at each interior point of I, since that point has positive distance from \RR\setminus I. This contradicts q_0(E+i0)>0 for almost every E\in I. In particular \chi_I(H)\ne0.□

Proof of Theorem 1.1. For d=3, fix the interval I_3=(-6+e_2/2,-6+e_2) and set \lambda_3=\lambda_0(3)>0 from Proposition 2.1. For each fixed 0<\lambda<\lambda_3, that proposition supplies the certificate with probability tending to one, hence at least 1/2 for all sufficiently large m, and with constants uniform in E\in I_3. Proposition 9.2 therefore proves the two spectral assertions on I_3.

For d\ge4, set

where n_*(d) is specified in Definition 8.2. For each fixed 0<\lambda<\lambda_d, put

The existence assertion in Proposition 8.3 follows from the construction proving Proposition 2.1. Its final assertion applies the defining feasibility criterion directly: for this coupling and each m\ge4n_0 and E\in I_d, the true-variable certificate event has probability at least 1/2. With \eta_m=s_m=u4^{-m}, its fixed bounds B=p_+=n_*(d) and p_-=q_*=1/n_*(d) satisfy precisely the hypotheses of Proposition 9.2 on I_d. Hence almost surely I_d is contained in the spectrum, and its spectral restriction is purely absolutely continuous and has nonzero weight. In either dimension case, conjugating by \psi(x)\mapsto(-1)^{x_1+\cdots+x_d}\psi(x) gives the identical conclusion for positive hopping and leaves every iid diagonal variable unchanged. The interval is independent of \lambda, and the probability-one event is asserted separately for each fixed disorder parameter. This proves the stated spectral conclusion; no diffusion assertion is used or deduced.□