Invertibility, absorption, and Schur complements

The inverses in [mat:response] exist almost surely under any absolutely continuous joint law of the true site potentials. Indeed, if \(\mathscr L u=0\), taking the imaginary part of \(u^*\mathscr L u\) gives \(Y^\top u=0\). Thus \(u\) vanishes at every port site, so restoring the true diagonal entries leaves its equation unchanged: \((H-E)u=0\). For a fixed finite site set, \(\det(H-E)\) is a polynomial in its true diagonal variables whose coefficient of their full product is one. It is not identically zero. A nonzero polynomial has a Lebesgue-null zero set, as follows by induction on the number of variables and one-variable slicing. This proves the assertion even when the port set and reference potentials depend on those variables. The same argument applies to deterministic restrictions to physical sites. We always discard the resulting null exceptional sets when forming such inverses.

A matrix \(D\) is passive if \(\operatorname{Im}D\le0\), with the Hermitian imaginary part fixed in 2. The absorption identities below apply to any real rectangular \(Y\), including real coordinate or subspace inclusions.

We will also use the elementary kernel consequence of passivity: if \(Du=0\) and \(D\) is passive, then \(\operatorname{Im}D\,u=\operatorname{Re}D\,u=0\). Indeed, the nonnegative matrix \(-\operatorname{Im}D\) has zero quadratic form on \(u\), so it annihilates \(u\); the assertion about the real part follows from \(Du=0\).

Closing ports and comparing energies

Symmetry and unitarity make \(\left\lvert S_{ja}\right\rvert^2\) symmetric nonnegative weights with unit row sums. For real scalar or real Euclidean-vector data \(f\) on the ports, their Dirichlet energy and its single-port contribution are \[ \mathcal Q_S(f)=\frac12\sum_{a,j}\left\lvert S_{ja}\right\rvert^2\left\lvert f_j-f_a\right\rvert^2, \qquad \mathcal M_a(f)=\sum_j\left\lvert S_{ja}\right\rvert^2\left\lvert f_j-f_a\right\rvert^2. \tag{6}\] For scalar data, writing \(F=\mathop{\mathrm{diag}}(f)\), unitarity gives \[ \mathcal Q_S(f)=\mathop{\mathrm{tr}}(F^2-FSFS^*). \tag{7}\] For vector data, one sums this identity over real components.

The use of the conditional mean \(\mu\), rather than \(\mathbb E[\left\lvert\gamma\right\rvert\mid\mathcal F]\), is important. In later applications the open system and the data must remain measurable without exposing the hidden true value at the port to be closed.

A standard Cauchy law for \(w\) makes this conditional mean vanish. To restore a potential within a bounded interval, we truncate that law and estimate the resulting error.

Dimension-independent analytic bounds

The common-phase closure formula contains the resolvent \((\mathbf 1-zB)^{-1}\). We need weak-\(L^1\) bounds for its sandwiched and trace expressions, with no dimensional factor beyond the displayed norms. For a nonnegative random variable \(X\) on the circle, write \[\left\lVert X\right\rVert_{L^{1,\infty}} =\sup_{t>0}t\,m_\mathbb T\{X>t\}.\] The Hilbert–Schmidt estimate below has the same dimension-independent scale as the dissipative-resolvent bound in [2]. We prove the disk estimates directly using a Hilbert-space-valued Calderón–Zygmund decomposition [6].

We record a useful extension of the same proof. If a rational analytic \(d\times d\) matrix function \(F\) has nonnegative Hermitian real part in the disk, its representing positive measures have total mass \(\operatorname{Re}F(0)\). Their Hilbert–Schmidt variation is at most \(\mathop{\mathrm{tr}}(\operatorname{Re}F(0))\le d\left\lVert F(0)\right\rVert\). Restoring the constant \(i\operatorname{Im}F(0)\), the preceding argument gives \[ \left\|\left\lVert F(e^{it})\right\rVert\right\|_{L^{1,\infty}} \le C d\left\lVert F(0)\right\rVert. \tag{16}\] We need only the resulting polynomial dependence on dimension and the norm at the center.

Determinant phases and artificial absorptions

To average inverse traces over a real diagonal shift, we integrate the phase of the determinant. The estimate must also allow artificial absorptions that vary with that shift.

Two variants will be used without changing this argument. If the real derivative is a Hermitian matrix \(K(s)\ge cP_J\) for a fixed \(c>0\), its contribution is \(\mathop{\mathrm{tr}}(K\operatorname{Im}G)\ge c\mathop{\mathrm{tr}}_J\operatorname{Im}G\). Thus [mat:phase-integral] holds with constants also depending on \(c\). Additional real diagonal derivatives can instead be retained explicitly in [mat:phase-derivative] and estimated using available absorption bounds. This is how the calibrated root derivative will be handled.

The artificial probe profile

Separate from the cell ports, we allow at most one artificial probe at each site. A prescribed mask chooses among absence, strength \(\tau\), and strength \(\tau\vartheta(v_i)\), where \(v_i\) is always the true site potential, including when the real reference value differs from it. Masks do not depend on the potentials. They may be fixed after conditioning on spatial offsets or other independent prescriptions. Zero strength means absence.

Here \(\vartheta:[-h,h]\to[0,1]\) is fixed, of class \(C^{1,1}\), polynomial on finitely many pieces, positive on \((-h,h)\), and identically one on a plateau containing \([-h/2,h/2]\) and the compact interior potential settings of the cell construction. Such a profile can be chosen after those compact settings are fixed and before the later scale and width parameters are selected. To construct it, choose \(a<h\) large enough that the required compact set lies in \([-a,a]\), and use \(3t^2-2t^3\) on each endpoint transition, with \(t=(v+h)/(h-a)\) on \([-h,-a]\) and \(t=(h-v)/(h-a)\) on \([a,h]\); put \(\vartheta=1\) on \([-a,a]\). The endpoint values are zero, the derivatives match at the joins, and \[ \left\lvert\vartheta'(v)\right\rvert^2\le C\vartheta(v),\qquad \vartheta(v)\asymp\mathop{\mathrm{dist}}(v,\{-h,h\})^2 \quad\hbox{near the endpoints}. \tag{20}\] Constants may depend on the fixed profile. In particular, \(\vartheta^{-q}\) is integrable on \([-h,h]\) for every \(q<1/2\).

For estimates at a single cell of size \(L\), artificial strengths will obey \(0\le\tau\le e^{-c_*L}\) with a sufficiently large fixed \(c_*>0\). The screening estimates will also allow \(0\le\tau\le1\). All spatial masks remain arbitrary subject to the independence convention above.

Clipping the corners

Fix an integer margin \(b\ge3\). In a tile of odd side \(b_l=3^l\), use coordinates centered at its central site and put \[ L=\frac{b_l-1}{2}-b,\qquad V=\{-L,\ldots,L\}^2\setminus\{(\pm L,\pm L)\},\qquad J=V\setminus\{0\}. \tag{21}\] Only sufficiently large \(l\) are used. Let \[B=\{x\notin V:\mathop{\mathrm{dist}}_1(x,V)=1\}, \qquad p_x=\mathbf 1_{\{|x|_1=1\}}\quad(x\in J).\] Thus \(p\) is the adjacency column from the root \(0\) to \(J\). Let \(M:\mathbb R^B\longrightarrow\mathbb R^J\) be the adjacency matrix from \(B\) to \(J\); when used on \(V\), it has an additional zero row at the root. Define \[ Z=\{x\in J:M_{x,\cdot}\ne0\},\qquad Z^+=\{x\in J:\mathop{\mathrm{dist}}_1(x,Z)\le1\},\qquad P=\operatorname{proj}_{(\ker M)^\perp}. \tag{22}\] We identify a projection with its range when specifying a matrix domain. The cell and its corner coupling are shown in 1.

Calibrating the root with a decaying profile

We choose the root reference value so that the Dirichlet cell has a mode at energy \(E\). For arbitrary real values \(v|_J\), write \[A=\operatorname{Adj}_J+\mathop{\mathrm{diag}}_J(v-E).\] Whenever \(A\) is invertible, define \[ c(v)=E+p^\top A^{-1}p. \tag{25}\] The Dirichlet matrix on \(V\), with root potential \(c(v)\), then satisfies \[ \begin{pmatrix}c(v)-E&p^\top\\p&A\end{pmatrix} \begin{pmatrix}1\\-A^{-1}p\end{pmatrix}=0. \tag{26}\] Thus invertibility of the punctured matrix \(A\) permits resonance of the full Dirichlet cell. We must choose the interior law so that \(A^{-1}\) is controlled, \(c(v)\) stays inside the physical potential interval, and this root-normalized mode is small at the boundary. The transmission estimates will later control how it couples to an arbitrary passive exterior.

We first construct a comparison profile \(\psi\), normalized by \(\psi_0=1\), and potentials \(v^0\) satisfying its local equation. The profile is defined also on \(B\), where it is small but nonzero, so it need not be a Dirichlet eigenvector. A deformation of this profile will give a polynomial inverse bound for the punctured matrix at \(v^0\). These two properties will control the exact calibrated mode after the interior values are randomly perturbed.

Choose a rational multiple \(\theta\) of \(\pi\) and a real phase \(\phi\) such that \[ |E-4\cos\theta|<h, \qquad m_\theta:=\min_{k\in\mathbb Z}|\cos(\theta k+\phi)|>0. \tag{27}\] The first choice is possible because the energy is in \((-4-h,4+h)\) and rational multiples of \(\pi\) are dense. The sequence of cosine values is periodic, so the second condition excludes only finitely many phases modulo \(2\pi\). Put \[d=\frac{h-|E-4\cos\theta|}{8}>0.\] Define the real sequence \(f\) on all integers by \[ \begin{aligned} f_0&=0,\\ f_k+f_{k+1} &=\operatorname{sign}\bigl(\cos(\theta k+\phi) \cos(\theta(k+1)+\phi)\bigr). \end{aligned} \tag{28}\] Recursion in either direction gives \(|f_k|\le|k|\). For small fixed \(\nu,c_u>0\), and \(|u|\le c_u/L\), set \[ \begin{aligned} \psi_x(u)&=e^{-\nu|x|_\infty} \frac{\cos(\theta(x_1+x_2)+\phi)}{\cos\phi} e^{u f_{x_1+x_2}}\quad(x\in V\cup B),\\ v_x^0(u)&=E-\sum_{y\sim x}\frac{\psi_y(u)}{\psi_x(u)} \quad(x\in V). \end{aligned} \tag{29}\] Every neighbor in the last sum belongs to \(V\cup B\) and \(\psi_0=1\). Uniformly over the allowed parameters, \[ c e^{-\nu|x|_\infty}\le |\psi_x(u)| \le C e^{-\nu|x|_\infty}. \tag{30}\] Indeed \(|u f_{x_1+x_2}|\) is bounded by a fixed multiple of \(c_u\) and the periodic carrier is bounded away from zero. For neighbors \(x,y\) the additional ratio factors differ from \(1\) by \(O(\nu+c_u)\): their radial exponents differ by at most \(\nu\), and \(|u(f_{y_1+y_2}-f_{x_1+x_2})|\le Cc_u\). At \(\nu=u=0\), the cosine addition formula gives \(v_x^0=E-4\cos\theta\). We therefore fix \(\nu,c_u\) small enough, once and for all, that \[ |v_x^0(u)-(E-4\cos\theta)|\le d, \qquad |v_x^0(u)|\le h-7d. \tag{31}\]

From now on \(u=u_L\) is fixed and suppressed in the notation. Set \[ \epsilon=e^{-\nu L},\qquad \sigma=\epsilon^2. \tag{35}\] In particular, \(|\psi_x|\) is comparable to \(\epsilon\) on \(B\) and on \(Z^+\), with constants independent of \(L\). We have chosen the port strength \(\sigma\) as the square of this designed boundary-profile scale. The exact calibrated mode will satisfy the polynomially lossy comparison in [cell:residual].

The branch sampler

We first describe the boundary conditionings used by the sampler. For each corner triple use the fixed basis in [cell:corner-basis] and put \[ \mathsf D_j=\mathop{\mathrm{diag}}\left(\frac{(r_j)_i}{n_i}:1\le i\le3\right) \quad(j=1,2). \tag{36}\] These matrices satisfy \[\mathsf D_jn=r_j,\qquad \mathop{\mathrm{tr}}\mathsf D_j=0, \qquad n^\top\mathsf D_jn=0.\] Thus a common shift changes the real diagonal in the missing direction \(n\), while a shift along \(\mathsf D_j\) changes its coupling to \(r_j\) without changing that diagonal. The three diagonal directions \(I,\mathsf D_1,\mathsf D_2\) are linearly independent: multiplying a vanishing linear combination by \(n\) gives a vanishing linear combination of the orthonormal vectors \(n,r_1,r_2\). Thus centered coordinates \((s,t_1,t_2)\) specify physical potentials by \[ \mathop{\mathrm{diag}}(v)=\mathop{\mathrm{diag}}(\bar v)+sI+t_1\mathsf D_1+t_2\mathsf D_2. \tag{37}\] The map has a fixed nonzero Jacobian. In fact its three columns, as vectors of diagonal entries, are orthogonal and have length \(\sqrt3\). Use this same coordinate convention at all levels, including at exactly coincident triples.

A prescribed corner box is the image under [cell:box-coordinates] of a product of three centered intervals. Fix an exponent \(\kappa\). The boxes allowed below have all three side lengths at least \(c(1+L)^{-\kappa}\) and at most \(C\), lie at distance at least \(c(1+L)^{-\kappa}\) from the boundary of \((-h,h)^3\), and keep each physical coordinate in a single polynomial piece of the fixed artificial profile \(\vartheta\). Constants may depend on \(\kappa\); in the hierarchy this exponent is fixed by \(K_0\). Uniform measure in these box coordinates is also uniform measure in the physical parallelepiped. Because \(\vartheta\) is positive in the interior and comparable to squared distance from the support endpoints, every nonzero artificial profile \(a_i\in\{1,\vartheta\}\) on a permitted box obeys \[ a_i(v_i)\ge c(1+L)^{-2\kappa}. \tag{38}\] Increasing the exponent absorbs any fixed comparison constants.

Choose \(D_1>C_2+2\). A shell-depth parameter \(D_2\) will be fixed after the polynomial thresholds in the boundary and zero-jitter transmission estimates have been fixed. Finally choose \(D_3\) sufficiently large depending on \(D_1,D_2\) and these thresholds. On \(J\) sample \[ v_x=v_x^0+r\mathbf 1_{Z^+}(x)+\zeta_x, \qquad r\sim\operatorname{Unif}[-L^{-D_1},L^{-D_1}], \tag{39}\] where the \(\zeta_x\) are independent centered uniforms of half-width \[ w_x= \begin{cases} L^{-D_3},& L-|x|_\infty\le D_2\log L+5,\\ \epsilon L^{-D_3},& L-|x|_\infty>D_2\log L+5. \end{cases} \tag{40}\] They are independent of \(r\) and all boundary sampling.

For these interior draws, use the matrix \(A\) and the calibrated value \(c(v)\) from [cell:calibrated-root]. At the root, put an open cell port \(a\) of strength \(\sigma\), and use \(c(v)\) as its real reference potential. To define the hidden true potential, fix a smooth cutoff \(\chi:\mathbb R\to[0,1]\) with \(\chi=1\) on \([-d,d]\) and \(\mathop{\mathrm{supp}}\chi\subset[-2d,2d]\), and sample \[ \begin{gathered} v_0=c(v)+\delta,\qquad \mathbb P(\delta\in\,\mathrm dt\mid v|_J) =\frac{1}{Z_\sigma}\chi(t)\frac{\sigma}{\pi(t^2+\sigma^2)}\,\,\mathrm dt,\\ Z_\sigma=\int_{\mathbb R}\chi(t)\frac{\sigma}{\pi(t^2+\sigma^2)}\,\,\mathrm dt. \end{gathered} \tag{41}\] This is exactly the cutoff Cauchy law used in 5. The open reference \(c(v)\) does not depend on the hidden draw \(\delta\).

The boundary law is as follows. On each corner triple sample uniformly in its prescribed box, independently between the four triples. On every single side site \(x\) sample \[ v_x=\xi_x+s_0,\qquad \xi_x\sim\operatorname{Unif}[-h/4,h/4],\qquad s_0\sim\operatorname{Unif}[-h/4,h/4], \tag{42}\] using independent \(\xi_x\) and one common \(s_0\). These draws are independent of the triples and the interior draws. Sites of the tile outside \(V\cup B\) retain their independent uniform \([-h,h]\) laws, independently of all draws just described. Apart from the common shifts \(r,s_0\) and the dependence of the root center \(c(v)\) on \(v|_J\), all sampling is independent.

We record explicitly the order of choices used later. The carrier, \(d,\nu,c_u,C_1,C_2\), the compact cutoff and plateau, and the corner-box exponent are fixed first. Next fix \(D_1\). The polynomial threshold in 11 and the regular inverse thresholds in the zero-jitter part of 17 can then be fixed without \(D_2,D_3\). At depth \(L-|x|_\infty=\lfloor D_2\log L\rfloor\), [cell:profile-size] gives \[\frac{|\psi_x|}{\epsilon}\ge cL^{\nu D_2}.\] Choose \(D_2\) large enough that this dominates the already fixed polynomial errors and reflected-field bounds in that argument. Finally choose \(D_3\) to satisfy both the residual requirements above and the inverse and path-entry stability requirements of 17. Enlarging \(D_3\) changes neither the exponents in [cell:residual] nor the earlier zero-jitter bounds. All parameters are now fixed; only after that do we increase the lower cell-size threshold. This order prevents the depth choice from depending on a polynomial exponent that itself grows with the depth.

Elementary reductions and polynomial estimates

We first record the comparison that permits bounded changes of the absorbed coordinates.

The inverses used in this section exist almost everywhere in the stated boundary law. To see this without any nonsingularity assumption on \(T\), recall that a kernel vector of a passive matrix is annihilated by both its Hermitian real and imaginary parts. Absorption on \(P\) restricts such a vector to the four missing directions. On that space the Hermitian real part has the form of a fixed real symmetric matrix plus the independent common corner shifts on its diagonal. Its determinant is a nonzero polynomial, with the product of those shifts as its leading monomial. The same reasoning applies after one missing direction is deleted or after bounded changes in plane blocks. With absorption only on \(P_0\), a possible kernel lies on the full corner triples. The four common shifts act there as scalar shifts on their respective three-dimensional blocks; the determinant again has a nonzero leading monomial. Finally, \(\mathcal T\) itself has physical real diagonal variables with a joint density, so its Hermitian real part is nonsingular almost everywhere. These observations also justify almost-everywhere slicing below.

The response weights are integrable. For example, writing \(G=(\mathcal T-iP)^{-1}\), the Ward identity implies \[\left\lVert PGk\right\rVert^2\le k^*(\operatorname{Im}G)k \le \left\lVert k\right\rVert^2\mathop{\mathrm{tr}}\operatorname{Im}G.\] The joint law of the physical boundary diagonals has a bounded density on a compact set; the bound here may depend on \(L\). Slice that set along a common real shift of all the physical diagonals. The phase estimate 7, including its artificial-diagonal derivative term, bounds the integral of \(\mathop{\mathrm{tr}}\operatorname{Im}G\) along each slice. Integration over the transverse coordinates proves finiteness. One may add \(-i\zeta\mathbf 1\) first and pass to the limit by Fatou. In particular, after some boundary variables are frozen, all integrals below are finite for almost every such choice.

We shall use two elementary facts about polynomials. Their degree bounds are fixed throughout this section.

The following one-dimensional observation will prevent an extremely narrow response peak from defeating a short-exception estimate.

Eight corner tests

Remove the common shift from the single side variables and omit their artificial absorptions. Denote the resulting boundary matrix by \(\mathcal T_0\), and set \[w_0=\left\lVert P(\mathcal T_0-iP)^{-1}k\right\rVert^2.\] The independent single side uniforms are still present. By 12, \(w_0\) and \(\rho\) are pointwise comparable by fixed factors. We now hold those independent side uniforms fixed.

Choose an exponent \(H\), to be fixed after the polynomial losses below, and put \[\eta=L^{-H}, \qquad G_\eta=(\mathcal T_0-iP_0-i\eta Q)^{-1}.\] For each vector \(b'\) in the eight prescribed real orthonormal corner-plane bases, define \[ \mathcal E_{b'}= \left\{\eta\operatorname{Im}(b'^\top G_\eta b')\ge\frac1{40}\right\}. \tag{51}\] Outside their union, \[ \left\lVert Q(\mathcal T_0-iP_0)^{-1}Q\right\rVert\le\eta^{-1}. \tag{52}\] Indeed, if \(\left\lVert QG_\eta Q\right\rVert<(2\eta)^{-1}\), inverse difference and a Neumann series give the bound in [imp:Qgood]. Hence failure of that bound forces \(\left\lVert QG_\eta Q\right\rVert\ge(2\eta)^{-1}\). Ward on \(Q\) then gives \[\eta\mathop{\mathrm{tr}}(Q\operatorname{Im}G_\eta Q) \ge \eta^2\left\lVert QG_\eta Q\right\rVert_{\mathrm{HS}}^2\ge\frac14.\] If all eight tests fail to occur, the left side is less than \(8/40<1/4\), a contradiction.

Fix one test and condition on all boundary variables outside its corner triple. Use centered box coordinates \((s,t_1,t_2)\) there, where \(s\) is the common shift and \(t_j\) is the shift whose diagonal matrix maps the missing direction \(n\) to the plane basis vector \(r_j\). In particular, the \(t_j\) shift has zero \(n,n\) entry. Write \(t=(t_1,t_2)\). Every side length of the box is at least \(\ell\ge(1+L)^{-C}\) and at most a fixed constant. All probabilities and averages on this box are normalized uniform ones.

There is first an unweighted bound \[ \mathbb P_{s,t}(\mathcal E_{b'})\le\operatorname{poly}(L)\eta. \tag{53}\] For fixed \(t\), apply 7 while varying \(s\). Its real derivative is the identity on the tested triple. The artificial derivatives satisfy the profile condition and are controlled by their actual absorption, so the integral of the triple partial trace of \(\operatorname{Im}G_\eta\) is polynomially bounded. Since \(b'\) is a unit vector in that triple, its quadratic form is bounded by this trace. Divide by the interval length, then average over \(t\), to obtain [imp:unweighted]. Neither the phase range nor this polynomial loss depends on \(H\) or on the exterior norm.

We must strengthen [imp:unweighted] to a bound for \(w_0\)-mass.

A fixed remainder at one corner

The exceptional test still uses the actual inverse \(G_\eta\). To measure its weight, we replace \(w_0\) by a comparable response whose remainder after deleting the tested missing direction \(n\) is fixed. In \(\mathcal T_0-iP\), freeze the real variations of the tested triple within its plane-plane block at the box center. Replace the artificial absorption in that plane-plane block by \(\tau\) times the plane identity. Retain every fixed term from \(T\) and retain all actual mixed terms and the missing-direction diagonal. Call this modified matrix \(\widetilde D\).

The modification is bounded and supported on the tested plane. It also preserves absorption at least \(P\) plus the original artificial absorption of the tested triple. In fact the change of artificial absorption is \[\tau Q_a\bigl(\mathbf 1-\mathop{\mathrm{diag}}(a_i(v_i))\bigr)Q_a\ge0,\] because \(0\le a_i\le1\). Consequently \[ C^{-1}w_0\le W:=\left\lVert P\widetilde D^{-1}k\right\rVert^2\le Cw_0. \tag{54}\] The constant is fixed by 12.

Delete the tested missing direction \(n\), and denote the inverse of the remaining block of \(\widetilde D\) by \(G_{\mathrm a}\). This inverse is independent of \((s,t)\). The off-diagonal column from \(n\) and its Schur denominator have the form \[\begin{align*} p_1(s,t)&=p_{10}+t_1r_1+t_2r_2 -i\tau\sum_{i\text{ in the triple}}a_i(v_i)n_iQe_i, \tag{55}\\ s-e(s,t)&=s-e_{\mathrm{fix}} -p_1^\top G_{\mathrm a}p_1 -i\tau\sum_{i\text{ in the triple}}a_i(v_i)n_i^2. \tag{56}\end{align*}\] The fixed vector \(p_{10}\) may have components outside the tested triple; no bound on it is assumed. All variable parts of \(p_1\) lie in the tested plane. The constants in [imp:denominator] include the fixed diagonal at \(n\); there is no real \(t\)-dependent contribution to that diagonal.

Write \(k_{\widehat n}\) for \(k\) with its \(n\) coordinate deleted, and set \[ a=PG_{\mathrm a}k_{\widehat n},\qquad d_1=PG_{\mathrm a}p_1,\qquad \beta=p_1^\top G_{\mathrm a}k_{\widehat n}-k_n. \tag{57}\] The block inverse formula yields \[ W=\left\lVert a+\frac{d_1\beta}{s-e}\right\rVert^2 =\frac{\left\lVert N_1\right\rVert^2}{\left\lvert s-e\right\rvert^2}, \qquad N_1=a(s-e)+d_1\beta. \tag{58}\] The norm of \(d_1\) measures the response on the absorbed coordinates induced by unit amplitude in the missing direction. The scalar Schur denominator \(s-e\) controls how sharply \(W\) can concentrate. It need not agree with the corresponding denominator in the actual test \(G_\eta\). Each profile lies on one fixed polynomial piece throughout the corner box. Hence \(p_1,d_1,\beta,e,N_1\) are polynomials in \((s,t)\) of bounded degree, independently of all exterior data. Their coefficients need not be bounded.

Weighted exceptions when the coupling is appreciable

Put \[D=\max_{s,t}\left\lVert d_1(s,t)\right\rVert,\qquad \alpha_0=\frac1{100}.\] All maxima are taken over the closed box, which does not affect its conditional law. The bound \(\operatorname{Im}e\ge\left\lVert d_1\right\rVert^2\) ties the width of the response to this coupling. When \(D\ge\eta^{\alpha_0}\), we can isolate the few transverse parameters where the coupling becomes small and use the unweighted test bound elsewhere. The case where \(D\) is small throughout the box will instead require comparing the length of each exceptional slice with its response width.

We first treat \(D\ge\eta^{\alpha_0}\).

Suppose initially that \(\left\lvert s-e\right\rvert\ge1\) throughout the box. The maximum and minimum of this denominator are then comparable by a fixed factor. Indeed, [imp:derivatives-b,imp:derivatives-c] show that \(d_1\) varies by at most a fixed constant. If \(D\) is large, then \(\operatorname{Im}e\ge cD^2\) throughout, whereas the variation of \(s-e\) is at most \(C(1+D)\). If \(D\) is bounded, the same variation bound and the assumed lower bound one suffice. By [imp:poly-average,imp:unweighted], \[ \mathbb E_{s,t}[W\mathbf 1_{\mathcal E_{b'}}] \le\operatorname{poly}(L)\eta\,\mathbb E_{s,t}W. \tag{64}\]

It remains to consider the case where \(\left\lvert s-e\right\rvert<1\) somewhere. At that point \(\left\lVert d_1\right\rVert<1\); the derivative bounds then imply that \(D\) is bounded throughout the box. Since \(s\) is bounded, the same argument shows that \(e\) and \(s-e\) are bounded as well. Set \[M_1=\max\left\lVert N_1\right\rVert,\qquad B_1=\max\left\lvert\beta\right\rvert.\] The bounded denominator and [imp:poly-average] give \[ \mathbb E_{s,t}W\ge cM_1^2. \tag{65}\] Furthermore, \[ \left\lVert a\right\rVert+DB_1\le\operatorname{poly}(L)M_1. \tag{66}\] Here is the required quantitative argument. By the product estimate, \[DB_1\le C\max\left\lVert d_1\beta\right\rVert \le C(M_1+\left\lVert a\right\rVert).\] Differentiate \(N_1=a(s-e)+d_1\beta\) in \(s\) and use [imp:poly-derivative,imp:derivatives-a,imp:derivatives-b]. As \(D\) is bounded, this gives \[\left\lVert a\right\rVert\le C\ell^{-1}M_1+C\tau(\left\lVert a\right\rVert+B_1).\] Substitute the preceding estimate for \(B_1\). Since \(D\ge\eta^{\alpha_0}\), the coefficient \(C\tau(1+D^{-1})\) is less than one half for sufficiently large \(L\) after \(H\) is fixed. Absorption proves [imp:numerator-control]. Its polynomial exponent is independent of \(H\).

Choose \(\Delta=\eta^{1/8}\) and let \[\mathcal A=\{t:\min_s\left\lVert d_1(s,t)\right\rVert<\Delta\}.\] Then \[ \mathbb P_t(\mathcal A)\le C\frac{\Delta}{D}. \tag{67}\] To prove this, [imp:p1] and Ward show that \(d_1(s,t)\) is within \(C\tau\) of an affine function of \(t\), uniformly in \(s\). Rescale the two \(t\) intervals to unit intervals. If \(\mathcal A\) is nonempty, the affine function has a value of norm at most \(\Delta+C\tau\), and its maximum norm is at least \(D-C\tau\). Project it by a real unit linear functional attaining that latter norm. Because \(\Delta/D\to0\) and \(\tau\ll\Delta\), one of the two scalar slopes has magnitude at least \(cD\). Slicing in that coordinate bounds the fraction where the scalar magnitude is at most \(\Delta+C\tau\) by \(C\Delta/D\). This includes \(\mathcal A\).

For each \(t\in\mathcal A\), [imp:W,imp:s-integral] gives \[\mathbb E_s W\le\operatorname{poly}(L)(\left\lVert a\right\rVert^2+B_1^2) \le\operatorname{poly}(L)(1+D^{-2})M_1^2.\] On the complement of \(\mathcal A\), absorption gives \(\left\lvert s-e\right\rvert\ge\left\lVert d_1\right\rVert^2\ge\Delta^2\), so \(W\le\Delta^{-4}M_1^2\). Using [imp:unweighted] there and [imp:affine-sublevel] on \(\mathcal A\), we obtain \[ \mathbb E_{s,t}[W\mathbf 1_{\mathcal E_{b'}}] \le\operatorname{poly}(L) \left(\frac{\Delta}{D^3}+\frac{\eta}{\Delta^4}\right) \mathbb E_{s,t}W. \tag{68}\] We used the boundedness of \(D\) and [imp:weight-lower] in the last step. If \(M_1=0\), all the weighted integrals vanish and the inequality holds directly. In particular, this argument never divides by a possibly zero expected weight. The powers in [imp:first-case] satisfy \[\frac{\Delta}{D^3}\le\eta^{1/8-3\alpha_0}, \qquad \frac\eta{\Delta^4}=\eta^{1/2}, \qquad \frac18-3\alpha_0>0.\]

Weighted exceptions when the coupling is small

We now assume \(D<\eta^{\alpha_0}\) and write \(b'=r_j\) for the tested basis vector. Vary \(t_j\) between two points separated by at least half its interval length. From [imp:p1,imp:adb], \[ a_{b'}:=PG_{\mathrm a}b', \qquad \left\lVert a_{b'}\right\rVert\le C\ell^{-1}(D+\tau). \tag{69}\] This can be made as small as a prescribed fixed constant by choosing \(H\) sufficiently large, then \(L\) sufficiently large. The inference uses variation over the whole \(t_j\) interval; it does not assume that the exterior has a lower bound on its coupling.

Next delete \(n\) in the actual matrix \(G_\eta^{-1}\), without freezing its plane block. Let \(G_{\eta,\mathrm{rem}}\) denote the remaining inverse, put \[d_{1,\eta}=PG_{\eta,\mathrm{rem}}p_1,\] and write \(s-e_\eta\) for its Schur denominator at \(n\). Its off-diagonal column is exactly the same \(p_1\) as before. The remaining matrices for \(G_{\mathrm a}\) and \(G_{\eta,\mathrm{rem}}\) differ by a bounded matrix supported on \(Q\). Moreover, \(\left\lVert QG_{\eta,\mathrm{rem}}Q\right\rVert\le\eta^{-1}\) by Ward. Inverse difference and symmetry therefore give \[\begin{align*} \eta\left\lvert b'^\top G_{\eta,\mathrm{rem}}b'\right\rvert &\le C\left\lVert a_{b'}\right\rVert, \tag{70}\\ \left\lvert b'^\top d_{1,\eta}\right\rvert &\le\left\lvert b'^\top d_1\right\rvert +C\left\lVert a_{b'}\right\rVert\left\lVert Qd_{1,\eta}\right\rVert. \tag{71}\end{align*}\] For clarity, the inverse identity used in both estimates is \(G_{\eta,\mathrm{rem}} =G_{\mathrm a}+G_{\mathrm a}\Delta G_{\eta,\mathrm{rem}}\), where \(\Delta=Q\Delta Q\) has bounded norm. The first left factor is thus controlled by \(PG_{\mathrm a}b'\), regardless of the exterior norm.

Take the bound in [imp:small-b] small enough that the first term in the Schur inverse formula cannot by itself cause \(\mathcal E_{b'}\). On that exception, \[ \frac{\eta\left\lvert b'^\top d_{1,\eta}\right\rvert^2}{\left\lvert s-e_\eta\right\rvert} \ge c. \tag{72}\] On the other hand, absorption on the actual Schur extension yields \[\operatorname{Im}e_\eta\ge\eta\left\lVert Qd_{1,\eta}\right\rVert^2.\] Together with [imp:exception-schur], this implies \(\left\lVert Qd_{1,\eta}\right\rVert\le C\left\lvert b'^\top d_{1,\eta}\right\rvert\). Insert [imp:small-comparison-b] and absorb its small last coefficient. We conclude that, on the exception, \[ \left\lVert Qd_{1,\eta}\right\rVert\le C\left\lVert d_1\right\rVert, \qquad \left\lvert s-e_\eta\right\rvert\le C\eta\left\lVert d_1\right\rVert^2. \tag{73}\]

The exceptional set is short on every almost-everywhere \(s\) slice. To establish this rigorously, let \(D_t=\max_s\left\lVert d_1(s,t)\right\rVert\). Differentiating the actual Schur formula shows that, on the exception, \[\left\lvert\partial_s e_\eta\right\rvert \le C\left\lVert Qd_{1,\eta}\right\rVert^2 +C\tau(1+\left\lVert Qd_{1,\eta}\right\rVert) \le C D^2+C\tau(1+D)<\frac12.\] The first term comes from the real derivative of the tested plane block. Its artificial derivative is absorbed in the same bound; the mixed and missing-direction derivatives give the terms containing \(\tau\). Thus \(\operatorname{Re}(s-e_\eta)\) has derivative at least one half on the exception.

There are only boundedly many components of that exceptional slice away from poles. Indeed, with all other variables fixed, the matrix varies polynomially only on the three coordinates of the tested triple. In a basis containing those coordinates, expansion of the determinant uses variable factors from at most three rows. Its degree, and the degrees of the relevant cofactors, are bounded by a constant depending only on the profile degree, not on the dimension or norm of the exterior. The same assertion holds after deleting \(n\). Clearing the squared absolute value of a denominator converts [imp:test] into a real polynomial inequality of bounded degree. Such a set has a bounded number of intervals and isolated points, except for poles, which also have bounded number on an almost-everywhere slice.

On each interval component, [imp:small-exception] confines \(\operatorname{Re}(s-e_\eta)\) to an interval of length \(C\eta D_t^2\), while its derivative is at least one half. Summing over the bounded number of components gives \[ \left\lvert\{s:(s,t)\in\mathcal E_{b'}\}\right\rvert \le C\eta D_t^2. \tag{74}\] If \(D_t=0\), [imp:small-exception] and the Schur formula rule out the exception off its null set of poles.

For \(D_t>0\), the denominator of the modified weight has a matching minimum scale: \[ \operatorname{Im}e(s,t)\ge cD_t^2 \quad\hbox{throughout the slice}, \qquad \left\lvert\partial_s(s-e)\right\rvert\le C. \tag{75}\] Here is the point requiring the artificial-absorption floor. By [imp:derivatives-b], the oscillation of \(d_1\) on the slice is \(O(\tau)\). If its norm is everywhere at least \(D_t/2\), absorption proves the first bound immediately. Otherwise \(D_t\le C\tau\). In this latter case an artificial term must be present, since without one \(d_1\) is independent of \(s\). Now [imp:probe-floor] gives \(\operatorname{Im}e\ge\tau/\operatorname{poly}(L)\ge cD_t^2\) for sufficiently large \(L\). The derivative bound follows from [imp:derivatives-a] and \(D<\eta^{\alpha_0}\).

The exceptional length \(O(\eta D_t^2)\) must be compared with the resolution \(\min(\ell,D_t^2)\) supplied by the modified denominator. The real parts of the two Schur denominators need not vanish at the same point. Apply 14 to \(N_1\) and \(s-e\), using [imp:linewidth]. It gives \[\max_s W(s,t) \le\frac{C}{\min(\ell,D_t^2)}\int W(s,t)\,\,\mathrm ds.\] Combining with [imp:slice-length], \[\int W(s,t)\mathbf 1_{\mathcal E_{b'}}\,\,\mathrm ds \le\frac{C\eta D_t^2}{\min(\ell,D_t^2)} \int W(s,t)\,\,\mathrm ds \le\operatorname{poly}(L)\eta\int W(s,t)\,\,\mathrm ds.\] The last inequality uses \(D_t\le D<1\) and the inverse-polynomial lower bound on \(\ell\). Averaging over \(t\) proves \[ \mathbb E_{s,t}[W\mathbf 1_{\mathcal E_{b'}}] \le\operatorname{poly}(L)\eta\,\mathbb E_{s,t}W \tag{76}\] in the small-coupling case. This estimate remains uniform as the coupling vanishes; no positive lower bound for \(D\) was used here.

Restoring the side shift

We now choose the parameters. All polynomial losses in [imp:comparable-case,imp:first-case,imp:second-case], and the box width exponent, have been fixed independently of \(H\). Choose \(H\) large enough that their products with \[\eta^{1/8-3\alpha_0},\qquad \eta^{1/2},\qquad\eta\] tend to zero, and that [imp:small-b] is uniformly small in its case. With this \(H\) fixed, take \(L\) sufficiently large that \(\tau\ll\Delta\), \(\tau/\eta^{\alpha_0}\ll1\), and all preceding absorptions are valid. The assumption \(\tau\le e^{-c_*L}\) ensures every one of these requirements for any fixed \(c_*>0\).

For each of the eight tests, condition on the other corner variables, apply the appropriate estimate, and compare \(W\) with the same weight \(w_0\) by [imp:modified-weight]. Integration and summation give \[ \mathbb E[w_0\mathbf 1_{\mathcal G}]\ge\frac34\mathbb Ew_0, \qquad \mathcal G=\bigcap_{b'}\mathcal E_{b'}^c, \tag{77}\] for sufficiently large \(L\). This is still conditional on the independent side uniforms. The threshold can be chosen uniformly in those frozen values and in all exterior data. Zero conditional weights cause no difficulty, since every estimate was an inequality between integrals.

Fix data in \(\mathcal G\), before restoring the common side shift. On the space \(P\) consider the rational matrix function \[F(z)=P(\mathcal T_0+zP_0)^{-1}P, \qquad\operatorname{Im}z<0.\] It is analytic there and has nonnegative imaginary part. To check analyticity, a kernel at one point of the lower half-plane would be annihilated by \(P_0\) and by \(\mathcal T_0\), hence would also be a kernel at \(z=-i\), where the inverse exists. On \(\mathcal G\), [imp:Qgood] bounds the \(Q,Q\) block of \(F(-i)\) by \(\eta^{-1}\). Ward gives a bound one on the \(P_0,P_0\) block and \[\left\lVert P_0(\mathcal T_0-iP_0)^{-1}Q\right\rVert^2 \le\left\lVert Q\operatorname{Im}(\mathcal T_0-iP_0)^{-1}Q\right\rVert \le\eta^{-1}.\] Transpose symmetry gives the same bound on the other mixed block. Thus \[ \left\lVert F(-i)\right\rVert\le C\eta^{-1}. \tag{78}\]

For completeness, map the unit disk to the lower half-plane by \(z=-i(1+w)/(1-w)\). Then \(-iF(z(w))\) has positive semidefinite Hermitian real part, and its value at zero obeys [imp:center-bound]. The positive-real-part weak estimate [mat:positive-real-extension] gives a boundary weak-\(L^1\) bound at most \(C(\mathop{\mathrm{rank}}P)\eta^{-1}\). One can obtain this bound directly from the positive matrix measure in its Herglotz representation: its total trace mass is at most \((\mathop{\mathrm{rank}}P)\left\lVert F(-i)\right\rVert\), while the constant imaginary part has the same polynomial bound. The boundary change of variable sends Haar measure to \(\,\mathrm dx/(\pi(1+x^2))\).

The Cauchy density has a positive lower bound on the fixed interval \([-h/4,h/4]\). Therefore there is a polynomial \(R_L=C(1+L)^C\), now allowed to depend on \(H\), such that for at least half the common side shifts \(x\) in their prescribed uniform law, \[ \left\lVert P(\mathcal T_0+xP_0)^{-1}P\right\rVert\le R_L. \tag{79}\] This conclusion uses only the norm at \(-i\), not the norm of \(T\).

Restore the artificial absorptions at the single side sites. Their change has norm at most \(\tau\) and is supported on \(P_0\). For a shift satisfying [imp:side-good], the inverse difference identity on \(P\) and a Neumann series give \[\left\lVert P\mathcal T^{-1}P\right\rVert\le2R_L\] as soon as \(\tau R_L\le1/2\). This holds for sufficiently large \(L\). Throughout the common-shift average, with or without these side absorptions, 12 compares the weight to \(w_0\) by fixed factors. Hence at least a fixed fraction of the \(\rho\)-weight survives the good shifts for every corner setting in \(\mathcal G\). Integrate this statement, use [imp:weighted-Qgood], and finally average over the initially frozen independent side uniforms. Since \(\mathbb E\rho\) and \(\mathbb Ew_0\) are comparable, this proves [imp:weighted-estimate]. The pointwise comparison assertion in 11 is already contained in 12. \(\square\)

The Schur form used by the interior

The following formulation makes the interface with the transmission calculation explicit.

Exterior data and the two branch estimates

Suppose that the root is the only cell port in \(V\cup B\), and that the source port lies outside this set. Artificial probes with a prescribed mask may be present. Eliminate the exterior physical sites. Its effect on \(B\) is a fixed complex-symmetric passive matrix \(T\), and the source gives a fixed vector \(k\), including its input multiplier. The remaining boundary matrix is \[\mathcal T=T+\mathop{\mathrm{diag}}_B\bigl(v_i-i\tau a_i(v_i)\bigr).\] The exterior is held fixed throughout the cell integration, as are the four corner boxes. In particular, no bound on \(\left\lVert T\right\rVert\) or \(\left\lVert k\right\rVert\) is assumed.

Let \(P\) also denote its range, equipped with a real orthonormal basis. Eliminating the four missing boundary directions gives \[ \widehat T=(P\mathcal T^{-1}P)^{-1},\qquad \widehat k=\widehat T P\mathcal T^{-1}k,\qquad k_*=(\widehat T-iI_P)^{-1}\widehat k,\qquad \rho=\left\lVert k_*\right\rVert^2. \tag{80}\] These formulas are used at their almost-everywhere inverses. The missing-direction block is generically invertible by 16. Schur complementation preserves symmetry and passivity, and gives the identity \[k_*=P(\mathcal T-iP)^{-1}k.\] Thus \(\rho\) is exactly the weight in 11, and \[ \left\lVert(\widehat T-iI_P)^{-1}\right\rVert\le1. \tag{81}\] Hereafter \(M\) is restricted to \(P\), and is extended by a zero row at the root when it is regarded as a map into \(V\).

Let \(G\) be the inverse on \(V\oplus P\) with the root open, reference potential \(c(v)\), and root strength \(\sigma=\epsilon^2\). Its internal artificial absorptions are constant during the branch integration: the true values lie on the plateau specified in 10. Set \[ u=\sqrt\sigma\,Ge_0,\qquad g=\sigma G_{00},\qquad t=\widehat k^\top u_P. \tag{82}\] Since \(k\) includes the source input multiplier, the root-to-source scattering amplitude satisfies \(\left\lvert S_{\mathrm{source},a}\right\rvert^2=4\left\lvert t\right\rvert^2\). The sign convention for the source does not affect this identity. The inverse exists almost everywhere even though \(c(v)\) depends on other potentials. Indeed, absorption forces a kernel vector to vanish at the root. Restoring its true real diagonal therefore preserves that vector. After eliminating the fixed boundary data, the real part is a fixed matrix plus the true real diagonals on \(V\); their absolutely continuous joint law makes its determinant nonzero almost everywhere. This also justifies the almost-everywhere slices used below.

The cutoff error and the small-response region

We first derive [trans:relative] assuming [trans:lower]. Fix the boundary data and all jitters, and vary the common real shift \(r\) on \(Z^+\). Write \(P_+\) for the coordinate projection onto \(Z^+\), extended by zero on the root and on \(P\). Since \(c(v)=E+p^\top A^{-1}p\) and \(A'=P_+\) on \(J\), differentiation gives \[ c'=-\left\lVert P_+A^{-1}p\right\rVert^2,\qquad g'=-u^\top P_+u-\sigma^{-1}c'g^2. \tag{85}\] The profile estimate in [cell:residual], together with \(\left\lvert\psi_i\right\rvert\le C\epsilon\) on \(Z^+\), implies \(\left\lvert c'\right\rvert/\sigma\le C(1+L)^C\). Root absorption and 2 imply \[ \left\lvert g\right\rvert^2\le\operatorname{Im}g\le1. \tag{86}\]

Let \(I_r=[-L^{-D_1},L^{-D_1}]\). The derivative of a continuous determinant phase of \(G^{-1}\) is \[\Phi'(r)=\mathop{\mathrm{tr}}(P_+\operatorname{Im}G P_+) +\frac{c'}{\sigma}\operatorname{Im}g.\] The phase has range at most \(\pi\dim(V\oplus P)\) after strict regularization. Integrating, using [trans:g-ward], and then removing the regularization as in 7 gives \[ \int_{I_r}\mathop{\mathrm{tr}}(P_+\operatorname{Im}G P_+)\,\,\mathrm dr \le C(1+L)^C. \tag{87}\] In particular, Ward and symmetry bound the integral of \(\left\lVert P_+u\right\rVert^2\) by the same polynomial. We need an additional factor when \(g\) is small.

Put \(D=-\operatorname{Im}G^{-1}\). Complex symmetry makes \(D\) real symmetric and gives \[u-\overline u=2iG^*Du,\qquad u^*Du=\operatorname{Im}g.\] For each coordinate, Cauchy–Schwarz in the form \(D\) bounds the square of the corresponding entry of \(G^*Du\) by \((\operatorname{Im}G)_{ii}\operatorname{Im}g\). Consequently, with entrywise imaginary part on vectors, \[ \left\lVert P_+\operatorname{Im}u\right\rVert^2 \le (\operatorname{Im}g) \mathop{\mathrm{tr}}(P_+\operatorname{Im}G P_+). \tag{88}\] Take \(q=\sqrt\sigma=\epsilon\), and set \(\Omega_q=\{r\in I_r:\left\lvert g(r)\right\rvert\le q\}\), excluding null singular sets. [trans:phase-trace,trans:imaginary-vector] give \[\int_{\Omega_q}\left\lVert P_+\operatorname{Im}u\right\rVert^2\,\,\mathrm dr \le C(1+L)^Cq.\]

There are only polynomially many component intervals of \(\Omega_q\). To see this, \(A\) is affine in \(r\), \(c\) is a rational function obtained by inverting \(A\), and \(G\) is a further matrix inverse in cell dimension \(O(L^2)\). Thus numerator and denominator degrees of \(g\) are bounded by a fixed polynomial in \(L\). The inequality \(\left\lvert g\right\rvert^2\le q^2\) becomes a real polynomial inequality after clearing the squared denominator. Its intervals and the poles are therefore polynomially many; the values of the exterior coefficients do not affect this degree bound. Integrate the real part of [trans:shift-derivatives] on these intervals, first truncating away from any endpoints where necessary. Every endpoint value of \(\operatorname{Re}g\) has modulus at most \(q\). Hence \[\int_{\Omega_q}\operatorname{Re}(u^\top P_+u)\,\,\mathrm dr \le C(1+L)^Cq+C(1+L)^Cq^2\left\lvert I_r\right\rvert.\] Using \(\left\lVert P_+u\right\rVert^2=\operatorname{Re}(u^\top P_+u) +2\left\lVert P_+\operatorname{Im}u\right\rVert^2\), we obtain \[ \int_{\Omega_q}\left\lVert P_+u\right\rVert^2\,\,\mathrm dr \le C(1+L)^Cq. \tag{89}\] Normalized uniform integration in \(r\) only introduces the polynomial factor \((2L^{-D_1})^{-1}\).

The \(Z\) rows of the equation \(G^{-1}u=\sqrt\sigma e_0\) involve only \(u_P\) and \(P_+u\). Their coefficients are uniformly bounded, and \(M\) is bounded below on \(P\). The \(P\) rows then give, respectively, \[\left\lVert u_P\right\rVert\le C\left\lVert P_+u\right\rVert,\qquad \widehat T u_P=-M^\top u_J,\qquad \left\lVert\widehat T u_P\right\rVert\le C\left\lVert P_+u\right\rVert.\] It follows from symmetry and [trans:normalization] that \[ \left\lvert t\right\rvert =\left\lvert k_*^\top(\widehat T-iI_P)u_P\right\rvert \le C\sqrt\rho\,\left\lVert P_+u\right\rVert. \tag{90}\] The cutoff bounds in 5 now yield \[ \mathbb E\bigl[\left\lvert\mu\right\rvert\,\left\lvert t\right\rvert^2\bigr] \le Cq\mathbb E\left\lvert t\right\rvert^2+C(1+L)^Cq\mathbb E_B\rho. \tag{91}\] The first term treats \(\left\lvert g\right\rvert>q\); the second follows from [trans:small-response,trans:source-control] on \(\Omega_q\). Since \(q=e^{-\nu L}\), substitution of [trans:lower] absorbs the factor \(e^{C(\log L)^2}\) and proves [trans:relative] for some fixed \(c>0\).

Regular boundary and shift data

We turn to [trans:lower]. By 11, it is enough to work on boundary data satisfying \[ \left\lVert\widehat T^{-1}\right\rVert\le C(1+L)^C, \tag{92}\] since this event carries a fixed positive fraction of \(\mathbb E_B\rho\). Fix such data, assume \(\rho>0\), and initially set every jitter to zero. Define the boundary Schur matrix \[\mathcal B=\widehat T-M^\top A^{-1}M.\] We claim that at least half of the uniform \(r\)-interval satisfies \[ \left\lVert\mathcal B^{-1}\right\rVert\le C(1+L)^C. \tag{93}\] All exponents in this assertion can be fixed before \(D_2,D_3\).

Indeed, \[\mathcal B'=M^\top A^{-1}P_+A^{-1}M\ge cI_P.\] For the last inequality, put \(w=A^{-1}Mx\). The \(Z\) rows of \(Aw=Mx\) have bounded coefficients, use only \(P_+w\), and give \(\left\lVert Mx\right\rVert\le C\left\lVert P_+w\right\rVert\). The lower singular-value bound for \(M\) gives \(\left\lVert P_+w\right\rVert\ge c\left\lVert x\right\rVert\), which proves the assertion. For \(\xi>0\) let \(K_\xi=(\mathcal B-i\xi I_P)^{-1}\). If \(\left\lVert\mathcal B^{-1}\right\rVert>\xi^{-1}\), inverse difference implies \(\left\lVert K_\xi\right\rVert\ge(2\xi)^{-1}\). Ward then implies \[ \xi\mathop{\mathrm{tr}}\operatorname{Im}K_\xi\ge\frac14. \tag{94}\] The same conclusion follows at a singular matrix from a null vector. The determinant phase derivative of \(\mathcal B-i\xi I_P\) is at least \(c\mathop{\mathrm{tr}}\operatorname{Im}K_\xi\). Thus phase integration bounds the Lebesgue measure of the failed event in [trans:regular-r] by \(C\xi\dim P\). Dividing by \(2L^{-D_1}\) and choosing \(\xi\) to be a sufficiently small inverse polynomial proves the claim. The only dependence on the exterior here is through coefficients of a passive matrix; its norm never enters.

For regular zero-jitter data, omit both the root and artificial absorption in \(J\), and write \[ R_0=\begin{pmatrix}A&M\\M^\top&\widehat T\end{pmatrix}^{-1}, \qquad \Gamma=(R_0)_{JJ},\qquad z_0=(R_0)_{JP}\widehat k,\qquad U_0=\Gamma p/\epsilon. \tag{95}\] The subscript \(0\) denotes the absence of internal artificial absorption; the boundary matrix \(\widehat T\) is unchanged. These definitions will also be used after the jitters have been varied. The field \(z_0\) is the response in \(J\) to the exterior source. Define its normalized coupling to the root by \[F_0=p^\top z_0/\epsilon.\] If we adjoin the calibrated open root, still omitting the artificial absorptions on \(V\), block inversion gives the analogue of \(t\) for this auxiliary system: \[-\frac{F_0}{(c(v)-E-i\sigma-p^\top\Gamma p)/\sigma}.\] The prescribed internal absorptions will be restored below. We first obtain a lower bound for this numerator using the shell jitters, then control the denominator. A diagonal change at a site \(j\) changes \(F_0\) through the product \((U_0)_j(z_0)_j\). We will find a source component near the boundary and a large component of \(U_0\) at logarithmic depth, and use the jitters to propagate the source component to that deeper site.

Block inversion through \(A\) and \(\mathcal B\) gives polynomial bounds on \(R_0\) and \(\Gamma\). Moreover, \[ \Gamma^{-1}=A-M\widehat T^{-1}M^\top \tag{96}\] has polynomial norm by [trans:good-boundary].

The source normalization requires some care because \(\widehat k\) and \(\widehat T\) need not be bounded. Put \(C_{\rm ext}=\widehat T-iI_P\), \(\Delta=\mathcal B-C_{\rm ext} =iI_P-M^\top A^{-1}M\), and \(b_0=\mathcal B^{-1}\widehat k\). The identities \[ b_0=(I_P-\mathcal B^{-1}\Delta)k_*,\qquad k_*=(I_P+C_{\rm ext}^{-1}\Delta)b_0 \tag{97}\] use only polynomially bounded \(\Delta,\mathcal B^{-1}\) and the norm bound [trans:absorbed-exterior]. Consequently, \[ (1+L)^{-C}\sqrt\rho\le\left\lVert b_0\right\rVert \le C(1+L)^C\sqrt\rho,\qquad \left\lVert z_0\right\rVert\le C(1+L)^C\sqrt\rho. \tag{98}\] Since \(Az_0=-Mb_0\), the sparse \(Z\) rows also imply \(\left\lVert P_+z_0\right\rVert\ge c\left\lVert b_0\right\rVert\). There is therefore a site \(j_0\in Z^+\) with \[ \left\lvert(z_0)_{j_0}\right\rvert\ge(1+L)^{-C}\sqrt\rho. \tag{99}\]

The other field has the representation \[ U_0=\frac{A^{-1}p}{\epsilon} +A^{-1}M\mathcal B^{-1} \frac{M^\top A^{-1}p}{\epsilon}. \tag{100}\] The profile estimate and \(\left\lvert\psi_i\right\rvert\le C\epsilon\) on \(Z\) show \(\left\lVert M^\top A^{-1}p\right\rVert\le C(1+L)^C\epsilon\). Thus the reflected term in [trans:reflected-field] has polynomial norm, with exponent fixed independently of \(D_2\). At depth \(d=\lfloor D_2\log L\rfloor\), the profile has \[\frac{\left\lvert\psi_i\right\rvert}\epsilon\ge c e^{\nu d} \ge c' L^{\nu D_2}.\] Choose \(D_2\) large enough that this exceeds both the reflected term and the profile error in units of \(\epsilon\), with fixed slack. Every site \(i\) at that depth then satisfies \[ \left\lvert(U_0)_i\right\rvert\ge2. \tag{101}\]

A path of resolvent entries

Join \(j_0\) to a site at depth \(d\) by a physical nearest-neighbor path of length \(O(1+\log L)\) in \(J\), moving inward by decreasing absolute coordinates. The corner clipping guarantees that the path can avoid \(Z\) after its first vertex. For each successive pair \(y,j\), with \(j\notin Z\), [trans:gamma-inverse] gives \[(\Gamma^{-1})_{yj}=1.\] We next replace this physical edge by a uniformly bounded number of large entries of \(\Gamma\). This replacement is needed because \(\Gamma\) need not be spatially local.

Write \(B_\Gamma=\Gamma^{-1}\), \(n_J=\left\lvert J\right\rvert\), and \(K_\Gamma=\max(1,\left\lVert B_\Gamma\right\rVert)\). Let \[S_y=\mathop{\mathrm{supp}}(B_\Gamma e_y),\qquad S_j=\mathop{\mathrm{supp}}(B_\Gamma e_j),\qquad \delta_\Gamma=(2n_JK_\Gamma^2)^{-1}.\] Form the undirected graph on \(S_y\cup S_j\) with an edge between \(a\in S_y\) and \(b\in S_j\) whenever \(\left\lvert\Gamma_{ab}\right\rvert\ge\delta_\Gamma\). The endpoints \(y,j\) belong to this vertex set because \((B_\Gamma)_{yj}=1\). If they were not connected, let \(F\) be the diagonal indicator of the component of \(y\), extended by zero outside the graph. The inverse commutator identity would give \[\begin{align*} 1 &=\left\lvert[F,B_\Gamma]_{yj}\right\rvert =\left\lvert(B_\Gamma[F,\Gamma]B_\Gamma)_{yj}\right\rvert\\ &\le\delta_\Gamma \left\lVert(B_\Gamma)_{y,:}\right\rVert_1 \left\lVert(B_\Gamma)_{:,j}\right\rVert_1 \le\delta_\Gamma n_JK_\Gamma^2=\frac12. \end{align*}\] Indeed every nonzero summand with \(F_{aa}\ne F_{bb}\) must cross components, so its \(\Gamma\) entry is below threshold. This contradiction proves connectivity.

Every graph edge meets \(S_j\), which consists of \(j\) and at most its four physical neighbors: \(j\notin Z\) makes the nonlocal term in [trans:gamma-inverse] vanish in its column. Thus \(S_j\) is a vertex cover of size at most five. A simple connecting path has at most ten edges, since a simple path with no two successive vertices outside a set of size five has at most eleven vertices. The other support \(S_y\) lies in the physical star of \(y\) together with \(Z\). All these vertices lie in the shell \(L-\left\lvert i\right\rvert_\infty\le D_2\log L+5\); the fixed buffer accommodates the physical stars.

Concatenate the paths for the physical edges and erase loops. We obtain distinct sites \[ j_0,j_1,\ldots,j_s,\qquad s\le C(1+\log L),\qquad \left\lvert\Gamma_{j_a j_{a+1}}\right\rvert\ge(1+L)^{-C}, \tag{102}\] all carrying shell jitters, with \(j_s\) at depth \(d\). Fixed tie rules make all choices measurable functions of the regular zero-jitter data alone.

Propagation by the independent jitters

Choose \(D_3\) sufficiently large after \(D_2\). For every allowed simultaneous jitter perturbation, its diagonal operator norm is at most \(L^{-D_3}\). Inverse difference applied to \(R_0\) therefore preserves its polynomial norm, the bounds for \(\Gamma^{-1}\), and the path-edge lower bounds, with constant-factor slack. In particular, if superscripts denote perturbed and zero-jitter values, then \[z_0^{\rm new}-z_0^{\rm old} =-\Gamma^{\rm new}\Delta_J z_0^{\rm old},\] where \(\Delta_J\) is the jitter diagonal. [trans:source-size,trans:initial-component] show that the norm and starting-component bounds also retain constant-factor slack. The terminal bound retains, say, \(\left\lvert(U_0)_{j_s}\right\rvert\ge1\) throughout these perturbations. For this last assertion use [trans:reflected-field] and the uniform profile estimate again: the large total size of \(U_0\) is not controlled by an additive small-perturbation argument.

Condition on all nonpath jitters. Expose the path jitters successively, setting unexposed ones to zero only in the intermediate calculations. Let \(w=L^{-D_3}\) be their common half-width. At step \(a<s\), changing the diagonal at \(j_a\) by \(\zeta\in[-w,w]\) gives the exact rank-one formula \[ (z_0^{\rm new})_{j_{a+1}} =(z_0)_{j_{a+1}} -\frac{\Gamma_{j_{a+1}j_a}(z_0)_{j_a}\zeta} {1+\Gamma_{j_a j_a}\zeta}. \tag{103}\] Values on the right refer to the state just before this update. The denominator has modulus between \(1/2\) and \(3/2\) after increasing \(D_3\) if necessary. For \(\zeta\in[-w,-w/2]\) and \(\zeta'\in[w/2,w]\), the two values on the right differ by \[\left\lvert\Gamma_{j_{a+1}j_a}(z_0)_{j_a}\right\rvert \frac{\left\lvert\zeta-\zeta'\right\rvert} {\left\lvert 1+\Gamma_{j_a j_a}\zeta\right\rvert \left\lvert 1+\Gamma_{j_a j_a}\zeta'\right\rvert} \ge c w(1+L)^{-C}\left\lvert(z_0)_{j_a}\right\rvert.\] If one of the two intervals contains a value below one quarter of this separation in modulus, every value in the other interval is above that threshold. If it contains no such value, that interval itself is successful. Hence the conditional success probability is at least \(1/4\), and success gives \[ \left\lvert(z_0^{\rm new})_{j_{a+1}}\right\rvert \ge(1+L)^{-C}\left\lvert(z_0)_{j_a}\right\rvert. \tag{104}\] Constants and the exponent here include the fixed \(D_3\).

Finally expose the jitter at \(j_s\). Applied to the scalar \(F_0\), the update is \[F_0^{\rm new}=F_0- \frac{(U_0)_{j_s}(z_0)_{j_s}\zeta} {1+\Gamma_{j_sj_s}\zeta}.\] Since \(\left\lvert(U_0)_{j_s}\right\rvert\ge1\), the same two-interval argument gives a final success of probability at least \(1/4\) and size at least \((1+L)^{-C}\left\lvert(z_0)_{j_s}\right\rvert\). Only the current component is used at each step; later jitters need not preserve previously propagated components. There are \(O(1+\log L)\) updates, all at distinct sites and hence using independent uniforms. Combining [trans:initial-component,trans:jitter-gain], we conclude that \[ \mathbb P_{\rm jitters}\left\{ \left\lvert p^\top z_0/\epsilon\right\rvert \ge e^{-C(\log L)^2}\sqrt\rho\right\} \ge e^{-C(1+\log L)}. \tag{105}\] This holds conditionally on every regular zero-jitter boundary and \(r\) datum, and uniformly after conditioning on nonpath jitters.

Restoring absorption and the root

Reinsert the prescribed artificial absorptions in \(J\). If their diagonal is \(\tau A_{\rm art}\), with \(0\le A_{\rm art}\le I_J\), the rootless inverse becomes \[R_\tau= \left(R_0^{-1}-i\tau\begin{pmatrix}A_{\rm art}&0\\0&0\end{pmatrix} \right)^{-1}.\] Write \(\Gamma_\tau=(R_\tau)_{JJ}\) and \(z_\tau=(R_\tau)_{JP}\widehat k\). Inverse difference and the polynomial bounds already established give \[ \left\lVert\Gamma_\tau-\Gamma\right\rVert\le\tau C(1+L)^C,\qquad \left\lVert z_\tau-z_0\right\rVert\le\tau C(1+L)^C\sqrt\rho. \tag{106}\] Thus the error in \(p^\top z_0/\epsilon\) is at most \(\epsilon^{-1}\tau C(1+L)^C\sqrt\rho\).

Now restore the open root. Its scalar Schur denominator is \[d=c(v)-E-i\sigma-i\tau a_0-p^\top\Gamma_\tau p.\] For the auxiliary system with internal artificial absorptions omitted, the calibration and block formula for \(\Gamma\) give \[ \frac{c(v)-E-i\sigma-p^\top\Gamma p}{\sigma} =-i-\frac{p^\top A^{-1}M\mathcal B^{-1}M^\top A^{-1}p}{\sigma}. \tag{107}\] The boundary profile estimate gives \(\left\lVert M^\top A^{-1}p\right\rVert\le C(1+L)^C\epsilon\), and hence the second term is polynomially bounded. Reinstating absorption changes the normalized denominator by at most \(\sigma^{-1}\tau C(1+L)^C\), including the absorption at the root. Because \(c_*>2\nu\), this is small for large \(L\). We have therefore proved, uniformly on all the data being used, \[\left\lvert d/\sigma\right\rvert\le C(1+L)^C.\] Passivity and the open root also give \(-\operatorname{Im}d\ge\sigma\), so this denominator is nonzero.

Block inversion now identifies the amplitude, with an irrelevant sign, as \[ t=-\frac{p^\top z_\tau/\epsilon}{d/\sigma}. \tag{108}\] The numerator error in [trans:absorption-errors] is exponentially smaller than the successful lower bound in [trans:jitter-success]. The polynomial denominator and the success probability can therefore be incorporated into \(e^{-C(\log L)^2}\). For each good boundary datum, integration over the regular \(r\) event, of probability at least \(1/2\), and then all jitters gives \[\mathbb E_{r,\rm jitters}\left\lvert t\right\rvert^2 \ge e^{-C(\log L)^2}\rho.\] When \(\rho=0\), invertibility in [trans:normalization] gives \(\widehat k=0\), and the inequality is immediate. Finally integrate over the weighted good boundary event from 11. This proves [trans:lower] and completes the proof of 17.

For clarity, the choices above have no circular dependence. First fix the profile and its plateau, the box-width exponents, and \(D_1\) as in 10. The polynomial in 11, the regular zero-jitter inverse thresholds, and the exponents for the profile residual and reflected field can then be fixed without \(D_2,D_3\). Choose \(D_2\) to obtain the profile height in [trans:terminal-field]. Choose \(D_3\) afterwards to guarantee the profile estimates and all the perturbation margins used above. The constants in [trans:jitter-success] may depend on this last choice. Any fixed \(c_*>2\nu\) then makes the absorption errors small for sufficiently large \(L\); no polynomial threshold is chosen by using the norm of an exterior matrix.

An envelope for arbitrary interior choices

For comparison between different interior laws we need a bound that does not require the branch on \(V\). Keep the same geometry, boundary data, exterior source, and prescribed artificial mask. Allow any set \(D\subset V\) of internal cell ports, at most one per site, with strengths \(s_d\in[e^{-C_0L},1]\) and real reference potentials bounded in modulus by a fixed \(C_0\). All other internal real diagonals are their true values, and there are no cell ports on \(B\). The set \(D\), strengths, and references may depend arbitrarily on the true interior potentials and on auxiliary choices.

When this envelope is compared with the branch lower bound, the corner-box conditionings must coincide. The single-side laws on \(B\) may differ: their settings and artificial absorptions change \(\rho\) only by the pointwise bounded factors in 11. This is the precise boundary comparison needed when an adaptive collection of descendant ports is replaced by an open parent root.

An iid implementation of the rare cell laws

On the torus of side \(b_N\), choose one uniform lattice offset, independently of the true iid potentials. For each \(l_0\le l\le N\), partition the torus into squares of side \(b_l\) whose centers are congruent to that offset modulo \(b_l\). These partitions are nested. In each tile use the translated geometry and branch law of [cell:geometry,cell:branch], with \(L=L_l\). The word ancestor will always refer to strict containment in this nested family unless the contrary is stated.

We first record the geometric feature that makes conditioning compatible across levels. The corner triples belonging to different tiles and levels are disjoint except when they coincide exactly. At a coincident triple the ordering of its three sites, and the two real plane-basis vectors used for its box coordinates, are the same. In fact, the missing corner and its two companions have fixed displacements from the corresponding two ends of a tile. Passing to an extreme child preserves those positions: \[b_l+\frac{b_l-1}{2}-b =\frac{b_{l+1}-1}{2}-b.\] Consequently, if an ancestor triple meets a descendant tile, it is one of that descendant’s own triples. Also, the set \(V\cup B\) of a descendant is contained in that of its ancestor, and every descendant center belongs to the ancestor’s \(V\). The latter inclusion will ensure that all descendant reference-potential changes are interior changes when we use the parent-cell envelope.

Put \(K_0=10^6\). For each distinct corner triple, give its three true potentials a single box label as follows. In each product of polynomial pieces of \(\vartheta\) lying in \((-h,h)^3\), use the fixed linear box coordinates from 10. Partition the resulting region into maximal dyadic cubes whose concentric doubles lie in the region. Their inverse images are the corner boxes. All labels are defined from the true potentials, but we will expose only specified labels. At exact corner coincidences we use the same label, rather than successively refining it. Null boundaries may be assigned arbitrarily.

Conditional on a box label, the three true variables are uniform on that box: the original density is constant, and the coordinate change is linear with constant nonzero Jacobian. Labels of disjoint triples are independent. If a box has width less than \(\xi\), its points lie within \(C\xi\) of the boundary of one of the finitely many piece regions. Indeed, away from those boundary strips a dyadic cube of width comparable to \(\xi\) containing the point has its concentric double inside the region, so a maximal cube containing it cannot be smaller. Thus \[ \mathbb P(\text{width of a given triple's box}<\xi) \le C\xi. \tag{115}\] A level-\(l\) tile has valid own labels when the widths of all four corner boxes are at least \(b_l^{-K_0}\). The doubled-box requirement also provides the polynomial clearance from support endpoints and piece boundaries required in the cell estimates.

Assign to every tile \(U\) of level \(l\) an independent selector \(\eta_U\in\{0,1\}\) of success probability \(q_l\). These selectors are independent of all true potentials. A tile is eligible if it is selected and has no selected strict ancestor. Eligible tiles are pairwise disjoint, even when a selected tile later fails its branch test. An eligible tile is allowed if its own labels and those of every strict ancestor are valid at their respective levels.

For an allowed tile \(U\), let \(\mathcal L_U\) be the likelihood of its cell branch law relative to the true product law with its four corner boxes fixed. It changes the law on \(V\) and the noncorner single-side coordinates of \(B\), leaves the conditional laws on the four corner triples unchanged, and leaves all remaining sites of the tile unchanged. Its root factor is the conditional density centered at the reference \(c(v)\) constructed from the true values on \(J\). Apart from the allowance gate, the likelihood uses only data in \(U\) and its own corner boxes. In particular it does not use ancestor labels in any other way. The density bound of 10 gives, uniformly over valid boxes, \[\mathcal L_U\le \exp\bigl(C L_l^3+C L_l^2\log(2+L_l)\bigr).\] After increasing \(l_0\), we have \(p_l\mathcal L_U\le1\).

Give each tile a further independent uniform variable \(t_U\) on \([0,1]\). Its acceptance indicator is \[ A_U=\mathbf 1_{\{U\text{ eligible and allowed}\}} \mathbf 1_{\{t_U\le p_l\mathcal L_U\}}. \tag{116}\] The selectors, acceptance uniforms, and offset are auxiliary randomness; the true potentials are sampled first from their unchanged iid law.

For a stage \(n\), \(l_0\le n\le N\), place active cell ports at precisely the accepted roots of levels at least \(n\). Use their branch strengths and reference potentials. Every site without an active cell port has its true real potential. Because eligible tiles are disjoint, there is at most one cell port at a site. Denote the scattering matrix without artificial probes by \(S_{n,N}\).

We may also prescribe at most one artificial probe per site, absent or of strength \(\tau\) or \(\tau\vartheta(v_i)\). Its mask may depend on the offset and on external randomness independent of the potentials and branch decisions. The profile is always evaluated at the true value, including at an active cell root. When comparing stages \(n<m\), we assume \[ 0\le\tau\le\exp(-c_*b_m). \tag{118}\] This is sufficient for the cell estimate at every removed level and at its immediate parent. In particular all constants below are uniform over these masks. For notational convenience, expectations denoted by \(\mathbb E_{\rm off}\) may include any external independent randomness used in the geometry or deterministic data prescriptions.

Thinning passes and admissible data

Fix \(n<m\le N\). A thinning pass acts on a designated union of level-\(m\) tiles, chosen from the offset and any independent geometric randomness. In that region remove all accepted root ports of levels \(n,\ldots,m-1\), restoring their true real values. Process these levels in increasing order. At a given level interpolate by retaining each eligible accepted root independently with probability \(\theta\), with \(\theta\) decreasing from one to zero. Use fresh independent thinning marks; never recompute branch choices. One may subsequently perform a second pass on the complementary union of level-\(m\) tiles. Each such pass is covered separately by the estimates below, with its own starting system. A prefix may end between levels or at an intermediate value of \(\theta\).

Let \(f\) be deterministic finite real scalar or Euclidean-vector site data, possibly depending on the offset. Its restriction supplies data on every port set. Let \(g\) be data fixed at the beginning of the pass such that, for a deterministic \(c_g<\infty\), \[ \left\lvert g_i-g_j\right\rvert^2\le c_g^2\left\lvert f_i-f_j\right\rvert^2 \quad\text{for all sites }i,j. \tag{119}\] We permit \(g\) to depend on selectors, branch statuses, and the start-of-pass response, and on external independent randomness. The precise admissibility requirement is that, at each root closure to which the energy comparison is applied, \(g\) is measurable in the conditioning that preserves that root’s hidden law in 20. In particular it must not inspect still hidden true values needed for future closures, and it must not depend on the thinning marks of the pass. This formulation allows response-dependent adjustments while retaining the conditional closure identity.

For a level-\(l\) tile \(U\), let \(D(U)\) be its immediate parent, and put \[h_D=\sup_{i,j\in D}\left\lvert f_i-f_j\right\rvert.\] Define the nonnegative charge \[ \mathcal C_{n,m}(f) =\sum_{l=n}^{m-1} q_lp_l\, \mathbb E_{\rm off}\sum_{U:\,\operatorname{level}(U)=l} h_{D(U)}^2. \tag{120}\] The sum includes all tiles of a level, even when the pass acts on only part of the torus; this only enlarges the bound.

We prove the theorem by first establishing the estimate that controls the interpolation near \(\theta=0\). In a current level-\(l\) system, let \(\mathcal A_l\) be the thinnable roots in the pass region that are still present. For \(a\in\mathcal A_l\), write \(U(a)\) for its level-\(l\) tile. Set \[\begin{align*} Z_l(\theta) &=\mathbb E\sum_{a\in\mathcal A_l} \sum_{j:\,\operatorname{site}(j)\notin D(U(a))} \left\lvert S_{ja}\right\rvert^2\left\lvert f_j-f_a\right\rvert^2, \tag{122}\\ B_l(f) &=q_lp_l\,\mathbb E_{\rm off} \sum_{U:\,\operatorname{level}(U)=l}h_{D(U)}^2. \tag{123}\end{align*}\] Port data in these formulas are evaluated at their physical sites. Trivially, \[ Z_l(\theta)\le2\mathbb E\mathcal Q_{\rm current}(f). \tag{124}\]

The two useful scales of the additive charge

Prospective observations and a generic observation bound

Choose a large power of three \(Q_1\), and tile the torus by observation cores of side \(Q_1\). Set \(\beta=1/12\) and require \(Q_1^\beta\geq b_n\). For a core centered at \(z\), test candidate level-\(n\) roots in \(\Lambda_{3Q_1}(z)\) using the prospective rule from 19: retain all local acceptance rules, but omit ancestor selectors and validity tests at sides exceeding \(Q_1^\beta\). The resulting test for one candidate uses randomness within distance \(C Q_1^\beta\) of that candidate, conditional on the offset.

Each test succeeds with probability at least \(c q_n p_n>0\). To see this, first expose the required selector statuses and the needed corner-box identities. Conditional on the candidate being eligible and all required labels being valid, its acceptance likelihood integrates to one, so its selector and acceptance contribute \(q_np_n\). The probability of the required strict-ancestor selector absences and the own and ancestor validity conditions is bounded below by a positive absolute constant: their failures have a summable union bound along the ancestor chain, made small by the choice of the starting level. This also explains why the local likelihood is not additionally conditioned on finer corner data.

Cover \(\Lambda_{2Q_1}(z)\) by mesh squares of side comparable to \(Q_1^{1/4}\). In each such square choose candidates separated by a sufficiently large multiple of \(Q_1^\beta\). There are at least \(c Q_1^{2(1/4-\beta)}=cQ_1^{1/3}\) candidates with disjoint randomness neighborhoods, for sufficiently large \(Q_1\) at fixed \(n\). Their success indicators are independent conditional on the offset. The probability that some mesh square contains no successful root is therefore at most \[ C Q_1^{3/2}\exp(-c_n Q_1^{1/3}). \tag{133}\] In particular, with arbitrarily high probability, the prospective observation roots form a \(C Q_1^{1/4}\)-net of \(\Lambda_{2Q_1}(z)\), also for nearest-neighbor graph distance after changing the absolute constant.

The next argument applies to every possible observation set with this net property. Its equations always use the true potentials.

Projected resolvents and shield comparisons

The decay estimate below uses an exponential-conjugation argument of Combes–Thomas type [8]. We first establish the projected inverse bound on the observed coordinates; no bound on the full inverse is required.

Call an observation core good if the prospective roots form the specified net, the deterministic observation bound holds, and no hazardous tile of side greater than \(Q_1^\beta\) meets \(\Lambda_{4Q_1}\) about its center. The prospective roots of a good core are actual stage-\(n\) ports: all omitted ancestor requirements hold. Moreover, every real reference change in its tested rows occurs at a cell port of side at most \(Q_1^\beta\). Such a port has strength bounded below by a positive number depending on \(Q_1\).

We may count only a subset of these good cores. In a partly thinned or Dirichlet-restricted system, we count only cores for which the physical neighborhood \(\Lambda_{3Q_1+1}(z)\) remains present, and for which all cell ports, their strengths, and the real reference values in the tested rows \(\Lambda_{3Q_1}(z)\) agree with the full stage-\(n\) system. We call these tests intact. The neighbor layer preserves every hopping term in the true equations; artificial probes may still have any of the allowed masks. Let \(J_{\mathrm g}\) be the coordinate embedding of the sites in the counted cores, and set \[\Pi=[J_{\mathrm g}\ \ Y],\qquad G=(H^\#-E-iYY^\top)^{-1},\] where \(Y\) includes every current port, whether or not it lies in a counted core.

Crossing costs and the wall variant

By [screen:net-failure,screen:observation], the probability of a local test failure can be made arbitrarily small, first by taking \(Q_1\) large and then \(C_{\mathrm{obs}}\) large. The additional hazard cost for one core is at most \[C\sum_{b_l>Q_1^\beta}(1+Q_1/b_l)^2b_l^{-K_0},\] which tends to zero as \(Q_1\) grows. Consequently, in any neighborhood of radius a fixed multiple of \(Q_1\), the probability that some core meeting that neighborhood is bad is arbitrarily small. All these probabilities are conditional on the offset, uniformly in that offset.

Jump tails and fixed-stage limits

At full stage \(n\), with the possible extra probes, define \[ d_{n,N}(x)=\frac12 \sum_{a:\,\operatorname{site}(a)=x}\sum_j |S_{ja}|^2\mathop{\mathrm{dist}}_2(x,\operatorname{site}(j))^2. \tag{142}\] Here \(\mathop{\mathrm{dist}}_2\) is Euclidean distance formed from the shortest coordinate displacements. Let \(d_{n,N}^{\leq D}(x)\) and \(d_{n,N}^{>D}(x)\) denote the same sum restricted respectively to jump lengths at most \(D\) and greater than \(D\).

Compact profiles and convergence of affine energies

To approximate affine coordinate data on a torus without choosing a discontinuous coordinate branch, put \[ f_N(x)=\frac{b_N}{2\pi} \bigl(\cos(2\pi x_i/b_N),\sin(2\pi x_i/b_N)\bigr)_{i=1,2} \in\mathbb R^4. \tag{150}\] Its chord distances satisfy \[ \frac2\pi\mathop{\mathrm{dist}}_2(x,y)\leq|f_N(x)-f_N(y)|\leq\mathop{\mathrm{dist}}_2(x,y), \tag{151}\] and their ratio to \(\mathop{\mathrm{dist}}_2(x,y)\) tends to one uniformly for bounded nonzero jumps as \(N\to\infty\). Indeed, for shortest coordinate displacements \(\delta_i\), the squared chord distance is \(\sum_i(b_N\sin(\pi\delta_i/b_N)/\pi)^2\). Truncate the jumps, use this local limit, and then use [screen:jump-tail] for the tails. Translation after offset averaging and [screen:point-limit] give \[ b_N^{-2}\mathbb E\,\mathcal Q_{S_{n,N}}(f_N)\longrightarrow d_n. \tag{152}\] There are no artificial probes in this identity.

The order of parameters in this section will be used explicitly below. One fixes \(n\) before choosing observation scales, quantiles, jump cutoffs, or local approximation radii. Their possibly very large values impose only fixed thresholds on subsequent spatial scales. With fixed additional parameters, one can then take \(m>n\) so large that scales of order \(\exp(b_m^2)\) exceed every such threshold. A probe strength \(\tau=C_0/R\) at those scales tends to zero and eventually satisfies \(\tau\leq\exp(-c_*b_m)\), as required by 21. Finally \(N\) is increased after these choices. Neither this procedure nor [screen:d-limit] supplies or uses a rate of convergence uniform in \(n\).

Geometry and order of scales

Fix a number \(M_0>20\), to be enlarged below. After choosing \(n\), choose \(m>n\) and a power of three \(L_*\) such that \[ e^{b_m^2}\leq L_*<3e^{b_m^2}, \qquad R=\frac{L_*}{M_0^2}. \tag{155}\] The torus side \(b_N\) will be much larger than \(L_*\). Since \(L_*\) is a power of three, the torus admits an aligned partition into grid tiles \(D\) of side \(L_*\). The alignment is specified by the hierarchy offset. Write \(c_D\) for a tile center and use max-norm radii about this center.

Let \(I_D\) be the union of level-\(m\) tiles whose centers have radius at most \(R\), and let \(O_D\) be the union of level-\(m\) tiles whose centers have radius at least \(L_*/3\). Thus the inner square has radius \(R+O(b_m)\), and the outer walls begin at radius \(L_*/3+O(b_m)\). The remaining level-\(m\) tiles form the gap. Thin levels \(n,\ldots,m-1\) in all gaps, leaving the inner squares and walls at stage \(n\). The result is called the intermediate system. A complementary pass on the retained regions finishes the transition to uniform stage \(m\). Both pass regions are unions of whole level-\(m\) tiles, as required by 21.

Define the selector event \[ \mathcal P_D =\{\text{no selected tile of level at least $m$ meets $D$}\}. \tag{156}\] This event involves selectors only. Conditional on the offset, \[ \mathbb P(\mathcal P_D^c\mid\text{offset}) \leq C\sum_{l\geq m}(1+L_*/b_l)^2q_l=o(1). \tag{157}\] Indeed \(q_l=e^{-b_l^6}\) and \(L_*<3e^{b_m^2}\), so the right side tends to zero faster than any fixed negative power of \(R\). On \(\mathcal P_D\), the intermediate cell ports in \(D\) lie only in \(I_D\cup O_D\); after the complementary pass there are no cell ports in \(D\).

Independently of the potentials, selectors, and acceptance variables, choose a uniformly random grid tile, conditional on the offset. For this test tile write \(D,c,I,O,\mathcal P\) instead of \(D,c_D,I_D,O_D,\mathcal P_D\). All expectations below include this choice. It is a permitted independent prescription for the screening and comparison estimates. The retained regions and the probe annulus used below are shown in 2.

An adaptive adjustment of the affine data

For the moment use no artificial probes. Let \(S'\) be the intermediate scattering matrix and let \(f=f_N\) be the four-component torus embedding used to define the affine energy in 30. Its chord distances are globally comparable to shortest Euclidean torus distances, and its nearest-neighbor increments are bounded. Define four-component data on the intermediate cell ports by \[ F=\frac{f-f(c)}{R},\qquad F_I=\mathbf 1_I F, \qquad H^0=\sum_{\nu=1}^{4} \mathop{\mathrm{tr}}\!\left[F_{I,\nu}(F_\nu-S'F_\nu S'^*)\right]. \tag{158}\] As elsewhere, real scalar data inside a matrix expression denote the corresponding diagonal matrices. The trace \(H^0\) is real, since every diagonal entry of \(S'F_\nu S'^*\) is real. It is the local first variation of the affine energy: for fixed \(S'\), \[\left.\frac{\,\mathrm d}{\,\mathrm dt}\mathcal Q_{S'}(f-tRF_I)\right|_{t=0} =-2R^2H^0.\] Here the perturbation is data on the intermediate ports. The proof below extends it by a cutoff to site data with bounded increments and chooses its sign separately in each grid tile. On the selector event, the affected ports disappear in the complementary pass.

An annulus of artificial probes

Add probes at all sites of \[C=\{x:M_0R\leq |x-c|_\infty\leq 2M_0R\},\] and nowhere else, using strength \(\tau\vartheta(v_x)\), where \[ \tau=\frac{C_0}{R}. \tag{163}\] The constant \(C_0\) will be fixed before \(n,m,N\) are chosen. For sufficiently large \(m\), [van:scales] implies \(\tau\leq e^{-c_*b_m}\), so this mask is allowed throughout the first comparison pass. Use the same underlying hierarchy choices as before. The resulting intermediate scattering matrix has block form \[ S=\begin{pmatrix}A&T\\T^\top&B\end{pmatrix}. \tag{164}\] The first block consists of all cell ports, and the second consists of the new probes. We also use \(C\) for their index set. In this section \(I\) denotes the cell ports in the retained inner square and \(J\) denotes all other cell ports; these symbols are local to this section. On \(\mathcal P\), every port in \(J\) is outside the probe annulus toward the exterior. Set \[K=\left\lVert T\right\rVert_{\mathrm{HS}}^2.\] The data \(F,F_I\) still refer to [van:trace-data], now restricted to the cell ports of [van:block].

We first give the deterministic use of a good annular barrier. A site is called good for this purpose when \(\vartheta(v_x)\geq d_0\), where \(d_0>0\) is fixed sufficiently small. Count only edges with both endpoints good and lying in a buffered square band strictly inside \(C\), for example with both endpoint radii in \([1.2M_0R,1.8M_0R]\). Give such an edge cost one, and all other edges cost zero. Denote by \(\ell(x)\) the resulting cost distance from the region \(|x-c|_\infty\geq 2M_0R\). Let \(\mathcal B\) be the event \[ \ell(x)\geq\gamma M_0R \quad\text{whenever }|x-c|_\infty\leq M_0R, \tag{165}\] where \(\gamma>0\) is a sufficiently small fixed constant.

The next event combines small total transmission with positive internal jump energy and a bound on transmission weighted by \(F\). The weighted bound is needed because \(F\) grows on distant cell ports: small \(K\) alone does not control those coordinate factors in the trace.

An analytic trace with nonzero central value

Close all inserted probes with the common phase \(z\). For real \(s\), the corresponding physical shifts and the phase are \[ v_x\longmapsto v_x+s\vartheta(v_x)\quad (x\in C), \qquad z=\frac{s+i\tau}{s-i\tau}. \tag{174}\] By 3, the scattering matrix on cell ports is \[ S_z=A+\Delta(z),\qquad \Delta(z)=Tz(1-zB)^{-1}T^\top. \tag{175}\] Define \(H(z)\) on the unit circle by replacing \(S'\) with \(S_z\) in [van:trace-data]. Its matrix expression contains the adjoint \(S_z^*\). To remove that obstruction to analyticity, replace the adjoint by the adjugate and multiply the constant term by the determinant: \[ \mathfrak h(z)=\sum_\nu\mathop{\mathrm{tr}}\!\left[ F_{I,\nu}\{F_\nu\det S_z-S_zF_\nu\operatorname{adj}S_z\}\right] \tag{176}\] This function is analytic in the disk. Indeed \(B\) is a contraction, and the scattering entries in [van:closed-family] are rational analytic functions there. Possible boundary poles are removable: away from finitely many points their boundary values are unitary, which excludes a genuine pole of a rational entry on the circle. Thus \(\mathfrak h\) extends continuously to the boundary. There, \[ \mathfrak h(z)=\det(S_z)H(z),\qquad |\mathfrak h(z)|=|H(z)|, \tag{177}\] because \(\operatorname{adj}S_z=(\det S_z)S_z^*\).

Changing the potential law under a bounded twist

The artificial profile vanishes at the support endpoints and belongs to \(C^{1,1}([-h,h])\). For sufficiently small \(|s|\), the map \[T_s(v)=v+s\vartheta(v)\] is therefore a diffeomorphism of \([-h,h]\) onto itself, with derivative at least \(1/2\). If \(r_s\) is the density of its pushforward of the uniform law relative to that same law, then \[r_s(y)=\frac{1}{1+s\vartheta'(T_s^{-1}(y))}, \qquad |r_s-1|\leq C|s|,\qquad \int r_s\,\,\mathrm d\mu=1.\] Consequently \[ \int r_s^2\,\,\mathrm d\mu =1+\int(r_s-1)^2\,\,\mathrm d\mu\leq1+Cs^2. \tag{183}\] Here \(\mu\) is the uniform law on \([-h,h]\).

Fix \(W<\infty\) and assume \(|s|/\tau\leq W\). Conditional on the offset and the test tile, change every true variable in \(C\) by \(T_s\) and leave all other true variables, selectors, and auxiliary uniforms unchanged. There are at most \(C(M_0R)^2\) changed sites. By independence and [van:single-density], the resulting joint law has a density \(\mathscr D_s\) relative to the original joint law with \[ \left\lVert\mathscr D_s\right\rVert_{L^2}^2 \leq \exp(CM_0^2R^2s^2) \leq \exp(CM_0^2C_0^2W^2)=D_W^2. \tag{184}\] This also holds after averaging over the offset and the test tile. The bound is uniform in \(m,N\) once they are sufficiently large for the maps to be diffeomorphisms. Its constant may be large; only its independence of the later scales matters.

We next check why this change of true potentials reproduces the closed-probe system despite the potential-dependent hierarchy. This check concerns recomputed statuses, not an assumption that statuses are fixed by definition.

On \(\mathcal P\), selected tiles of level at least \(m\) do not meet the test tile. Every tile of level below \(m\) touching \(C\) lies wholly in the thinned gap, since \(b_m\ll R\) and the annulus is separated from both retained regions by a distance comparable to \(R\). Thus a selected tile capable of carrying an intermediate cell port is disjoint from \(C\). Its own true variables, own corner labels, local likelihood, and reference potential are unchanged by the shift. Eligibility depends only on selectors and is unchanged as well.

There is one remaining possibility: a changed coordinate may alter the validity of an ancestor label used by a retained tile. Any such ancestor contains both the retained root and a coordinate in \(C\). Their separation is at least a fixed multiple of \(R\), so the ancestor side is at least \(cR\). The probability that any such large tile has an invalid own label is bounded by \[ u_R=C_{M_0}\sum_{b_l\geq cR}(1+R/b_l)^2b_l^{-K_0}=o(1). \tag{185}\] This counts only tiles meeting \(C\), which also covers every relevant changed triple. The original bound is uniform in the offset and torus size. After the shift the probability is at most \(D_W\sqrt{u_R}\) by [van:joint-density]. Exclude both events. All potentially changed ancestor validity tests are then valid before and after the shift. Ancestor box identities themselves need not be the same: the acceptance rules use those identities only through the validity gates, whereas each retained tile’s own box and likelihood are unchanged.

On the remaining part of \(\mathcal P\), all intermediate cell ports and references are therefore unchanged when the hierarchy is recomputed with the same selectors and auxiliary uniforms. Changes of acceptance inside the gap have no effect, because those ports have already been thinned, and \(\mathcal P\) excludes the larger ones. Before closure the real potentials on \(C\) were the true ones; the only added terms there were the artificial absorbers. Hence [mat:closure,van:twist-parameter] imply that \(H(z)\) is exactly the no-extra-probe intermediate trace \(H^0\) evaluated at the shifted data. Almost-everywhere inverse statements suffice throughout.

Let \(a_{n,m,N}=\mathbb E[\mathbf 1_{\mathcal P}|H^0|]\) in the original law. Since \(\mathcal P\) uses only unchanged selectors, Markov’s inequality and [van:joint-density] give, for every fixed \(\delta>0\), \[ \mathbb P\{\mathcal G,\ |H(z)|>\delta\} \leq e_R(W)+D_W\left(\frac{a_{n,m,N}}{\delta}\right)^{1/2}, \qquad |s|/\tau\leq W, \tag{186}\] where one may take \(e_R(W)=u_R+D_W\sqrt{u_R}\). This tends to zero as \(m\) grows at fixed \(W,M_0,C_0\). The estimate is uniform over bounded twists; it does not assert simultaneous invariance for all twists on a single realization. Only the probability bound for each twist will be integrated.

The contradiction

Filling and twisting an annulus

Let \(X,Y'\) be arbitrary deterministic subsets of the probe sites in the radius-\(R\) squares centered at \(x,y\), respectively, where \(\mathop{\mathrm{dist}}(x,y)\ge10R\). Let \[C=\{i:2R\le\mathop{\mathrm{dist}}(i,x)\le4R\}.\] Fill every previously unoccupied site of \(C\) with a probe of strength \(\tau\vartheta(v_i)\), keeping existing probes and their strengths. The filled mask is again deterministic. Superscripts \({\rm old}\) and \({\rm fill}\) distinguish the resulting scattering matrices. Identify the probes in \(C\) with their sites and put \[U=\left\lVert S^{\rm fill}_{CX}\right\rVert_{\rm HS},\qquad V=\left\lVert S^{\rm fill}_{CY'}\right\rVert_{\rm HS},\qquad D=\left\lVert S^{\rm fill}_{Y'X}\right\rVert_{\rm HS}.\]

We seek a bound by an arbitrarily small multiple of \(\mathbb EV^s\). Thus [boot:twist-ineq] leaves two tasks: attenuate \(D\) across the filled annulus, and combine a small mean return \(U\) with a resampling bound for \(U^sV^s\). An unweighted estimate for barrier failure would not suffice when \(\mathbb EV^s\) is small. We therefore control that failure also under the \(V^s\)-weighted law.

Two barriers under a transmission tilt

Call a site good when \(\vartheta(v_i)\ge d_0\). Give cost one to an edge precisely when both endpoints are good and both radii about \(x\) belong to \([2.2R,2.8R]\); all other edges have cost zero. Let \(d_{\rm in}\) be the associated path-cost distance from the radius-\(2R\) square. Independently of this definition, count good-good edges with both radii in \([3.2R,3.8R]\), and let \(d_{\rm out}\) be the resulting distance from \(\{i:\mathop{\mathrm{dist}}(i,x)\ge4R\}\). Integer roundings of the four band boundaries make no difference to the constants. For a sufficiently small absolute \(\gamma>0\), define \[ \mathcal B=\left\{ \begin{array}{ll} d_{\rm in}(i)\ge\gamma R&\text{if }\mathop{\mathrm{dist}}(i,x)\ge3R,\\ d_{\rm out}(i)\ge\gamma R&\text{if }\mathop{\mathrm{dist}}(i,x)\le3R. \end{array}\right\}. \tag{191}\] These two strictly buffered bands will allow separate resampling of the inside and outside of radius \(3R\).

Cutoffs and the small return estimate

We record explicitly the consequence of 33 that will be used several times. Suppose \(\Psi=GY_J\) is a matrix of fields sourced at a probe set \(J\), and a scalar cutoff \(\chi\) changes only across edges whose two endpoints belong to \(C\) and carry probes of strengths at least \(\tau d\). If \(\left\lvert\chi_i-\chi_j\right\rvert\le L\), then the hopping commutator can be written as a probe source: \[ [H_N,\chi]\Psi=Y_C B_\chi, \qquad \left\lVert B_\chi\right\rVert_{\rm HS} \le\frac{K L}{\tau d} \left\lVert Y_C^\top\Psi\right\rVert_{\rm HS}. \tag{195}\] Only endpoints of changing edges need be included on the right. Indeed, in the coefficient at an endpoint one divides by its square root strength, and replaces each neighboring field by its absorbed field divided by that neighbor’s square root strength. Bounded lattice degree gives the stated operator bound. Ward then yields \[ \left\lVert Y^\top G[H_N,\chi]\Psi\right\rVert_{\rm HS} \le\left\lVert B_\chi\right\rVert_{\rm HS}. \tag{196}\] The output here includes all probes. These estimates also compare two systems agreeing on the cutoff support and its changing edges: cut off a field of the first system and correct its commutator residual in the second. No estimate on the exterior resolvent is involved.

Direct transmission and independent resampling

We next prove two pointwise estimates: \[ D\le Kd_{\min}^{-1}V\quad\hbox{almost surely},\qquad D\le K e^{-c C_0}V\quad\hbox{on }\mathcal B. \tag{198}\] For the first, use fields sourced at \(Y'\), symmetry of \(S\), and a cutoff equal to one inside radius \(2R\), zero outside radius \(4R\), with gradient at most \(K/R\). Its source term vanishes, while both endpoints of every changing edge have strength at least \(\tau d_{\min}\). [boot:commutator,boot:cutoff-ward] give \(D\le K(R\tau d_{\min})^{-1}V\); here \(C_0\ge1\). In particular, if \(\mathbb EV^s=0\), then \(D=0\) almost surely, and no tilted law is required.

For the improved estimate choose an integer \[\ell=\left\lceil\frac{K_1}{\tau d_0}\right\rceil\] with \(K_1\) sufficiently large. For \(j\ge1\) use the cutoff \[\chi_j(i)=\min\left\{1, \frac{(d_{\rm out}(i)-(j-1)\ell)_+}{\ell}\right\}.\] All original sources have \(d_{\rm out}=0\). Nonzero gradients have both endpoint distances at least \((j-1)\ell\), because cost distances are integer valued and change by at most one across an edge. Gradients only occur on counted good-good edges in the outer band. For the field \(\Psi=G^{\rm fill}Y_{Y'}\), put \[A_j=\left\lVert\mathbf 1_{\{d_{\rm out}\ge j\ell\}}Y^\top\Psi\right\rVert_{\rm HS}, \qquad B_j=\left\lVert\mathbf 1_{C\cap\{d_{\rm out}\ge j\ell\}} Y^\top\Psi\right\rVert_{\rm HS}.\] The cutoff equations and Ward give \[A_j\le\frac K{\ell\tau d_0}B_{j-1} \le\tfrac12 B_{j-1},\qquad B_j\le A_j.\] Start with \(B_0=V/2\). On \(\mathcal B\), the set \(X\) has \(d_{\rm out}\ge\gamma R\), so it remains beyond every one of the \(\lfloor\gamma R/\ell\rfloor\) cutoffs. For sufficiently large fixed \(C_0\), followed by sufficiently large \(R\), this number is at least \(c(d_0)C_0\). The iteration proves the second estimate in [boot:direct]. Notice that \(A_j\) includes every absorbed output, including \(X\); the proof does not require \(X\subset C\).

To separate the product \(V^sU^s\), enlarge the probability space with two independent fresh families of true potentials. Let \(A=\{i:\mathop{\mathrm{dist}}(i,x)<3R\}\). Form one comparison configuration by retaining the original potentials on \(A^c\) and replacing those on \(A\) by the first fresh family. Form a second by retaining the original potentials on \(A\) and replacing those on \(A^c\) by the second fresh family. Keep the filled deterministic mask in both configurations, and recompute its strengths from the corresponding potentials. Denote the first configuration’s transmission to \(Y'\) by \(V^{\rm ext}\), and the second’s transmission from \(X\) by \(U^{\rm int}\).

On \(\mathcal B\), the cutoff \((1-d_{\rm out}/(\gamma R))_+\) is supported in the shared exterior \(A^c\), equals one on \(Y'\), and has all changing edges in its buffered outer band. Applying it to the fresh-interior fields and correcting in the original system proves \[V\le\left(1+\frac K{C_0d_0}\right)V^{\rm ext}.\] Likewise \((1-d_{\rm in}/(\gamma R))_+\) is supported in the shared interior \(A\) and gives \[U\le\left(1+\frac K{C_0d_0}\right)U^{\rm int}.\] Take \(C_0\) large enough that both factors are at most two. The two comparison configurations are independent unconditionally: they use disjoint parts of the original product sample and independent fresh families. Each has the original true product distribution. Dropping \(\mathbf 1_{\mathcal B}\) before factoring therefore yields \[ \mathbb E[\mathbf 1_{\mathcal B}V^sU^s] \le 2^{2s}\mathbb EV^s\,\mathbb EU^s \le 2^{2s}\mathbb EV^s(\mathbb EU^2)^{s/2}. \tag{199}\] This factorization makes no assertion of independence conditional on the original barrier event.

The exceptional event is controlled under the tilted law from 38. Unitarity gives \(U\le K R\), and [boot:direct,boot:tilted-barriers] give, respectively, \[\begin{align*} \frac{\mathbb E[\mathbf 1_{\mathcal B^c}V^sU^s]}{\mathbb EV^s} &\le K R^s\nu(\mathcal B^c),\\ \frac{\mathbb E[\mathbf 1_{\mathcal B^c}D^s]}{\mathbb EV^s} &\le K\mathbb E_\nu[\mathbf 1_{\mathcal B^c}d_{\min}^{-s}]\\ &\le K(\mathbb E_\nu d_{\min}^{-2s})^{1/2} \nu(\mathcal B^c)^{1/2} \le K R e^{-cR/2}. \end{align*}\] Both quantities tend to zero uniformly over all the prescriptions. Combining these estimates with 37, we obtain \[ \mathbb E\left\lVert S^{\rm old}_{Y'X}\right\rVert_{\rm HS}^s \le K_s(1+C_0) \left[e^{-scC_0}+(\mathbb EU^2)^{s/2}+\epsilon_R\right]\mathbb EV^s, \qquad \epsilon_R\longrightarrow0. \tag{200}\] The constants \(K_s,c>0\) are independent of large \(C_0\). First choose \(C_0\) to make its exponential term small, then use 39 with \(n,m,N\) in order to make the return term small, and enlarge \(m,N\) as necessary for \(\epsilon_R\). Thus the entire multiplier can be made arbitrarily small. These are uniform expectation bounds for each choice of mask and centers; no simultaneous good event over all choices is needed.

Contraction over all masks

The passage from finite-scale smallness to exponential fractional-moment decay parallels the finite-volume criteria of [4]. We prove the contraction here for the probe transmission moments and deterministic masks.

Removing the two endpoint probes

Choose a mask consisting only of two constant-strength probes \(\tau\) at distinct sites \(x,y\). Let \(G^0=(H_N-E)^{-1}\), and let \(g\) be its two-by-two compression to \(\{x,y\}\). Resolvent subtraction gives the damped compression \[g^{\rm damp}=(1-i\tau g)^{-1}g.\] For a symmetric two-by-two matrix, direct multiplication by the adjugate shows \[(g^{\rm damp})_{xy} =\frac{g_{xy}}{\det(1-i\tau g)}.\] Since \(S_{xy}=2i\tau(g^{\rm damp})_{xy}\), the exact identity is \[ G^0_{xy}=\frac{S_{xy}}{2i\tau}\det(1-i\tau g). \tag{202}\]

For completeness, the moments needed to remove the determinant are uniform in the exterior and the volume. Conditional on all other potentials, Schur complementation gives \[g=(A+\mathop{\mathrm{diag}}(v_x,v_y))^{-1}\] with \(A\) a fixed real symmetric matrix. Put \(t=(v_x+v_y)/2\), \(u=(v_x-v_y)/2\). The integration square becomes \(\left\lvert u\right\rvert\le h\), \(\left\lvert t\right\rvert\le h-\left\lvert u\right\rvert\), with Jacobian two. For each fixed \(u\), let \(\lambda_1(u),\lambda_2(u)\) be the real eigenvalues of \(A+\mathop{\mathrm{diag}}(u,-u)\). If \(0<q<1\), \[\int_{-(h-\left\lvert u\right\rvert)}^{h-\left\lvert u\right\rvert} \left\lVert(A+\mathop{\mathrm{diag}}(u,-u)+t1)^{-1}\right\rVert^q\,\,\mathrm dt \le\sum_{j=1}^2\int_{-h}^h\left\lvert t+\lambda_j(u)\right\rvert^{-q}\,\,\mathrm dt \le K_q h^{1-q}.\] Integration in \(u\), with the normalized joint uniform density, therefore proves \[ \mathbb E\left\lVert g\right\rVert^q\le K_{h,q},\qquad 0<q<1. \tag{203}\] The bound is uniform over all real symmetric \(A\), so no exterior norm enters. The deleted system is generically invertible; its exceptional set has zero product measure. A one-site Schur complement and the same scalar integral also give \(\mathbb E\left\lvert G^0_{xx}\right\rvert^q\le K_{h,q}\).

Since \(\left\lvert\det(1-i\tau g)\right\rvert\le(1+\tau\left\lVert g\right\rVert)^2\), [boot:two-site] with \(q=2s<1\) gives a uniform bound on \(\mathbb E\left\lvert\det(1-i\tau g)\right\rvert^s\). Apply Cauchy–Schwarz to [boot:endpoint-identity]: \[\mathbb E\left\lvert G^0_{xy}\right\rvert^{s/2} \le(2\tau)^{-s/2} \bigl(\mathbb E\left\lvert S_{xy}\right\rvert^s\bigr)^{1/2} \bigl(\mathbb E\left\lvert\det(1-i\tau g)\right\rvert^s\bigr)^{1/2}.\] The first factor of expectations decays exponentially by [boot:mask-decay], applied to singleton subsets. The diagonal is covered by the one-site bound. The now fixed \(R,\tau\) and all constants are absorbed into \(A(h,E,s)\) and \(a(h,E,s)>0\), proving 36. This conclusion remains a fixed-real-energy finite-volume estimate; the spectral passage uses it separately at each energy.