The model and the result

Fix an integer \(d\ge3\). On the product probability space \[\Omega=[-1,1]^{\mathbb Z^d},\qquad \mathbb P=\left(\tfrac12\mathbf1_{[-1,1]}(v)\,dv\right)^{\otimes\mathbb Z^d},\] let \(V_x(\omega)=\omega_x\). For \(\lambda>0\), define the bounded self-adjoint operator on \(\ell^2(\mathbb Z^d)\) by \[ (H_{\lambda,\omega}\psi)(x) =-\sum_{j=1}^d\bigl(\psi(x+\mathbf e_j)+\psi(x-\mathbf e_j)\bigr) +\lambda V_x(\omega)\psi(x). \tag{1}\] Here \(\mathbf e_1,\ldots,\mathbf e_d\) are the coordinate vectors. Multiplication by \((-1)^{x_1+\cdots+x_d}\) is a unitary operator that reverses the hopping sign and leaves the diagonal potential unchanged. Thus the positive-hopping convention gives the identical conclusion. For a Borel set \(B\), write \(\chi_B(H)\) for the spectral projection, and let \(\mathcal H_{\mathrm{ac}}(H)\) be the subspace of vectors whose spectral measures are absolutely continuous with respect to Lebesgue measure.

The interval is independent of \(\lambda\). The probability-one event may depend on the fixed coupling, and all estimates may depend on the fixed dimension. No uniformity as \(d\to\infty\) is asserted. The integer \(n_*(d)\) for \(d\ge4\) is specified by an infinite family of probability conditions on finite matrices. The proof establishes its existence but supplies no effective numerical bound.

Historical context and scope

Anderson’s random-site-energy model explains how spatial disorder can suppress quantum transport (Anderson 1958). The competing weak-disorder question is whether extended states survive on a Euclidean lattice of dimension at least three. Simon’s Problem 1 asks for a purely absolutely continuous energy range for the nearest-neighbor model with iid uniform potentials in dimensions at least three, at suitable disorder widths (Simon 2000). Theorem 1 supplies the purely absolutely continuous energy range requested in that problem for every sufficiently small fixed disorder width in each fixed dimension \(d\ge3\). It does not locate the full mobility edge discussed after Problem 1 or resolve the diffusion question stated separately as Problem 3.

Rigorous localization was developed through several complementary methods. Fröhlich and Spencer introduced a multiscale Green-function analysis in strong-disorder and suitable extreme-energy regimes (Fröhlich and Spencer 1983); Fröhlich, Martinelli, Scoppola and Spencer subsequently established pure-point spectrum with exponentially decaying eigenfunctions in such regimes (Fröhlich et al. 1985). The fractional-moment method of Aizenman and Molchanov gives another route to spectral localization at large disorder and extreme energies (Aizenman and Molchanov 1993). Corollary 2 combines a fixed-disorder edge-localization theorem with Theorem 1 at the same coupling. A localized edge interval alone does not determine the spectral type throughout the rest of the band.

On the Bethe lattice, Klein proved purely absolutely continuous spectrum on compact intervals inside the free band at sufficiently weak disorder (Klein 1994, 1998). Aizenman, Sims and Warzel established stability of the absolutely continuous component by a fluctuation-based method (Aizenman et al. 2006); that result does not itself assert spectral purity. Aizenman and Warzel used resonant delocalization to obtain an absolutely continuous component beyond the free band for regular full-support laws at weak disorder, and proved pure absolute continuity near the spectral edges in a weak bounded-disorder regime that includes the uniform law (Aizenman and Warzel 2013). For sufficiently high-degree trees, at a specified hopping scale and for a regular class of full-support single-site densities, Aggarwal and Lopatto obtained mobility edges separating absolutely continuous and pure-point intervals (Aggarwal and Lopatto 2025). The branching geometry distinguishes these results from the model on \(\mathbb Z^d\); the last result also has different law and scale assumptions.

Jakšić, Last and Warzel gave a further tree argument and a conditional lattice criterion through the Schur complement of a two-site Poisson-transformed spectral measure (Jakšić et al. 2026). Their criterion assumes a positive lower limit of the expected energy-integrated quantity and concludes a nontrivial absolutely continuous component. It does not establish that lower bound for the present weak-disorder lattice model.

Erdős, Salmhofer and Yau proved a weak-coupling diffusion limit for the lattice model: at disorder-dependent long times and spatial scales, the expected Wigner distribution converges to a heat-equation solution as the coupling tends to zero (Erdős et al. 2007). For Gaussian disorder, Black, Drogin and Hernández obtained estimates at long finite times and positive resolvent heights in a weak-coupling regime (Black et al. 2025). Here the boundary limit is taken at each fixed sufficiently small positive coupling, and the conclusion also excludes singular spectral mass.

The proof mechanism

The proof first establishes a finite-volume resolvent certificate. On a large periodic lattice, we construct a set containing at least half the sites and add small complex damping only on that set. The diagonal imaginary parts there remain bounded below as the damping tends to zero. At each site, the final real potential is its original independent uniform variable. Proposition 3 states the certificate precisely.

We begin with the free operator and write each uniform potential as a series of independent binary signs. Revealing a sign replaces part of the artificial damping by true randomness. Active sites retain damping; terminal sites have their complete true potentials and have been eliminated by a Schur complement. During a stage, a diagonal self-consistency parameter remains frozen until the stage-end reset. We carry three controls for the rescaled inverse on active physical sites. Its complex entrywise square controls the linearized equation for the eventual diagonal reset. A bound on row fourth moments prevents the mass of a resolvent row from concentrating at a few sites, limiting the influence of individual updates. The Ward identity turns squared absolute resolvent entries into a Laplacian whose inverse form supplies a spatial energy norm for accumulating local changes. Fourier estimates for the free resolvent initialize these bounds with slack on the chosen energy interval.

During the one-digit stages, ordinary updates are tested on finite tiles, since a union bound over all sites at all scales would be too expensive. A coupled active site carries a true prefix with an untouched suffix, and a first tile trial uses its next true sign when that site is reached. If an ordinary tile trial fails, its numerical changes are rolled back and every answer remains in the retained information. All active sites in that tile are marked pending, including sites not reached by the trial. The mark is conservative: at a pending site the construction ceases to rely on the true-prefix and untouched-suffix guarantees. Ordinary updates at pending sites use independent auxiliary digits; a retry uses new ones throughout the tile.

At later scales we clear small components of pending sites. A unitary scattering matrix records the interaction between a fixed core and its exterior. The range of its cross block contains the survivor directions through which completing the core can act. Unitarity bounds the size of that change, so small diagonal entries of the projection onto the cross-block range make its effect small in every surviving row. This gives a deterministic target for preparing the core, with no freshness assumption on the core values.

Regularized projection weights guide exposures of true residuals at exterior sites. A spatial estimate controls the entire cross-block range even when its nonzero singular values are arbitrarily small; together with component separation and the retained marking rules, it keeps selected exterior pivots close to their core and ensures that their true residuals are still untouched. A negative drift bounds the number of exposures, while stopped martingale estimates control their accumulated effect, including a first violating update. After successful preparation, any still-unknown true core values are revealed and recorded, and the deterministic completion is applied. The core and exposed exterior sites then become committed terminals. A failed clearing rolls back its provisional matrix changes, keeps every answer, and marks its core and every inspected exterior site pending.

Two inductions connect these operations to the certificate. A deterministic induction combines committed local changes, controls the geometry of terminal sites, keeps at least half the sites active, and closes the self-consistent reset. A probabilistic induction tracks the retained information and bounds failures at deterministically ordered labels by successive conditioning. A surviving pending site yields either a recent failure label or many separated older witnesses whose leaf labels are distinct. With high probability the cutoff finds no pending sites and the later sweeps succeed. The last sweep then inserts the entire remaining true suffix at every active site. Those sites keep the final small damping, although their real potentials are now complete true values.

The spectral argument converts the certificate to information about the ordinary resolvent of the true operator. A cut by its spectral projection onto a narrow energy window controls the comparison through that projection’s rank for every qualifying certificate. Own-site averaging under the original product law of the true potentials bounds the expected rank, giving a positive expectation for a bounded function of the true scalar diagonal imaginary part. That estimate passes to infinite volume. At almost every fixed energy, ergodicity and rank-one changes of a site potential then propagate boundary positivity to every site almost surely. Conditioning on the operator with one site removed produces an exceptional energy set independent of the remaining site variable. Absolute continuity holds on its complement, and spectral averaging removes mass on the exceptional set itself. This final step gives purity.

The form of the construction is common to all fixed \(d\ge3\). Dimension enters through three estimates. A deliberately weaker Fourier bound retains the three-dimensional interpolation gains in every dimension, with a convex-slice argument handling two-resolvent shell tangencies. The allowed size exponent for pending components is chosen inversely proportional to \(d\), keeping spatial packing losses below the analytic gains. The history recursion uses a dimension-dependent separation of scales to pay for the possible positions of spatial witnesses.

Related resolvent controls appear in Black, Drogin and Hernández’s work on the Gaussian Anderson model: a self-consistent diagonal shift, concentration governed by fourth-power row sums, and an elliptic equation for averaged squared resolvent entries (Black et al. 2025, secs. 1.1–1.2). Their spatial equation develops the \(T\)-equation approach of Erdős, Knowles, Yau and Yin for random band matrices (Erdős et al. 2013). Here the Ward identity gives an exact Laplacian for each current matrix state, and we transport its inverse form and the other controls through the adaptive revelations.

Coexistence with edge localization

The interval \(I_d\) lies above the lower free band edge \(-2d\). At the lower edge of the almost-sure spectrum, fixed-disorder localization gives a separate interval. Combining that result with Theorem 1 yields the following consequence at the same fixed coupling.

The certificate

Fix the interval \(I_d\) from (2), a coupling \(0<\lambda<1\), and integers \[ \begin{aligned} n_0&=\lfloor\log_4(3/\lambda^2)\rfloor,\qquad m\ge4n_0,\\ s_n&=\frac{\lambda^2}{3}4^{n_0-n}\quad(n_0\le n\le m),\\ \eta&=s_m,\qquad t_n=s_n+\eta. \end{aligned} \tag{3}\] Equivalently, if \(u=(\lambda^2/3)4^{n_0}\in(1/4,1]\), then \(s_n=u4^{-n}\) and \(\eta=u4^{-m}\). Thus \(\frac14 4^{-n}<s_n\le4^{-n}\) and \(t_n\asymp4^{-n}\), with absolute comparison constants. Let \[L=4^{20m},\qquad \mathbb T_L=(\mathbb Z/L\mathbb Z)^d,\qquad N=L^d,\] and let \(H_m\) be the periodic version of (1), in the negative-hopping convention: \[(H_m f)(i)=-\sum_{|e|_1=1}f(i+e)+\lambda V_i f(i), \qquad V_i\ \text{independent and uniform on }[-1,1].\] There is an independent potential at every point of this \(d\)-dimensional torus. For \(A\subset\mathbb T_L\), write \(P_A\) for coordinate restriction or projection, as indicated by its domain, and let \(\mathop{\mathrm{diag}}_A r\) be the full matrix with diagonal \(r_i\) on \(A\) and zero elsewhere.

The proof occupies Sections 3–8. Section 9 proves that this certificate implies Theorem 1. The true potential vector retains its original product law as an unconditional marginal of the enlarged probability space. The success event may depend on that vector. The spectral proof uses a feasibility event in the true variables and unconditional averaging.

True digits, site states, and retained information

A uniform random variable \(\lambda V_i\) has the representation \[ \lambda V_i=\sum_{k\ge1}\lambda2^{-k}\varepsilon_{i,k}, \qquad \mathbb P(\varepsilon_{i,k}=1) =\mathbb P(\varepsilon_{i,k}=-1)=\tfrac12, \tag{4}\] with independent signs over all sites and indices. Resolve the possible binary ambiguity on its null set by a fixed convention. After the first \(h\) signs at a site have been exposed, and before any further information about that site has been queried, its remaining sum is uniform on \([-\lambda2^{-h},\lambda2^{-h}]\), with mean zero and variance \(\lambda^2 4^{-h}/3\). In particular, after \(n-n_0\) digits its variance is \(s_n\), and its next digit has amplitude \(\sqrt{3s_n/4}\).

At each committed state the physical sites are partitioned into an active set \(A\) and a terminal set \(T\). Active sites retain artificial damping. A committed terminal site has its exact true potential, has been Schur-eliminated, and never changes again. During the one-digit stages an active site’s real potential value is partial. At the final endpoint, the active sites carry complete true potentials while retaining damping.

A subset of the active sites is marked pending; a nonpending active site is coupled. Before the final residual sweep, the construction maintains the following implication: at a coupled site the installed partial value is exactly the true revealed prefix, and no digit of its remaining suffix has been queried. At a stage boundary this prefix has \(n-n_0\) digits; during a one-digit sweep it has one more digit at sites already processed. The final sweep installs the entire remaining true suffix. A pending site’s partial value may contain auxiliary digits, and the construction makes no freshness claim for its true suffix. The mark is conservative: the marking rule can cover an active site whose own next digit was not queried. A pending site can leave this class only by becoming terminal.

The numerical matrices and the information already revealed are separate records. Inside a trial, newly terminal physical sites and other matrix changes are provisional until the trial commits. A failure restores the pretrial potential values and partition into active and terminal sites, together with the corresponding matrices, and then applies the trial’s marking rule for pending sites. All answers already obtained remain in the retained history. Previously committed terminal sites remain permanent.

Initially every site is active and coupled, and every partial value is zero. The retained-information lemma below supplies the conditional law of an unqueried input. Sections 5 and 7 verify that the inputs selected by ordinary trials and clearings are eligible for that law. In the matrix estimates, “fresh” means a centered variable with the stated law conditional on the history available before the chosen update. It is an explicit input until the corresponding eligibility check has been made.

Schur reduction and the Ward Laplacian

For the fixed \(E\) and \(\lambda>0\), almost surely every principal matrix of the true \(H_m-E\) is invertible. Indeed, for a nonempty subset its determinant is a polynomial in the independent continuous potentials, with nonzero highest monomial \(\prod_{i}\lambda V_i\). Its zero set has Lebesgue measure zero. There are finitely many subsets on the torus. We henceforth work on this event, which also covers subsets chosen adaptively later. This probability-one property is not a preliminary test of unrevealed potentials and supplies no information to the algorithm’s filtration. The empty terminal block requires no inverse.

Let \(\mathcal H\) be the real symmetric matrix with the current real potential values on \(A\) and exact true values on \(T\). Eliminate the terminal block at the real energy \(E\), obtaining \[M=(\mathcal H-E)_{AA} -(\mathcal H-E)_{AT}(\mathcal H-E)_{TT}^{-1} (\mathcal H-E)_{TA}.\] The matrix \(M\) is real symmetric. Its off-diagonal entries connect nearest neighbors or sites joined through one connected component of terminal sites. Throughout, the subscript in a Schur complement names the eliminated indices: if \(R\) is the complement of \(\mathcal E\), \[\mathop{\mathrm{Schur}}_{\mathcal E} B =B_{RR}-B_{R\mathcal E}B_{\mathcal E\mathcal E}^{-1}B_{\mathcal ER}.\] We use this notation only when the displayed inverse exists. Put \(r=(r_i)_{i\in A}\in\mathbb C^A\), \(p_i=\Im r_i>0\), and, for a scale \(t>0\), define \[g=(M-t\mathop{\mathrm{diag}}r)^{-1},\qquad h=\sqrt t\,g,\qquad K_{ij}=h_{ij}^2,\quad F_{ij}=|h_{ij}|^2,\quad q_i=\Im g_{ii}.\] The inverse exists because its matrix has strictly negative imaginary part. In particular \(\|g\|\le(t\min p_i)^{-1}\). Both \(g\) and \(h\) are complex symmetric. For a matrix \(B\), \(\Im B=(B-B^*)/(2i)\); here it is also the entrywise imaginary part. The resolvent identity gives the Ward identity \[ h\mathop{\mathrm{diag}}(p)h^*=\Im g,\qquad Fp=q. \tag{5}\] Consequently \[D=\mathop{\mathrm{diag}}(pq)-\mathop{\mathrm{diag}}(p)F\mathop{\mathrm{diag}}(p)\] is a real graph Laplacian, with nonnegative edge weights \(p_i p_jF_{ij}\). Its quadratic form is \[D(f)=\frac12\sum_{i,j}p_i p_jF_{ij}|f_i-f_j|^2.\] The nonnegative weights \(F_{ij}=|h_{ij}|^2\) describe how the Ward mass is distributed across a row. We use \(B(f)=\langle f,Bf\rangle\) for the quadratic form of a Hermitian matrix. At the committed states below, the kernel of \(D\) is precisely the constants. Its pseudoinverse \(X=D^+\) is then an inverse on neutral charges, namely vectors \(y\) with \(\sum_i y_i=0\).

We record the precise Schur identity for these inverse forms. If \(U\) is eliminated and \(R=A\setminus U\ne\varnothing\), then \(D_{UU}\) is positive definite: extending a zero-energy vector by zero on \(R\) would otherwise give a nonconstant vector in \(\ker D\). Moreover, \[(\mathop{\mathrm{Schur}}_U D)(f_R)=\min_{f_U}D(f_R,f_U).\] It follows that \(\mathop{\mathrm{Schur}}_U D\) has only the constant kernel. For a neutral \(y\) supported on \(R\), extend it by zero to \(A\) and use the variational formula \[X(y)=\sup_f\{2\Re\langle y,f\rangle-D(f)\}.\] Minimizing first over \(f_U\) proves \[ (\mathop{\mathrm{Schur}}_U D)^+(y)=D^+(y). \tag{6}\] All later inverse-form comparisons across deletion are understood in this sense, on survivor-supported neutral charges, not as an equality of full pseudoinverse matrices.

At the beginning of a stage the state satisfies \(r_i=g_{ii}\). During a sweep or clearing trial, \(r\) remains frozen. Thus \(q=p\) at the stage boundary, and the difference \(g_{ii}-r_i\) measures the subsequent diagonal lag. Because \(g\) itself depends on \(r\), restoring the equality requires a small self-consistent correction. The stage-end reset in Section 7 constructs that correction.

The role of \(K\) is visible in this self-consistency equation. For the physical active inverse, differentiation with respect to the complex diagonal parameter gives \[\frac{\partial g_{ii}}{\partial r_j}=t g_{ij}^2=K_{ij}.\] Thus the linearization of \(r-(g_{ii}(r))_{i\in A}=0\) is \(I-K\), which is invertible whenever \(\|K\|<1\).

For \(n_0\le n<m\), a transition from scale \(n\) to \(n+1\) consists of an ordinary sweep, any scheduled clearings at \(t_{n+1}\), and a diagonal reset. Clearings run only through \(J=\lfloor m/2\rfloor\). At the end of stage \(J\), the construction aborts if any site remains pending; after \(J\) it aborts if the first trial at any tile fails. At scale \(m\), a final residual sweep inserts each remaining true suffix, leaves damping \(\eta\), and has no reset. Sections 7 and 8 specify the trial, marking, and abort rules.

The bounds carried between stages

The full-torus positive nearest-neighbor Laplacian is denoted by \(\Lambda\). Thus \(\Lambda(v)\) is the sum of \(|v_i-v_j|^2\) over unordered nearest-neighbor edges. Charges on a subset are extended by zero when inserted in \(\Lambda^+\). We fix \[\delta=0.01,\qquad \kappa=0.005\delta,\qquad \nu=10^{-5}\delta/d,\qquad H_*=t^{1+\delta},\qquad b=t_{n_0}t.\] Thus \(d\nu=10^{-5}\delta\); the dimension-scaled choice will absorb the overlap of component neighborhoods. Constants in \(O(\cdot)\) depend only on \(d\), the displayed fixed energy window, and the fixed broad bounds specified below. By making \(\lambda_0(d)\) smaller we may make \(n_0\) arbitrarily large. Every later use of a strict power margin is made after this single choice.

At a committed stage boundary, the entries of \(r\) are bounded and \(c\le p_i\le C\). We maintain \[\|K\|\le\tfrac13,\qquad \max_i\sum_j|h_{ij}|^4\le H_*,\qquad \ker D=\mathbb C\mathbf1.\] The fourth-power bound is the squared row norm \(\sum_jF_{ij}^2\). It limits concentration across the columns of each row. The local variance estimates need this control in addition to Ward’s bound on the weighted row sum. On neutral charges we maintain the three comparisons \[ \begin{aligned} X&\ge c_n b\,\Lambda^+,\\ X&\le C_0I+C_n b\,\Lambda^+,\\ X&\le(1+t^\kappa)\mathop{\mathrm{diag}}(p^{-2})+C_n t\,\Lambda^+. \end{aligned} \tag{7}\] Here \(C_0\) is a sufficiently large fixed constant, while \(c_n,C_n\) remain bounded above and below by positive constants, changing only by summable relative errors. Boundedness of the weighted row sums of \(F\) also gives \(X\ge cI\) on this subspace: the absolute row sums of \(D\) are at most \(2\max_i p_iq_i\), so \(D\le CI\).

The first two comparisons express an equivalence of spatial energies. To formulate it also for the duplicated sites used in a sweep, allow one or two ports at each active physical site. For a port charge \(y\), let \(y_\Sigma\) be the full-torus charge formed by summing its ports at each physical site and extending by zero. A full-torus function \(v\) is repeated at all ports at a site. Define the dual quadratic forms \[ \begin{aligned} W^*(y)&=\|y\|_2^2+b\,\Lambda^+(y_\Sigma),\qquad \sum_x y_x=0,\\ W(f)&=\inf_{f=\alpha+v|_{\mathrm{ports}}} \left(\|\alpha\|_2^2+b^{-1}\Lambda(v)\right). \end{aligned} \tag{8}\] The second form is on potentials modulo constants. Lemma 5 proves the duality and explains the resulting relations \(X\asymp W^*\) and \(D\asymp W\). The final comparison in (7) is sharper: its high-frequency diagonal coefficient tends to one. Retaining that coefficient will supply the extra gain in the row-fourth-moment estimates.

We also track geometry. Let \(d_T(i,j)\) be shortest-path pseudodistance in the physical torus when an edge touching a terminal site has cost zero and every other nearest-neighbor edge has cost one. The induction will maintain \[d_T(i,j)\ge\tfrac12\mathop{\mathrm{dist}}(i,j)-t^{-1.03}.\] The matrices \(M\) can have large entries after a Schur reduction. Nevertheless, every nonlocal entry joins points at zero \(d_T\)-distance. This permits the resolvent decay estimate needed for local monitoring. Section 7 proves the geometric bound, including the temporary terminal sites inside a clearing trial.

There are two distinct uses of these bounds. An ordinary trial stops at the first local budget violation or failure of a required prefix bound, including the global correction, row, and lag caps. The violating increment is included in the stopped estimates. Clearing prefixes are governed by their own stopped trial budgets, which yield the provisional diagonal and row bounds; geometry is supplied separately. For each successful clearing, the assembly argument proves the \(K\) and Laplacian comparisons for the committed output from its stated hypotheses. The trial statements make these two regimes explicit. The maximal concentration estimate used for their stopped probability bounds is proved in Section 5.

Duality and lattice estimates

Let \(\mathbb T_L=(\mathbb Z/L\mathbb Z)^d\), \(L\ge4\), and \(N=L^d\). An index set \(\mathcal A\) consists of sites of \(\mathbb T_L\), with at most two copies of each site permitted. Write \(\pi:\mathcal A\to\mathbb T_L\) for the underlying site map. For a function \(v\) on \(\mathbb T_L\), let \((\mathcal Cv)_x=v_{\pi(x)}\). The adjoint sends a charge \(y\) on \(\mathcal A\) to \[(y_\Sigma)_i=(\mathcal C^*y)_i =\sum_{x:\pi(x)=i}y_x,\] with value zero at sites not represented in \(\mathcal A\). The positive nearest-neighbor Laplacian is denoted by \(\Lambda\), and \(\Lambda^+\) is its inverse on mean-zero functions, extended by zero on constants. We normalize the gradient energy by \(\|\nabla v\|_2^2=\langle v,\Lambda v\rangle\).

We use unnormalized Fourier coefficients \(\widehat y(k)=\sum_x y_xe^{-ik\cdot x}\), with inverse normalization \(N^{-1}\). Frequencies belong to the grid \(\mathcal K_L=(2\pi/L)\mathbb Z^d\) modulo \(2\pi\mathbb Z^d\), represented in \([-\pi,\pi)^d\); the symbol of \(\Lambda\) is \[ \ell(k)=4\sum_{j=1}^d\sin^2(k_j/2)\asymp |k|^2. \tag{9}\]

These lattice estimates are uniform in the torus and cube sizes. They will allow local changes to be measured in the same forms as the initial state. We now construct that state and identify its Fourier comparison operator.

The fixed interval and the scalar diagonal equation

Self-consistent diagonal equations and free lattice resolvent estimates also play a central role in (Black et al. 2025, sec. 2.2 and Appendix A). We prove the specific finite-volume estimates needed here, including the gap at the smallest nonzero grid frequency.

The free dispersion in the negative-hopping convention is \[a(k)=-2\sum_{j=1}^d\cos k_j=-2d+\ell(k).\] Write \(a_d=e_2\) for \(d=3\) and \(a_d=1/100\) for \(d\ge4\), so that \(I_d=-2d+(a_d/2,a_d)\). Choose \(e_2>0\) fixed and sufficiently small, with \(e_2<1/100\). For estimates involving a small shift of energy, use the larger compact offset interval \[\mathcal J_d=[2a_d/5,5a_d/4]\subset(0,0.04).\] For \(d\ge4\) this is exactly \([1/250,1/80]\). The parameter \(a_d\) is fixed before the coupling or any scale is chosen. In the free-resolvent analysis below, constants may depend on \(d\) and this fixed \(a_d\); this dependence is included in the notation \(C_d\). In what follows, integrals over the frequency torus are normalized by \((2\pi)^{-d}\).

Proposition 10 supplies every initial comparison constant with fixed slack. Lemma 5 also gives \(X\asymp W^*\) and \(D\asymp W\) at this state. The subsequent construction transports these forms as sites are duplicated, updated, and eliminated; the weighted cube estimate provides the corresponding local energy control.

Splitting a site into two ports

At the beginning of a stage \(n<m\) that inserts one digit, put \(t=t_n\) and \[w=\frac{3s_n}{4t_n},\qquad \theta=1-w=\frac{t_{n+1}}{t_n}.\] For the final residual sweep at \(n=m\), take instead \(t=t_m=2\eta\) and \(w=\theta=1/2\). In both cases \(\theta\in[1/4,5/8]\). The one-step estimates below use only this range and the conditional moment assumptions stated below, so they apply to the final suffix as well as to a single digit.

Start from a committed physical state at a stage boundary satisfying the bounds of Section 2, including \(r_i^{\rm phys}=g_{ii}^{\rm phys}\). The terminal set stays fixed throughout the sweep. At a numerical prefix, let \(M_{\rm cur}\) be the real Schur matrix with the current real potential values, and let \(\mathcal I\) be the set of active physical sites whose current sweep increment has already been inserted. The physical inverse represented at that prefix is \[ g^{\rm phys} =\left(M_{\rm cur}-t\mathop{\mathrm{diag}}\bigl((\vartheta_i r_i^{\rm phys})_{i\in A}\bigr)\right)^{-1}, \qquad \vartheta_i=\theta+w\mathbf1_{\{i\notin\mathcal I\}}. \tag{27}\] Thus an unprocessed site retains the full frozen shift \(-tr_i^{\rm phys}\), whereas a processed site retains \(-\theta t r_i^{\rm phys}\). The matrix \(M_{\rm cur}\) records the real potential additions; \(r^{\rm phys}\) does not change. This description applies equally to a committed prefix and to the current provisional prefix of a tile trial.

Represent each active physical site initially by a bit port and a reserve port, with weights \(k_{\rm bit}=w^{1/4}\) and \(k_{\rm res}=\theta^{1/4}\). The bit port is deleted when that site is processed, and the reserve port remains throughout the sweep. In particular, deleting a bit port does not make its physical site terminal. For the ports remaining above \(i\), one has \(\sum_x k_x^4=\vartheta_i\). If \(x\) is a port above \(i\) and \(y\) is a port above \(j\), define \[ \begin{aligned} g_{xy}&=k_xk_y g^{\rm phys}_{ij},& r_x&=k_x^2r^{\rm phys}_i,& p_x&=\Im r_x,\\ h&=\sqrt t\,g,& K_{xy}&=h_{xy}^2,& F_{xy}&=|h_{xy}|^2. \end{aligned} \tag{28}\] After a deletion these formulas use the new physical inverse and the remaining ports. This virtual \(g\) is not an inverse on the duplicated space; its entries are repeated and rescaled physical resolvent entries. Put \(q_x=\Im g_{xx}\) and \(D=\mathop{\mathrm{diag}}(pq)-\mathop{\mathrm{diag}}(p)F\mathop{\mathrm{diag}}(p)\). When \(D\) has only the constant kernel, \(X=D^+\) denotes its inverse form on neutral port charges, as in Section 2.

The reserve part of this identity is the target of the sweep. If \(y\) is a neutral physical charge, let \(y_{\rm res}\) put that charge on the reserve ports and zero on the bit ports. The exact initial identity is \[ \theta X_0(y_{\rm res}) =\theta X^{\rm phys}(y) +(1-\theta)\sum_i\frac{|y_i|^2}{(p_i^{\rm phys})^2}. \tag{32}\] If all bit ports have been deleted, the definitions give \(h_{\rm cur}=\sqrt{\theta t}\,g^{\rm phys,new}\) and \(p_{\rm cur}=\sqrt\theta\,p^{\rm phys}\). When the current Laplacian has only the constant kernel, these scalings give \(X^{\rm phys,new}(y)=\theta X_{\rm cur}(y_{\rm res})\). We must therefore keep \(X_{\rm cur}\) close to \(X_0\) on the surviving reserves. In the target form (32), an old sharp diagonal coefficient \(1+t^\kappa\) becomes \(1+\theta t^\kappa\). Since \(\theta^\kappa>\theta\), this lies below the next coefficient \(1+(\theta t)^\kappa\), leaving room for later errors. The initial reserve-row fourth-power sum, \(\theta^2H_*\), likewise has strict slack below the cap at the new scale, \((\theta t)^{1+\delta}\). The estimates below control the corrections that could consume these two margins, as well as the \(K\) norm and the diagonal error.

One deletion and the transported matrices

Let \(\mathscr F\) denote the retained history just before choosing the next permutation slot, including the numerical prestate and all earlier queries. Conditional on choosing a particular remaining bit port \(x\) above a physical site \(i\), the real variable \(\xi_0\) is required to satisfy \[ \mathbb E(\xi_0\mid\mathscr F,x)=0,\qquad \mathbb E(\xi_0^2\mid\mathscr F,x)=t,\qquad |\xi_0|\le C\sqrt t. \tag{33}\] Deleting \(x\) adds the real potential increment \(k_x^2\xi_0=\sqrt w\,\xi_0\) to the partial potential at \(i\). It also changes \(\vartheta_i\) from \(1\) to \(\theta\) in (27), thereby adding \(tw r_i^{\rm phys}=k_x^2tr_x\) to the diagonal of the damped matrix. The complete change to that matrix is therefore \(k_x^2(\xi_0+tr_x)\). Only the first summand changes the real partial potential; the second removes the bit port’s share of the frozen shift and damping. For an ordinary stage, \(\xi_0\) is the next true digit, or an auxiliary digit when prescribed, divided by \(\sqrt w\); its physical variance is \(s_n-s_{n+1}=wt\). The expectation symbol in the remainder of Section 4 denotes this conditional expectation whenever the prestate and hypothetical pivot have been fixed. In such statements, “conditioning on the prestate” always includes the full retained history \(\mathscr F\), not just the matrix entries of the current state. At the final sweep, the untouched true suffix \(\zeta\) has conditional mean zero, variance \(s_m=\eta=wt\), and \(|\zeta|\le\sqrt{3\eta}\); thus \(\xi_0=\zeta/\sqrt w\) satisfies (33). We assume neither symmetry nor two-point support in any estimate below. The ordinary query rule is verified in Section 5 using Lemma 4; the conditional law is an input here, and no conclusion conditions on future success of the construction. The Laplacian pivot is strictly positive: by Ward, \(q_x-p_xF_{xx}=\sum_{y\ne x}p_yF_{xy}\), and the reserve port at the same site contributes a positive term because the physical diagonal inverse has positive imaginary part.

Pad \(J_x\) and \(L_x\) by zero on the deleted coordinate and on all earlier deleted ports. Adding such a survivor-supported matrix commutes with earlier elimination. Writing \(E_{\rm del}\) for the set of deleted ports, induction gives the exact cumulative identities \[ I-K_{\rm cur}=\mathop{\mathrm{Schur}}_{E_{\rm del}}\left(I-K_0-\sum J_x\right),\qquad D_{\rm cur}=\mathop{\mathrm{Schur}}_{E_{\rm del}}\left(D_0+\sum L_x\right). \tag{40}\] For later use impose the prefix bounds \[ \left\|\sum J_x\right\|\le t^{.10\delta},\qquad \left|\sum L_x\right|\le t^{.10\delta}D_0. \tag{41}\] The second bound is a two-sided quadratic-form bound. These caps imply \(D_{\rm cur}(f|_{\rm cur})\le C W(f)\) for every extension \(f\) to the initial ports, and \(X_{\rm cur}=(1+O(t^{.10\delta}))X_0\) on survivor-supported neutral charges.

They also control \(K\), despite the fact that \(I-K\) is complex symmetric rather than self-adjoint. Indeed, let \(A=I-B\) with \(\|B\|\le\rho<1\), and write \(I-\widehat B=\mathop{\mathrm{Schur}}_{E_{\rm del}}A\) on the survivors. The harmonic extension of a survivor vector \(v\) to \(u\) exists because \(\|B\|<1\); it satisfies \(u_{\rm cur}=v\) and \((Au)_{\rm deleted}=0\). Hence \((Bu)_{\rm deleted}=u_{\rm deleted}\) and \((Bu)_{\rm cur}=\widehat Bv\), so \[\|u_{\rm deleted}\|^2+\|\widehat Bv\|^2 \le\rho^2(\|u_{\rm deleted}\|^2+\|v\|^2).\] Consequently \(\|\widehat B\|\le\rho\). Apply this with \(B=K_0+\sum J_x\) and \(\widehat B=K_{\rm cur}\).

Consequences of the prestate caps

During a sweep set \[\beta=.42\delta,\qquad P=(I-K)^{-1},\qquad z=\sqrt t(g_{\rm diag}-r),\qquad U=Pz,\qquad \widetilde U=U/\sqrt t.\] We call \(g_{\rm diag}-r\) the raw diagonal error and \(\widetilde U=P(g_{\rm diag}-r)\) its transformed lag. On the current ports their exact relation is \[g_{\rm diag}-r=(I-K)\widetilde U.\] On the virtual space we use this algebraic identity. Comparing the equation \((I-K)U=z\) before and after a deletion will combine the diagonal change with the Schur change of \(I-K\). Lemma 16 uses the cancellation \(\mathbb E\gamma=-z_x+O(t)\) in that combined equation to control the transformed lag. At the start \(U=0\) by the physical fixed-point relation.

In addition to (41), the prestate caps are \[ \max_i\sum_jF_{ij}^2\le H_*=t^{1+\delta}, \qquad \|\widetilde U\|_\infty\le t^\beta. \tag{42}\] The frozen \(r\) has bounded modulus and its imaginary part lies between fixed positive constants. There are at least \(N/2\) active physical sites. The local stopping rule will retain an update only when the two displays of mutable caps hold at its prestate; the stage induction supplies the fixed bounds on \(r,p\) and the active-site count.

The small factor in (45) is the extra information provided by the sharp inverse-form comparison. Bounded row sums alone would not preserve the fourth-power cap. We next show how these prestate bounds control an arbitrary family of possible next deletions.

The correction to \(K\)

Fix one prestate satisfying Lemma 13, and let \(\mathcal B\) be any prestate-measurable subset of its remaining bit ports. For each \(x\in\mathcal B\), all matrices in the following estimates are evaluated at this same prestate; only its hypothetical fresh scalar \(\xi_{0,x}\) varies. There is no requirement that the hypothetical outcomes be mutually independent. Every subset and tracked column set below is fixed by the prestate. All increments are padded by zero at their pivot.

The Laplacian increment

We next control the change in the energy form. The estimates must hold when the next pivot is chosen from an arbitrary subset of the remaining bit ports. We prove a relative bound for the sum of the conditional means and two bounds for the sum of the squared output norms. The latter combine to give the dual norm needed for the local concentration argument.

The transformed diagonal lag

The next estimate controls increments of the transformed lag \(\widetilde U=(I-K)^{-1}(g_{\rm diag}-r)\). Its cap yields a small raw diagonal error through Lemma 13, preserving the small conditional means used in the preceding increment bounds. For brevity write \(u=\widetilde U\) in this subsection. At a deletion, we freeze the removed coordinate of \(u\), and continue to update its surviving coordinates. In particular, the increment is zero at the deleted coordinate.

Fix the prestate and let \(V\) be its set of ports. For a prestate-measurable subset \(S\subseteq V\) of eligible bit ports, an array \(Y=(Y_i^x)\) has row index \(i\in V\) and hypothetical deletion index \(x\in S\). Define \[A_j(Y)=\max_{i\in V} \left(\sum_{x\in S}|Y_i^x|^j\right)^{1/j}, \qquad j=1,2.\] Every survivor vector associated with deletion \(x\) is extended by zero at coordinate \(x\). This convention is used throughout this subsection.

The row and operator bounds in the statement are supplied by Lemma 13. The absolute row-sum bound for \(P\) is proved in Lemma 20; the present argument uses precisely that displayed bound. We give the details of the survivor restrictions and of the nonlinear remainder, since both affect the conditional mean.

Fourth powers of a row

The last estimate applies to two tracked column sets: all remaining columns, and reserve columns only. In the second case the row itself is a reserve port. A tracked row is frozen when its port is deleted. When another port \(x\) is deleted, define its proxy increment by taking the fourth-power change on the surviving tracked columns and ignoring the favorable loss of the removed column. Thus the accumulated proxy is an upper bound for the actual change of the tracked row sum.

The form needed for stopped random ordering

We collect the estimates in a format that separates algebra from locality and freshness. Set \(\ell=1+\log(1/t)\).

These estimates supply the local concentration inputs for an ordinary sweep. It remains to accumulate them with a failure bound independent of the torus volume. Section 5 proves the inverse row bound, constructs fixed local tests, and shows that their budgets imply the sweep caps. It then applies stopped concentration to those tests and returns to the physical endpoint described by (32).

Decay and the inverse of \(I-K\)

Assign cost zero to each nearest-neighbor edge meeting a terminal site, and cost one to every other nearest-neighbor edge. The minimum path cost is the pseudometric \(d_T\). Ports at the same physical site have distance zero in both metrics. The following proof uses the exponential-conjugation method of Combes and Thomas (Combes and Thomas 1973), applied to this collapsed metric.

The transformed-lag cap in Section 4 supplies precisely the hypothesis \(q=p+O(t^\beta)\) of Lemma 20. Lemma 20 therefore completes the inverse-row input used in the lag increment estimates. We shall also need decay for a single prospective update, even if that update violates one of the finer caps.

A finite-dimensional test for the energy increment

At ordinary stage \(n\), partition the torus into cubes of side \(L_{\mathrm{tile}}=4^{6n}\asymp t^{-6}\). Around each tile choose concentric cubes \(Q,Q'\) of side \(3L_{\mathrm{tile}},5L_{\mathrm{tile}}\). They fit in the torus, and each family has uniformly bounded overlap. Let \(\chi\) equal one on \(Q\), vanish at the boundary of \(Q'\), and have nearest-neighbor slopes at most \(C/L_{\mathrm{tile}}\).

Write \(\mathcal P_0\) for the ports present at the start of the duplicated sweep. The fixed real local test space \(\mathcal H_T\) consists of pairs \((\alpha,v)\), where \(\alpha\) is indexed by \(\mathcal P_0\cap Q'\), and \(v\) is indexed by all physical sites in \(Q'\), modulo constants. Its dimension is at most \(3|Q'|\le C_d L_{\mathrm{tile}}^d\le C_d t^{-6d}\). Its norm and extension map are \[ \| (\alpha,v)\|_{\rm loc}^2 =\|\alpha\|_{Q'}^2+b^{-1}\|\nabla v\|_{Q'}^2, \qquad B_T(\alpha,v)=\chi\big(\alpha+v-\operatorname{avg}_{Q'}v\big). \tag{73}\] The extension is zero outside \(Q'\), and physical values of \(v\) are repeated on ports. The subscript \(T\) here identifies the tile. The symbol \(B_T^*\) denotes the pullback into the dual of the local space. A pulled-back symmetric form is measured by the norm of its representing self-adjoint operator in the local inner product. The set \(\mathcal P_0\), the cubes, \(\chi\), \(t,b\), the local norm, and \(B_T\) stay fixed throughout the sweep, including retries. They may depend on the stage-start history, on which we condition. Each new correction is extended by zero to \(\mathcal P_0\) at its time of creation. The precise record convention below keeps this reference space fixed at every stopping step.

Trials and the transfer of local budgets

Start from a physical state satisfying the complete stage-boundary bounds of Section 2, including self-consistency, (7), the broad bounds on \(r,p\), the boundary cap \(\|K^{\rm phys}\|\le1/3\), the fourth-power row cap \(H_*=t^{1+\delta}\), and an active physical-site count of at least \(N/2\). Duplicate it as in Lemma 11. The resulting initial data include \[\widetilde U_0=0,\qquad D_0\asymp W,\qquad X_0\le(1+t^\kappa)\mathop{\mathrm{diag}}(p_x^{-2})+C_nt\Lambda^+_\Sigma,\] and the full duplicated fourth-power rows are at most \(.75H_*\). These initial comparisons, the broad bounds on the frozen \(r,p\), the active physical-site count, and (67) are standing assumptions for the rest of this section. Bit-port deletions leave the physical active and terminal sets unchanged, so the count and metric remain fixed; the initial comparison data and frozen parameters also remain fixed.

Process the tiles in a fixed order. A trial in a tile \(T\) starts from its current committed numerical pre-tile state and reveals a uniform permutation of all \(M=L_{\mathrm{tile}}^d\) physical sites in \(T\), one slot at a time. A terminal slot is skipped. At an active slot the trial deletes the bit port at that site and uses the update of Lemma 12. For each digit stage \(n<m\), tile \(T\), trial number \(a\ge1\), and physical site \(i\in T\), supply an auxiliary fair sign \(\varepsilon^{\rm aux}_{n,T,a,i}\). The auxiliary signs are independent over these indices and independent of the true digits. The corresponding physical digit is \(\sqrt{3s_n/4}\,\varepsilon^{\rm aux}_{n,T,a,i}\). Also supply independent uniform permutations \(\pi_{n,T,a}\) for \(n_0\le n\le m\), each tile \(T\) at that stage, and \(a\ge1\), independent of both sign families. Only the slots and auxiliary signs actually used are revealed. On a first digit trial, a coupled active site \(i\) uses \(\sqrt{3s_n/4}\,\varepsilon_{i,n-n_0+1}\), and a pending site uses its designated auxiliary digit. Later digit trials use their new auxiliary family at every active site. The actual final sweep is entered only with no pending site; its first trial at a tile uses \(\xi_i:=\sum_{k>m-n_0}\lambda2^{-k}\varepsilon_{i,k}\), the whole remaining true suffix, at every active site and aborts on failure. It uses only \(\pi_{m,T,1}\), samples no auxiliary suffix, and has no retry.

We now specify the numerical records and the test made after a slot. For each deletion \(x\), retain \(J_x,L_x\) on the fixed initial-port space \(\mathcal P_0\): pad the new increments by zero at \(x\) and at ports deleted before that update. Entries already recorded in an earlier increment are retained when a later port is deleted. This is the convention in the cumulative Schur identities (40).

There are also three scalar records on fixed coordinate sets. Let \(\Delta\mathsf u_i^x\) equal the transformed-lag increment \(\Delta\widetilde U_i^x\) on the survivors of deletion \(x\), and set it to zero at \(x\) and at every earlier deleted coordinate. For \(\sigma\in\{\mathrm{all},\mathrm{res}\}\), let \(\Delta\mathsf R_i^{\sigma,x}=R_i^{\sigma,x}/H_*\) be the normalized proxy increment of Lemma 17, with all current columns tracked when \(\sigma=\mathrm{all}\), and reserve columns tracked when \(\sigma=\mathrm{res}\). The latter record is defined only for reserve rows. Each proxy sums the fourth-power change over surviving tracked columns; it omits the favorable loss of the removed column. Its increment is zero when its own row is deleted and thereafter. Thus all three records have fixed coordinate sets in \(\mathcal P_0\), with a coordinate frozen at its deletion. The normalized proxy increments are the quantities denoted by \(\Delta R_i\) in (71).

Write \(x_\ell\) for the bit port at slot \(\ell\) when that slot is active, and interpret each summand below as zero at a terminal slot. The records for prefix \(k\) of a trial in \(T\) are \[\begin{aligned} J_{T,k}&=\sum_{\ell\le k}J_{x_\ell},& L_{T,k}&=\sum_{\ell\le k}L_{x_\ell},\\ \mathsf u_{T,k}&=\sum_{\ell\le k}\Delta\mathsf u^{x_\ell},& \mathsf R^\sigma_{T,k}&=\sum_{\ell\le k}\Delta\mathsf R^{\sigma,x_\ell}. \end{aligned}\] Every one of these records is zero at the start of each trial, including a retry. If an earlier tile \(S\) has been accepted, write \(J_S,L_S,\mathsf u_S,\mathsf R^\sigma_S\) for the endpoint records of its accepted trial. At a current prefix in \(T\), define the accumulated sweep records by \[ \begin{aligned} J^{\rm sw}_{T,k}&=\sum_{S\prec T}J_S+J_{T,k},& L^{\rm sw}_{T,k}&=\sum_{S\prec T}L_S+L_{T,k},\\ \mathsf u^{\rm sw}_{T,k}&=\sum_{S\prec T}\mathsf u_S+\mathsf u_{T,k},& \mathsf R^{\sigma,\rm sw}_{T,k} &=\sum_{S\prec T}\mathsf R^\sigma_S+\mathsf R^\sigma_{T,k}. \end{aligned} \tag{74}\] Only the accepted trial of each earlier tile enters these sums. A failed numerical prefix is absent. In this included numerical record, a physical site is used as a pivot at most once: the tiles are disjoint and each trial permutation visits a site once. The retained information may contain failed queries at those same physical sites.

Let \(\mathcal P_{\rm cur}\) be the current survivor ports, and let \(\mathcal P_{\rm res}\) be the reserve ports, which survive the whole sweep. Along any included sequence for which the increments are defined, the lag record has the exact restriction identity \[\mathsf u^{\rm sw}_{T,k}|_{\mathcal P_{\rm cur}} =\widetilde U_{\rm cur}.\] Indeed, both start at zero and have identical increments on each survivor; a deleted coordinate of the fixed record keeps its value immediately before deletion. The row records have a different role. For a surviving row, and respectively for a reserve row, they give only \[\begin{align*} \sum_{j\in\mathcal P_{\rm cur}}|h^{\rm cur}_{ij}|^4 &\le \sum_{j\in\mathcal P_0}|h^0_{ij}|^4 +H_*\mathsf R_i^{\mathrm{all},\rm sw},\\ \sum_{j\in\mathcal P_{\rm res}}|h^{\rm cur}_{ij}|^4 &\le \sum_{j\in\mathcal P_{\rm res}}|h^0_{ij}|^4 +H_*\mathsf R_i^{\mathrm{res},\rm sw}. \end{align*}\] Here and below the current \((T,k)\) subscripts are suppressed when unambiguous. The inequalities retain the favorable removed-column loss on the left and omit it from the proxy on the right.

Call the current prefix valid when it satisfies both (41) and (42), with the exact meanings \[ \begin{gathered} \|J^{\rm sw}_{T,k}\|\le t^{.10\delta},\qquad |L^{\rm sw}_{T,k}|\le t^{.10\delta}D_0,\\ \max_{i\in\mathcal P_{\rm cur}} \sum_{j\in\mathcal P_{\rm cur}}|h^{\rm cur}_{ij}|^4\le H_*, \qquad \|\widetilde U_{\rm cur}\|_{\ell^\infty(\mathcal P_{\rm cur})} \le t^\beta,\qquad \beta=.42\delta. \end{gathered} \tag{75}\] The first line tests the accumulated padded corrections on the initial space, and the second tests the actual current row and transformed lag on survivors. The two-sided form inequality for \(L^{\rm sw}_{T,k}\) is on the initial ports. The standing data above are not additional prefix tests. By the Section 4 Schur and prestate arguments, validity and those data supply the hypotheses of Proposition 18; Lemma 20 supplies its absolute inverse-row bound.

Let \(P_{Q_T}\) be the Euclidean projection onto \(\mathcal P_0\cap Q_T\), and let \(\mathcal R_T\) be the Riesz map of \(\mathcal H_T\). At every trial prefix impose the local budgets \[ \begin{gathered} \|\mathsf u_{T,k}\|_{\ell^\infty(\mathcal P_0\cap Q_T)} \le t^{.44\delta},\qquad \max_{\sigma\in\{\mathrm{all},\mathrm{res}\}} \|\mathsf R^\sigma_{T,k}\|_{\ell^\infty(Q_T)} \le t^{.0005\delta},\\ \|P_{Q_T}J_{T,k}P_{Q_T}\|\le t^{.125\delta},\qquad \|\mathcal R_T^{-1}B_T^*L_{T,k}B_T\|_{\mathcal H_T} \le t^{.125\delta}. \end{gathered} \tag{76}\] For a row record, the norm over \(Q_T\) uses its fixed row indices above that cube. The matrix norm is Euclidean and the form norm uses the fixed local inner product. These budgets apply to changes accumulated since this trial began, whereas (75) uses the full included sweep record.

After a slot, continue precisely when the local budgets and validity still hold. Stop and fail at their first violation, retaining the update that produces it. Its preceding prefix satisfies both cap displays, so Lemma 21 supplies the poststate inverse and a fixed \(I-K'\) invertibility margin. The transformed lag and this first possibly failing test are therefore defined without assuming fine caps at the poststate. If all \(M\) slots finish without a violation, accept the trial’s numerical changes and endpoint records. On a failure restore the pre-tile potential values, port set, and matrices, and discard the failed trial records. Retain every queried index and answer and every exposed permutation slot, including the answer producing the first violation. A failed first trial marks every active site of the tile pending, including active sites whose slots were not reached. Previously terminal sites remain terminal. When the digit retry construction is used, its numerical starting state has those restored values and all active labels in the tile pending. A later failure restores that retry starting state and retains its own queried information; the next retry uses a new auxiliary family and independent permutation. At the later cutoff where the construction requires the pending set to be empty, it begins aborting at any first-trial failure instead of entering a retry.

Here is the ordinary verification of the input law. At entry to scale \(n\), assume the retained-history invariant that each coupled active site has the first \(n-n_0\) true digits as its partial value and its remaining suffix is untouched. A digit trial consumes the next bit; the final trial at \(n=m\) consumes the whole suffix after those \(m-n_0\) digits. The permutation exposes a site before its scalar answer is queried, and every stopping and acceptance decision is a function of the resulting retained history. At a coupled first-trial slot the required true bit, or final suffix, is therefore unqueried; at a pending slot the designated auxiliary entry is unused. Lemma 4 gives the scalar law conditional on the retained history before the slot and on the newly exposed slot, and gives the conditionally uniform order on the unexposed slots of the new independent permutation. On success a coupled site consumes that true input and retains the unused suffix (empty after final completion), while a pending site stays pending. On first-trial failure the numerical rollback leaves every answer in the history, and the whole-tile marking rule removes all its active sites from the class at which true freshness is used. This rule also conservatively marks unreached active sites, whose own suffixes may still be untouched. Retries use only their unused auxiliary entries. Thus ordinary trials preserve the invariant at every coupled site and satisfy the conditional interface above after arbitrary previous failures. This verification uses the generic reveal lemma, without conditioning on future acceptance of a trial.

At a first possible failure with no local crossing, this transfer will force the four mutable caps to remain strict. We now estimate the probability of a local crossing in one tile.

A maximal concentration estimate

We will repeatedly apply the following finite-dimensional estimate inside a neighborhood whose dimension is polynomial in \(t^{-1}\) for fixed \(d\). The variance envelopes below are predictable scalars dominating the conditional squares in matrix order. This precise version follows from the finite-dimensional Golden–Thompson inequality, as the proof shows. For the general matrix-martingale framework, compare (Tropp 2011, Theorem 1.2 and Corollary 1.3); the scalar-envelope version used here is proved below.

A rectangular or complex increment \(B\) is handled by the Hermitian matrix \(\left(\begin{smallmatrix}0&B\\B^*&0\end{smallmatrix}\right)\). Its square requires bounds for both \(BB^*\) and \(B^*B\). Conditional centering doubles the jump bound at most. It does not increase either conditional square: if \(\bar B=\mathbb E(B\mid\mathcal F)\), then \[\mathbb E[(B-\bar B)(B-\bar B)^*\mid\mathcal F] =\mathbb E[BB^*\mid\mathcal F]-\bar B\bar B^* \le\mathbb E[BB^*\mid\mathcal F],\] and the same calculation holds in the other order. For vector increments, the squared Euclidean norm bounds both dilation blocks. All spaces and their inner products are fixed at the start of each trial; removed indices are padded by zeros.

Here is how scalar envelopes arise in a randomly ordered trial. If there are \(r\) equally likely next slots and their possible square-matrix increments \(B_x\) satisfy, for every vector \(v\) and every collection of outcomes, \[\sum_x\bigl(\|B_xv\|^2+\|B_x^*v\|^2\bigr) \le V_*\|v\|^2,\] then averaging first over the outcome in each slot and then over its uniform selection bounds each conditional dilation square by \((V_*/r)I\). Only the conditional marginal laws of these hypothetical outcomes are needed. Over \(M\) slots their envelopes sum to at most \(V_*\sum_{r=1}^M r^{-1}\). For \(M\le t^{-C_d}\) this costs \(O_d(\log(1/t))\). The localized energy operators and vector arrays used later have the corresponding square-stack bounds. In every application below, the predictable variance sum is bounded through the first prospective budget-breaking update. That increment is included whenever the trial’s tests permit another update at the preceding prefix; this decision is predictable, and all subsequent increments are zero. Thus the concentration hypotheses are verified before the outcome that might violate a budget is observed.

One trial and its retries

The same estimate applies after any number of failed retries whenever each retry is supplied with a new independent permutation and fresh scalar inputs satisfying the conditional moment interface. The indexed auxiliary signs supply these inputs for the digit sweeps. Each retry begins from the restored valid numerical pre-tile state, while its law is taken with respect to the larger retained history. Let \(\mathscr F_T\) be the retained pre-tile history and set \(p_{\rm fail}=\exp\{-t^{-c}\}<1\). For each fixed attempt number \(a\), stop at the first of the start of attempt \(a\) and completion of the tile by an earlier attempt. This stopping time of the retained filtration is almost surely finite: before it there are at most \(a-1\) attempted trials, each with at most \(M\) slots. Assign the \(a\)-th failure indicator value zero when an earlier attempt has completed the tile. The conditional failure bound then holds at every such stopped history, using Lemma 25 when attempt \(a\) is scheduled and the zero indicator otherwise. Successive conditioning gives \[\mathbb P(\text{the next }k\text{ trials all fail}\mid\mathscr F_T) \le p_{\rm fail}^k=e^{-kt^{-c}}.\] Hence a tile has infinitely many failures with probability zero. There are finitely many tiles at a fixed sweep, so any such supplied retry process terminates almost surely. The actual construction uses this retry conclusion for digit sweeps before the cutoff and uses the first-trial estimate after the cutoff, including the final sweep. This argument conditions only on histories already reached; no estimate within a trial conditions on its eventual acceptance.

The endpoint of an ordinary sweep

We have proved that accepted locally monitored trials maintain all four mutable caps. The remaining task is to express a completed sweep in the physical normalization at the smaller damping scale.

After the cutoff the actual construction requires that no active site is pending and aborts at any later first-trial failure. On a retained history that reaches and completes the final residual sweep without abort, every accepted final update therefore uses the whole true suffix; no auxiliary retry contributes to the certificate endpoint. Its law is used at its retained pre-query history, before final acceptance is known. With \(t=2\eta\) and \(w=1/2\), the physical update is \(\xi_i+\eta r_i\): it inserts the complete true suffix and changes the artificial term from \(-2\eta r_i\) to \(-\eta r_i\). Terminal sites already have their true values. Thus all physical potentials at this endpoint are the original true values, while the remaining active sites are still active and retain the damping \(-\eta\mathop{\mathrm{diag}}_A r\).

The scattering range and the completion target

Write \(D_p=\mathop{\mathrm{diag}}(p)\) and define \[ S=I+2itD_p^{1/2}gD_p^{1/2}. \tag{80}\] The matrix \(S\) is symmetric and unitary. Indeed, for \(Z=D_p^{1/2}gD_p^{1/2}\) the Ward identity gives \(\operatorname{Im}Z=tZZ^*\), and hence \[SS^*=I+2it(Z-Z^*)+4t^2ZZ^*=I.\] Write \[\mathcal C=S_{RB},\qquad \Pi=\Pi_{\mathop{\mathrm{ran}}\mathcal C}\] for the scattering cross block and the orthogonal projection onto its range in \(\mathbb C^R\). Its diagonal has the concrete meaning \[\Pi_{ii}=\sup_{v\in\mathop{\mathrm{ran}}\mathcal C,\ \|v\|=1}|v_i|^2,\] with value zero when the range is zero. Thus a small diagonal means that no unit vector in the cross-block range can concentrate at one exterior site. The next lemma shows why this is the condition needed to complete the core.

The middle coefficient in the core resolvent factorization can be large. The proof controls the resulting scattering change by its operator norm and its range instead. This explains why the raw row norms of \(\mathcal C\) are insufficient: a small singular amplitude can still correspond to a range direction concentrated at one exterior site. We must control the full range, including every nonzero singular direction. The exterior equation gives that range a spatial bound before any diagonal lag or comparison estimate is imposed.

Localization of the full scattering range

The same range estimate will locate the exterior exposures used to reach the projection bound. We now construct that bound while controlling the accumulated change of the inverse.

Exterior heats and regularized projection weights

An exterior update, called a heat step, replaces the partial potential at one exterior site \(x\) by its true value and removes that site from the trial’s active set. Such removal remains provisional until the clearing is committed; rollback restores the pretrial matrix and partial potential data while retaining the revealed values in the history. If its unexposed true residual is \(\xi\), the change of its diagonal in the damped matrix is \(\xi+tr_x\). The conditional hypotheses on the residual are \[ \mathbb E(\xi\mid\mathcal F)=0,\qquad \mathbb E(\xi^2\mid\mathcal F)=t-\eta,\qquad |\xi|\le C\sqrt t,\qquad \eta/t\le Ct. \tag{89}\] Here \(\mathcal F\) is the recorded history before the site is inspected. The selection of \(x\) must be \(\mathcal F\)-measurable. For the scheduled components, Section 7 combines the pivot geometry with Lemma 4 to verify these hypotheses.

There are two distinct sets of state bounds in the argument. At the beginning of a component trial we have fourth-power row slack and a small diagonal lag. At a provisional heat prefix we use only \[ \max_i|g_{ii}-r_i|\le C t^{\beta_w},\qquad \max_i\sum_j|h_{ij}|^4\le H_*. \tag{90}\] Together with (78), Ward’s identity gives \[ \sum_jp_j|h_{ij}|^2=\Im g_{ii},\qquad \max_i\sum_j|h_{ij}|^2\le C. \tag{91}\] The provisional estimates below use no bound on \(K\) and no comparison of the Laplacian \(D\) with a reference form. Section 7 transports those comparisons for the resulting committed operation.

At each heat the core \(B\) remains fixed, but its current exterior \(R\) and the cross block \(\mathcal C=S_{RB}\) change. Write \(S_B=S_{BB}\) and set \[\Gamma=\mathcal C^*\mathcal C.\] Unitarity gives \(\Gamma=I-S_B^*S_B\). For a ridge parameter \(\tau>0\), define \[ G_\tau=\Gamma+\tau I,\qquad U=\mathcal C G_\tau^{-1/2},\qquad m_i=\|u_i\|^2\quad(i\in R), \tag{92}\] where \(u_i\) is the \(i\)-th row of \(U\). Equivalently, \(m_i\) is the \(i\)-th diagonal entry of \[\mathcal C(\mathcal C^*\mathcal C+\tau I)^{-1}\mathcal C^*.\] These entries are the standard ridge leverage scores (Alaoui and Mahoney 2015, sec. 3.3, Definition 1); here we use them to select predictable exterior residual exposures while the ridge decreases toward the full range projection.

To see exactly what the regularization retains, write a singular-value decomposition \(\mathcal C=\sum_{\alpha:\sigma_\alpha>0}\sigma_\alpha v_\alpha w_\alpha^*\), with orthonormal left and right singular vectors. Then \[\mathcal C(\Gamma+\tau I)^{-1}\mathcal C^* =\sum_{\alpha:\sigma_\alpha>0} \frac{\sigma_\alpha^2}{\sigma_\alpha^2+\tau} v_\alpha v_\alpha^*.\] As \(\tau\downarrow0\), every nonzero singular direction receives weight one, including directions with arbitrarily small singular value. The matrix therefore increases to \(\Pi_{\mathop{\mathrm{ran}}\mathcal C}\), and each \(m_i\) is at most the corresponding projection diagonal. No zero singular direction contributes. For fixed \(\mathcal C\), each \(m_i\) is continuous for \(\tau>0\) and \[\frac{d}{d\tau}m_i(\tau) =-\mathcal C_i(\Gamma+\tau I)^{-2}\mathcal C_i^*\le0.\] Thus decreasing the ridge increases the regularized weights continuously toward the completion target.

Put \[m_*=\max_{i\in R}m_i,\qquad \mathcal A=\sum_{i\in R}m_i,\qquad \mathcal P=\sum_{i\in R}m_i^2.\] If \(R\) is empty, the maximum is zero. The matrices \[ Q=U^*U,\qquad V_\tau=I-Q=\tau G_\tau^{-1} \tag{93}\] satisfy \(0\le Q,V_\tau\le I\), so \(\mathcal A=\mathop{\mathrm{Tr}}Q\le\operatorname{rank}\mathcal C\le |B|\le r_*\).

Whenever the maximum regularized weight is at least \(2l_*\) and at most \(4l_*\), the procedure heats a site of maximum weight, chosen from the recorded history before its residual is inspected. Removing that pivot deletes its contribution \(m_*^2\) from \(\mathcal P\); the issue is to show that redistribution to the surviving rows is smaller. Maximality also gives \(m_i\le m_*\) in the estimates for that redistribution. The next lemma establishes this fixed-ridge drift under the provisional prestate bounds.

A heat holds the scalar \(\tau\) fixed, changes the damped diagonal at its chosen site \(x\) by \(\xi+tr_x\), and provisionally eliminates \(x\) as a true terminal. The standing invertibility of true terminal principal matrices makes the elimination well-defined. The new cross block is recomputed from the new active inverse at the same scalar ridge. Use primes for quantities on \(R'=R\setminus\{x\}\).

Lowering the ridge and counting heats

The procedure combines two motions: a fixed-ridge heat redistributes weight while reducing \(\mathcal P\) in conditional expectation, and a ridge decrease raises the regularized weights toward the projection. To control both with one quantity, define \[\Phi=\mathcal P-10l_*\mathcal A.\] During a ridge decrease for which every \(m_i\le2l_*\), each \(dm_i\) is nonnegative. Hence \[ d\mathcal P=2\sum_i m_i\,dm_i\le4l_*\,d\mathcal A, \qquad d\Phi\le-6l_*\,d\mathcal A\le0. \tag{111}\] The same conclusion holds at the zero-ridge limit by continuity. At a heat satisfying Lemma 30, with \(m=m_*\ge2l_*\), the correction to the squared-weight drift obeys \[10l_*|\mathbb E(\mathcal A'-\mathcal A)|=o(l_*m)=o(m^2).\] Combining this with the drift of \(\mathcal P\) gives, for small \(t\), \[ \mathbb E(\Phi'-\Phi)\le-\frac12m_*^2. \tag{112}\] Thus a ridge decrease contributes no positive drift, while an admissible heat with \(2l_*\le m_*\le4l_*\) decreases \(\Phi\) in conditional expectation by at least \(2l_*^2\).

Start with \(\tau=l_*^{-2}\). Since \(\mathcal C\) is a block of a unitary matrix, \(m_*\le\tau^{-1}=l_*^2<2l_*\). Whenever \(m_*<2l_*\), decrease \(\tau\) until one of two events occurs. If the maximum diagonal of \(\Pi_{\mathop{\mathrm{ran}}\mathcal C}\) is at most \(2l_*\), declare completion at the zero-ridge limit. Otherwise there is a unique positive ridge at which \(m_*=2l_*\), and the decrease stops there. For a nonzero row the displayed derivative of \(m_i(\tau)\) is strictly negative, which proves uniqueness. The ridge is measurable: its comparison with any positive rational is determined by the measurable value of \(\max_i m_i(\tau)\) there. The zero-ridge projection is measurable as the pointwise limit of the positive-ridge matrices. At a heat, choose the least site among those of maximum weight. This tie rule makes the pivot predictable.

For the chosen parameters, \[\beta_w-\frac{e_0}{2}=.14\delta,\qquad \delta-e_0-\nu=(.56-10^{-5}/d)\delta,\qquad e_h-e_0-\nu=(.02-10^{-5}/d)\delta\] are all positive. The leverage argument therefore bounds the probability of exhausting the heat limit. It remains to control the other stopping conditions, so that the procedure reaches completion.

The stopped trial and its local inputs

We now state the local input that turns the range estimate into a trial whose probability bound is independent of the torus volume. There is a scale \(\ell\asymp t^{-4}\) such that, as long as earlier heat pivots lie within ordinary distance \(\ell/2\) of \(B\), the current collapsed distance, and the distance after one further such deletion, satisfy \[ d_T(i,k)\ge\tfrac12\mathop{\mathrm{dist}}(i,k)-t^{-1.03}. \tag{114}\] The core lies in an ordinary-distance ball of radius \(t^{-C_{\mathrm{loc}}}\), for a fixed \(C_{\mathrm{loc}}\). Every exterior site within distance \(\ell/2\) of \(B\) which has not already been deleted is coupled and has a still unexposed true residual. A predictable choice of such a site therefore satisfies (89). These are hypotheses about geometry and recorded information; they contain no provisional bound on \(K\) or comparison for \(D\).

Here is the selection consequence of the geometric input. By the singular-value formula, a regularized weight is at most its projection diagonal. For \(\mathop{\mathrm{dist}}(i,B)\ge\ell/2\), Lemma 28 and (114) bound that diagonal by \(C\exp(-ct^{-3})<2l_*\). Consequently \[ m_x=m_*\ge2l_*\quad\Longrightarrow\quad \mathop{\mathrm{dist}}(x,B)<\ell/2. \tag{115}\] Starting with no earlier pivots, this implication proves location inductively whenever the stated metric input is available under the preceding-location hypothesis. It applies to the pivot chosen before a first breaking heat. The input for the distance after that deletion will also control its prospective output. Section 7 supplies this geometric input for the scheduled components.

For bookkeeping, keep a matrix \(\hat h\) indexed by the sites active at trial start. A pair of entries is frozen immediately before the first of its two endpoints is removed. Only survivor–survivor entries receive a heat increment. Put \(\hat g=\hat h/\sqrt t\), and write \(h^0,g^0\) for the trial-start matrices. Thus each original row of \(\hat h-h^0\), and the whole matrix viewed as a vector, has a fixed index set throughout the trial. The three distortion budgets are \[ \begin{split} \max_{i,k}|\hat g_{ik}-g^0_{ik}|&\le t^{.42\delta},\\ \max_i\sum_k|\hat h_{ik}-h^0_{ik}|^2&\le t^{.25\delta},\\ \sum_{i,k}|\hat h_{ik}-h^0_{ik}|^2&\le n_h t^{-.01\delta}. \end{split} \tag{116}\]

If \(B\) is empty, declare DONE without changing the state. Otherwise the trial starts at ridge \(\tau=l_*^{-2}\). If the maximum ridge leverage is below \(2l_*\), lower the ridge to the first value where the maximum equals \(2l_*\); if the zero-ridge projection already has maximum leverage at most \(2l_*\), stop with outcome DONE. Otherwise choose an exterior site of maximum regularized weight, using a fixed ordering to break ties, perform one heat step, and recompute the cross block and its leverages at the same scalar ridge. Stop with outcome FAIL if a distortion budget is exceeded, if a leverage exceeds \(4l_*\), or if \(n_h\) heats have been performed. Repeat the ridge lowering and heating rule until one of these outcomes occurs. True values exposed during a trial are part of the recorded history even when the state is later rolled back.

Call the state just before a heat performed by this rule a running heat prestate. Its three distortion budgets and leverage cap still hold, fewer than \(n_h\) heats have been performed, and the rule has selected a positive ridge with \(2l_*\le m_*\le4l_*\). These are exactly the conditions under which the stopped rule performs another heat. The budget argument below derives (90) at these states from the starting slack. Write \(\mathcal F_0\) for the retained history at trial start.

The uniform little-oh in (117) is supplied by committed-state assembly. In the proof of Proposition 32 it is used only to obtain (90) from the entry budget. We next verify every other stopping probability. All estimates are stopped at the first failure and continued afterwards by zero increments; the step which first causes failure is included.

Distinct pivots and the distortion budgets

The deterministic consequences of the budgets are useful before applying concentration. First, \[ \sum_k|\hat h_{ik}-h^0_{ik}|^4 \le\max_k|\hat h_{ik}-h^0_{ik}|^2 \sum_k|\hat h_{ik}-h^0_{ik}|^2 \le t^{1+1.09\delta}. \tag{119}\] The triangle inequality in \(\ell^4\), the initial factor \(.8\), and the positive margin \(.09\delta\) imply the fourth-power part of (90). The entry budget and (117) imply its lag part. Thus those provisional hypotheses hold at every running heat prestate.

For a fixed original site \(i\), sum over pivots \(x\) for which \(i\) is still active immediately before the corresponding heat. Each summand is evaluated at that prestate. Distinctness of the pivots and the previous entry budgets imply, through a prospective breaking step, \[ \sum_x F_{ix}^2 \le C\sum_x|h^0_{ix}|^4 +Cn_h t^{2+1.68\delta} \le C(H_*+t^{1+1.22\delta})\le CH_*. \tag{120}\] The first sum uses only the corresponding distinct columns of the initial row. This is why a bound on the current row alone would not have been enough. Cauchy–Schwarz now gives the pathwise predictable budget \[ B_i:=\sum_x F_{ix}\le C\sqrt{n_hH_*} \le Ct^{.27\delta}. \tag{121}\]

For reference, the heat increment on surviving pairs is \[ \Delta\hat g_{ik}=\frac{c_x}{\sqrt t}h_{ix}h_{xk}, \qquad \Delta\hat h_{ik}=c_x h_{ix}h_{xk}. \tag{122}\] Entries with a removed endpoint have zero increment. The conditional mean of \(c_x\) is \(O(\sqrt t\,t^{\beta_w})\), its absolute value is bounded, and the Ward row bound (91) is available at the prestate. For a fixed entry, summing its conditional means in absolute value costs \[Ct^{\beta_w}\sum_x\sqrt{F_{ix}F_{kx}} \le Ct^{\beta_w}\sqrt{B_iB_k}.\] The conditional variance sum is at most \[\frac Ct\sum_xF_{ix}F_{kx}\le CH_*/t,\] by (120), and each jump is at most \(C\sqrt{H_*/t}=Ct^{\delta/2}\).

For a fixed row regarded as a vector, its increment has norm at most \(C|h_{ix}|\); the same estimate with \(C\) replaced by \(C\sqrt t\,t^{\beta_w}\) bounds its conditional mean. Consequently the total mean norm, predictable variance, and maximum jump are bounded respectively by \[C\sqrt t\,t^{\beta_w}\sqrt{n_hB_i},\qquad CB_i,\qquad CH_*^{1/4}.\] For the whole matrix regarded as a Frobenius vector, the increment norm is at most \(C\|h_{\cdot x}\|_2^2\le C\), where the dot denotes the full current active column. Its corresponding three bounds are \[C\sqrt t\,t^{\beta_w}n_h,\qquad Cn_h,\qquad C.\] The conditional means here are computed before the query: the endpoint mask is predictable, so the scalar \(\mathbb Ec_x\) can indeed be factored out of the increment. This mask is a contraction in the row and Frobenius norms and does not change any bound. Center each increment at its conditional mean. For a complex vector \(z\), its Hermitian dilation \[\mathscr D(z)=\begin{pmatrix}0&z^*\\z&0\end{pmatrix} \quad\hbox{satisfies}\quad \mathscr D(z)^2=\mathop{\mathrm{diag}}(\|z\|^2,zz^*)\le\|z\|^2I.\] Centering cannot increase the conditional second moment in matrix order. Thus the displayed squared-norm bounds give predictable scalar variance envelopes in the fixed original coordinate spaces. For entries and rows their pathwise sums follow from (120) and (121); for the Frobenius vector the Ward bound and the limit of \(n_h\) heats give the sum directly. All include the candidate first breaking heat. This verifies the precise hypotheses of Lemma 24, rather than requiring a matrix-valued variance theorem.

Here are the margins at the required thresholds. For entries, use \(a=t^{.42\delta}\); the total drift is \(O(t^{.63\delta})\) and \[\frac{a^2}{V}\ge ct^{-.16\delta},\qquad \frac aJ\ge ct^{-.08\delta}.\] For row norms, use \(a=t^{.125\delta}\); the drift is \(O(t^{.265\delta})\) and \(a^2/V\ge ct^{-.02\delta}\), with a larger margin for \(a/J\). For the Frobenius norm, use \(a=\sqrt{n_h}\,t^{-.005\delta}\); its drift divided by \(a\) is \(O(t^{.135\delta})\), and \(a^2/V\ge ct^{-.01\delta}\). Thus all three budget crossings have stretched-exponentially small probability after restriction to a polynomial number of coordinates. We justify that restriction below, before drawing the conclusion.

A two-sided bound on leverage changes

The drift calculation controls the number of heats, but it does not by itself keep individual leverages below \(4l_*\). We prove that cap separately. For each original exterior site \(i\), record the fixed-ridge heat change \(d_i=m'_i-m_i\) whenever \(i\) survives the heat. Set the recorded change to zero at its own deletion and after deletion. Ridge adjustments are not included. More precisely, number the heats by \(k\). While the trial is running, let \(\mathcal F_{k-1}\) be the full retained history immediately before the residual query for heat \(k\), including the already selected pivot and ridge \(\tau_k\). Put \[Z_{k,i}={\bf1}_{\{i\text{ survives heat }k\}} \{m_i^{\mathrm{post}}(\tau_k)-m_i^{\mathrm{pre}}(\tau_k)\}.\] The survival indicator is predictable, because the pivot is chosen before its residual is queried. 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. Thus \(Z_{k,i}-\mathbb E(Z_{k,i}\mid\mathcal F_{k-1})\) is one adapted martingale stream even though \(\tau_k\) can vary with \(k\). Every estimate below is uniform in that predictable positive ridge; no discretization or union over its possible values is needed. The notation \(d_i\) denotes \(Z_{k,i}\) at the heat under consideration.

Write \(\varepsilon_t=t^{\beta_w}+\sqrt t+r_*\sqrt{H_*}\). The same expansion of the Gram normalization used in Lemma 30 gives, at a running heat prestate, \[ |\mathbb E(d_i\mid\mathcal F)| \le Cl_*\left[ \sqrt t\,t^{\beta_w}|v_i|+|v_i|^2 +l_*\varepsilon_t+\frac{l_*|v_i|}{\sqrt t} +\frac{l_*^2}{t}\right], \tag{123}\] \[ |d_i|\le Cl_*\left(|v_i|+\frac{l_*}{\sqrt t}\right). \tag{124}\] For completeness, the first two terms of (123) are the conditional mean of the raw row-norm change. In the first-order normalization, keeping the old row gives \(Cl_*\|\mathbb ED_*\|_1\le Cl_*^2\varepsilon_t\). Replacing it by the new row costs \(Cl_*|v_i|\|D_*\|\le Cl_*^2|v_i|/\sqrt t\). The quadratic remainder is at most \(Cl_*\operatorname{Tr}D_*^2\le Cl_*^3/t\). The pathwise estimate follows by using \(|c_x|\le C\) and \(\|D_*\|\le Cl_*/\sqrt t\) instead of their means.

Since the weights \(p_i\) are bounded above and below, \[\sum|v_i|^2\le CB_i,\qquad \sum|v_i|\le C\sqrt{n_hB_i}.\] The sum of the absolute conditional biases in (123), divided by \(l_*\), is therefore bounded by \[ C\left[ t^{.265\delta}+t^{.27\delta}+t^{.34\delta} +t^{1/2-.02\delta}+t^{1/2+.48\delta-\nu} +t^{.345\delta}+t^{.42\delta}\right]=o(1). \tag{125}\] The conditional variance sum is at most \[ Cl_*^2\left(B_i+n_hl_*^2/t\right) \le Cl_*^2t^{.27\delta}. \tag{126}\] The jump divided by \(l_*\) is at most \(C(H_*^{1/4}+l_*/\sqrt t)\), a positive power of \(t\).

Apply both signs of the maximal martingale inequality at a sufficiently small fixed multiple of \(l_*\). After adding the absolute bias bound (125), except with stretched-exponentially small probability we have \[ \sup_s\left|\sum_{k\le s}d_i^{(k)}\right|\le l_*/2 \tag{127}\] at every prefix and every monitored row. The absolute value in (127) is essential: it bounds the change over any interval of heat steps by \(l_*\).

Suppose a surviving row first exceeded \(4l_*\). Immediately after the most recent ridge lowering, or at initialization if there was none, its leverage was at most \(2l_*\). Let \(k_0\) be the number of heats preceding that adjustment and \(s\) the current number. The ridge has been held constant for heats \(k_0+1,\ldots,s\), and the row has survived all of them. Its actual change is therefore exactly \[\sum_{k=k_0+1}^s Z_{k,i} =\sum_{k\le s}Z_{k,i}-\sum_{k\le k_0}Z_{k,i}.\] The two prefix sums use the single varying-ridge stream; their difference is the desired constant-ridge interval sum. Equation (127) bounds it in absolute value by \(l_*\), giving leverage at most \(3l_*\), a contradiction. This argument includes a prospective cap-breaking heat and uses only running heat prestates in its concentration estimates.

Finite monitoring and conclusion

It remains to make the stopped probability estimates independent of the ambient torus size. The location implication (115) keeps every selected pivot within \(\ell/2\) of \(B\), including the pivot of a prospective first failure. The residual hypothesis therefore applies at every heat.

Choose a fixed ordinary-distance cube \(Q\) containing \(B\), its \(\ell\)-neighborhood, and a further buffer of width \(t^{-5}\). By the bound on the diameter of \(B\), the number of sites in \(Q\) is polynomial in \(1/t\). Outside \(Q\), projection decay rules out a leverage-cap crossing deterministically, even at a prospective failing output. For the distortion processes, the collapsed-distance resolvent estimate gives \[|h_{ik}|\le Ct^{-1/2}e^{-ctd_T(i,k)}.\] Each heat increment with an endpoint outside \(Q\) has an exponentially small factor connecting that endpoint to the pivot. The other factor has bounded row square sum by (91). More explicitly, the squared Frobenius norm of the portion of one increment with at least one endpoint outside \(Q\) is at most \[C\left(\sum_{i\notin Q}|h_{ix}|^2\right) \left(\sum_k|h_{xk}|^2\right).\] The first sum has a dimension-dependent bound which is still independent of the torus volume. An ordinary shell of integer radius \(q\) has at most \(C_d(1+q)^{d-1}\) sites. For an endpoint outside \(Q\) its distance from the pivot is at least \(t^{-5}\), so (114) gives \(d_T(i,x)\ge\mathop{\mathrm{dist}}(i,x)/4\) for small \(t\). Therefore \[\sum_{i\notin Q}|h_{ix}|^2 \le C_dt^{-1}\sum_{q\ge t^{-5}}(1+q)^{d-1}e^{-ctq} \le C_dt^{-d-1}e^{-c't^{-4}}.\] In the last step, extract \(e^{-ct^{-4}/2}\) and bound the remaining exponential moment by \(C_dt^{-d}\). Summing at most \(n_h\) increments preserves an exponential bound by the triangle inequality. The same reasoning gives the entry and individual-row tails. It uses no factor proportional to the torus volume.

We may therefore apply the martingale inequalities to entries and row vectors compressed to \(Q\), and to the matrix compressed to \(Q\times Q\), and then add the deterministic tails. Compressions do not increase the conditional square bounds. If the core lies in a ball of radius \(t^{-C_{\mathrm{loc}}}\), then \[|Q|\le C_d(t^{-C_{\mathrm{loc}}}+t^{-4}+t^{-5})^d.\] Thus the largest vector-dilation dimension is at most \(1+|Q|^2\), and there are at most \(C|Q|^2\) entry, row, and leverage tests. These are fixed powers of \(1/t\) for each fixed \(d\). Take the martingale thresholds to be one quarter of the budget thresholds. Their predictable biases and deterministic tails are smaller than another quarter for sufficiently small \(t\). The strict margins proved above imply that the probability of a distortion crossing or leverage crossing is at most \(\exp(-t^{-c})\) for some fixed \(c>0\).

Finally, Lemma 31 bounds the probability of reaching \(n_h\) heats before another stop by \(C\exp[-c t^{-(e_h-e_0-\nu)}]\). Because \(e_h-e_0-\nu=(.02-10^{-5}/d)\delta>0\), this is of the same required form after decreasing the exponent. There are no other failure modes. The procedure uses measurable spectral functions, a measurable threshold ridge and a fixed tie rule; each real update is measurable as well. It must stop after at most \(n_h\) heats. On DONE the zero-ridge projection satisfies (118) by the stopping rule, and the budgets hold. This proves Proposition 32.

On DONE the preparation has supplied bounded accumulated heat distortion and a small diagonal for the final full-range projection. Lemma 27 turns that projection bound into deterministic control of the true core completion, and Corollary 29 gives its spatial decay. The next section specifies the successful commit and the rollback of a failed preparation, and accumulates successful operations using comparisons only at committed states.

Geometry and localization during a clearing

For a terminal set, give a nearest-neighbor edge cost zero if either endpoint is terminal, and cost one otherwise. Its path pseudometric is denoted by \(d_T\); \(\mathop{\mathrm{dist}}\) always denotes ordinary torus distance. In particular, the subscript \(T\) distinguishes the collapsed metric from the dimension \(d\).

Summing the committed changes

A successful clearing supplies one difference between its inverse before and after the operation, restricted to its surviving sites. We first state the deterministic result that sums these differences. The energy error has two parts: the change of the surviving edges, and the old energy carried through the sites that have been removed. Both will be compared in the spatial form fixed at the start of the stage.

Resetting the diagonal

The completed stage has changed the resolvent while keeping \(r\) frozen. We restore the relation \(r_i=g_{ii}\) by a small deterministic change in \(r\). This is the only reset between consecutive ordinary sweeps.

Closure of the deterministic induction

We can now return from a completed stage to the same assumptions with which the next stage begins. The probabilistic work in the next section will control how many pending sites remain; the result here applies pathwise to every finite history produced by the trial rules, with numerical rollback on failure.

Birth labels and chronological probability bounds

A birth is a first tile-trial failure in a one-digit sweep or a failed clearing trial. Give a one-digit sweep from scale \(j-1\) to \(j\) the post-sweep index \(j\). The late-failure bound below treats the final residual sweep separately. A birth label consists of this index, its type, and its tile or snapshot-component anchor. Tile and clearing labels are distinct even when their physical anchors coincide. Each birth assigns its label to every site covered by that failure’s marking rule, replacing any label the site already carried. At the end of each clearing stage, including the last stage before the cutoff check, every pending site thus carries its latest birth label.

There is a deterministic ordering of all possible labels: increasing scale, the prescribed tile order within each sweep, and then increasing site order for possible clearing anchors. At an anchor absent from the frozen snapshot, or belonging to a component too large to be attempted, insert a skipped slot. Auxiliary retries create no further birth labels. Use one designated site as the label anchor for every one-digit tile. After a global abort, continue this bookkeeping list by recording all remaining slots as skipped. The list of possible labels is deterministic even though eligibility of a particular clearing anchor and the time spent on earlier retries depend on history.

Let \(B_a\) be the indicator of a birth at a potential label \(a\), with value zero at a skipped slot, and let \(\mathcal F_a^-\) be the retained history just before that slot. In the operation filtration of Lemma 4, the time at which this slot is reached is a stopping time: the preceding queries, trial outcomes, and decisions to schedule or skip a slot are all recorded. It is almost surely finite by the retry bound above. Thus \(\mathcal F_a^-\) is the corresponding stopped history, and whether the slot is scheduled is \(\mathcal F_a^-\)-measurable. The uniform conditional trial estimates imply, for some fixed \(c>0\) and sufficiently large \(n_0\), \[ \mathbb E(B_a\mid\mathcal F_a^-) \le p_{j(a)},\qquad p_j=\exp(-4^{cj}). \tag{155}\] Here the exponent has been decreased to absorb fixed constants and the one-unit difference between sweep and post-sweep indices. Indeed, writing \(a_\lambda=(\lambda^2/3)4^{n_0}\) gives \[ \tfrac14<a_\lambda\le1,\qquad t_j=a_\lambda(4^{-j}+4^{-m}),\qquad \tfrac14 4^{-j}<t_j\le2\,4^{-j}\quad(j\le m). \tag{156}\]

For any distinct, deterministically specified labels \(a_1<\cdots<a_s\), \[ \mathbb P(B_{a_1}=\cdots=B_{a_s}=1) \le\prod_{b=1}^s p_{j(a_b)}. \tag{157}\] To prove this, the product of the first \(s-1\) indicators is measurable before the last slot. Apply (155) to the last indicator and repeat backwards. The stopped histories include every query made during the almost surely finite intervening retries. Hence their random durations do not affect the chronological argument. Equation (157) does not assert independence of births. Adaptive anchors will be handled by a union over deterministic possible labels.

Witnesses for a surviving pending site

The remaining task is to exploit the small conditional probabilities of births. Pending sites can survive a clearing stage without a new failure only by belonging to a component too large to clear. Such a component provides many separated pending sites at a much earlier scale.

An eligible component at scale \(j\) has at most \(t_j^{-\nu}\) sites. Its \(R_j\)-connectedness and the inspection radius imply that every site marked by its failed trial lies at distance at most \[ C t_j^{-\nu}R_j+\ell_j\le C4^{(8+\nu)j} \tag{158}\] from its anchor. The same bound holds for a tile birth, whose marks lie in a tile of side \(4^{6(j-1)}\).

After clearing stage \(j\), every pending site is either a mark of a failure at that stage or belongs to a huge snapshot component, meaning one with more than \(t_j^{-\nu}\) vertices. This includes pending marks made during the preceding sweep, since they were present when the snapshot was taken. Eligible components disappear on success and receive a new label on failure. Separation and the pivot-locality argument in Section 7 prevent another component’s trial from touching a huge component.

We also use the following persistence observation. If a site is pending at the end of stage \(j\) and its latest birth index is at most \(r<j\), then it was pending at the end of stage \(r\). It could not have become coupled in between, and if it had become terminal it could never have been marked again.

The cutoff and the last true-tail sweep

We can now finish the finite-volume construction. Write \(J=\lfloor m/2\rfloor\). Proposition 38 and the volume \(N=4^{20dm}\) give \[ \mathbb P(\text{some site is pending at end }J) \le\exp\{20dm\log4-4^{c_{\rm h}J}\}\longrightarrow0. \tag{162}\] Abort if this event occurs. Otherwise every active site is coupled. Thereafter no clearing is scheduled, so a new pending site would have no later route to a committed terminal before the final sweep. We therefore perform ordinary sweeps without clearing and abort at the first first-trial failure. There are at most \((m+1)N\) possible tile trials in these stages, including the final sweep. Give a trial after a previous abort failure indicator zero. Successive conditioning and the uniform trial bound yield \[ \mathbb P(\text{a later ordinary failure before abort}) \le C(m+1)N\exp(-4^{cJ})\longrightarrow0, \tag{163}\] decreasing \(c\) once more if necessary. No independence between the late trials is asserted.

These two errors are also small uniformly over every \(m\ge4n_0\), once \(n_0\) is sufficiently large. Indeed, with \(q=\min(c,c_{\rm h})>0\) and \(a=20d\log4\), their sum is at most \[ [1+C(m+1)]\exp\{am-4^{q(m/2-1)}\}. \tag{164}\] Choose a dimension-dependent \(M\) so large that for all \(m\ge M\), \(4^{q(m/2-1)}\ge(a+2)m\) and \(1+C(m+1)\le e^m\). Then (164) is at most \(e^{-m}\) whenever \(m\ge4n_0\ge M\). The scale comparison (156) makes this estimate uniform in \(a_\lambda\in(1/4,1]\) and \(E\in I_d\) as well.

Consider a retained history that reaches the start of scale \(m\) without abort. Every active site is still coupled, so its true residual \(\xi_i\) has not been queried. Lemma 4 gives its uniform law with variance \(s_m=\eta\), conditionally independent between sites. Reaching this stage is decided by retained information; the law does not condition on eventual success of the final sweep. For that sweep take \[t=2\eta,\qquad w=\theta=\tfrac12,\qquad \xi_{0,i}=\xi_i/\sqrt w=\sqrt2\,\xi_i.\] Then, conditional on its pre-query history, \[\mathbb E\xi_{0,i}=0,\qquad \mathbb E\xi_{0,i}^2=t,\qquad |\xi_{0,i}|\le\sqrt{3t}.\] These are the bounded-input moment hypotheses of Proposition 26. The entire residual, rather than one digit, may therefore be consumed in this final sweep. If \(k=w^{1/4}\) and \(r_{\rm bit}=k^2r_{\rm phys}\), the physical diagonal update is exactly \[k^2(\xi_{0,i}+t r_{\rm bit}) =\sqrt w\,(\xi_i/\sqrt w+t\sqrt w\,r_{\rm phys}) =\xi_i+\eta r_{\rm phys}.\] It inserts the full true residual and leaves the artificial term \(-\eta r_{\rm phys}\) from the previous \(-2\eta r_{\rm phys}\). The reserve normalization is \(\sqrt{\theta t}=\sqrt\eta\). No reset is needed after this sweep.

Although \(A\) and \(r\) may depend on both true and auxiliary variables, the true potential vector has its prescribed iid law as the unconditional marginal of the enlarged space. Each joint success witnesses the assertion in the true variables that a qualifying certificate exists for \(H_m\). Section 9 proves that this feasibility event is Borel. Its probability under the true product law is at least the joint success probability. The spectral averaging is then unconditional under that law.

An intrinsic positive disorder threshold

In this subsection only, fix \(d\ge4\). We specify its threshold by finite matrices and a fixed success probability. This avoids leaving an arbitrary choice among the many smallness constants in the preceding estimates. For \(d=3\) we retain the existential threshold in Proposition 3.

The existence proof supplies no effective numerical upper bound for \(n_*(d)\). The spectral readout uses the fixed probability \(1/2\) and the certificate bounds in (165) at the torus and damping scales prescribed in Definition 39. Its comparison holds for every qualifying pair \((A,r)\), so no measurable choice of a certificate and no independence between \(A\) and the true variables are required.

Conditional spectral averaging

We use own-site averaging twice: first to bound the expected number of eigenvalues in a narrow true spectral window, and later to remove spectral mass on an energy-null set fixed after freezing the other potentials. We record its conditioning explicitly and prove the precise form used below. For the general rank-one theory and spectral-averaging background, see (Simon and Wolff 1986, Theorem 5 and Section 7) and (Marx 2011, Theorem 1.1 and Proposition 3.1).

The spectral readout