There are constants \(C\ge 0\) and \(c{\gt}0\), depending only on \(p,R,v_R,b_0,J,\Delta \) and the growth of the spheres, such that

and for every integer \(l\ge 0\) the operator \(k_{i,l}=E_{i,l}(k_i)\) lies in \([0,I]\) and satisfies \(\| k_i-k_{i,l}\| \le Ce^{-cl^\alpha }\). Each \(k_i\) acts on the connected component of \(a_i\).

Use Theorem 28.2 with \(p=1\), so \(\alpha =1/2\), and fix \(C_0{\gt}0\). There is \(C_1\) such that for every real \(n\ge 2\) and every set \(S_0\) with \(|S_0|\le C_0n^2\) (the source states this for nonempty \(S_0\); for \(S_0=\varnothing \) every constraint is retained), with \(r_0=\lceil C_1(\log n)^2\rceil \) and \(\epsilon _n=\min \{ n^{-1000},g/4\} \): the truncated terms are positive contractions, \(\| \widetilde H-H_F\| \le \epsilon _n\), the ground energy of \(\widetilde H\) lies in \([0,\epsilon _n]\), its ground vector \(\widetilde\Omega \) is unique, its gap is at least \(g/2\), and after a choice of phase \(\| \widetilde\Omega -\Omega \| \le 2(\epsilon _n/g)^{1/2}\) and \(\frac12\| |\widetilde\Omega \rangle \langle \widetilde\Omega |-P_\Omega \| _1 \le (2\epsilon _n/g)^{1/2}\).

Let \(V\) be a finite set of physical sites with local dimension \(q{\gt}0\), and let \(\Omega \) be a unit vector satisfying \(H\Omega =E\Omega \). Fix \(X\subseteq V\). In a spectral basis of the reduced density on \(X\), select a set \(F\) of positive total mass \(z\), and write \(d=|F|\). Let \(\psi \in \mathcal H_{X^c}\otimes \mathbb C^d\) be the normalized selected Schmidt vector, with reduced density \(\rho \) on \(\mathbb C^d\). For each \(k\), the symmetric group \(S_k\) permutes the \(k\) copies of \(\mathbb C^d\); for an irreducible representation \(\lambda \) of \(S_k\), \(P_\lambda \) denotes the central projection of \((\mathbb C^d)^{\otimes k}\) onto its \(\lambda \)-isotypic component and \(\dim \lambda \) the dimension of \(\lambda \) [ 247 , Section 6 ] . Following the prevector and high-label comparators of [ 247 , Section 8 and Section 9, lines 50–59 ] , there is one sequence of irreducible representations \(\lambda _k\) of \(S_k\) such that the following assertions hold. Put

Here all factors are identified with the configurations of the original sites and two additional auxiliary sites. For every \(k{\gt}0\),

The symmetry projection permutes all physical and auxiliary copies simultaneously; the two other projections act on the complete auxiliary copy registers. Eventually in \(k\),

In the setting of Theorem 28.1.3, suppose also that \(H-EI\geq \gamma (I-|\Omega \rangle \langle \Omega |)\) for some \(\gamma {\gt}0\). There is one label sequence \(\lambda _k\) satisfying all the conclusions of that theorem. Write \(S'=S(\rho )\) and set

Then \(2kS'-\varepsilon _k=2\log \dim \lambda _k\) for every \(k\), and \(\varepsilon _k/k\to 0\).

Fix \(K\geq 8\) and put \(a=4/K\), \(t=2/K\). There are constants \(c,C_f\geq 0\) before the copy number, symmetric coordinates, regional family and iterated weighted geometric mean are chosen. Let \(Z^*Z=I\) and \(ZZ^*=P_{\mathrm{sym}}\). For a finite family of physical regions \(Q_{jg},Y_{jg}\), with \(0\leq g{\lt}K\), assume

Include the near auxiliary site \(C\) in \(L_{jg}=Q_{jg}\cup \{ C\} \) and the far auxiliary site \(R\) in \(F_{jg}=(Q_{jg}\cup Y_{jg})^c\cup \{ R\} \), where the complement is taken in the physical sites. For the replica metric \(W\) at parameter \(t\), put

Here \(T\) is any iterated weighted geometric mean of the family, and \(\xi _k\) is the same initial vector as in the preceding theorem. Put \(\varphi _k=w_k/\| w_k\| \), \(v_k=Z\varphi _k\), and

The sequence \(R_k\) tends to zero and is independent of the chosen coordinates, regions and geometric mean. For every \(k{\gt}0\), \(w_k\ne 0\), \(\| v_k\| =1\), and \(\delta _k\geq 0\). For \(0{\lt}\tau \leq 1/2\),

This is the lower comparison for the physical initial vector of [ 247 , Section 8 ] . Choosing a scan point and identifying its regions and geometric mean remain separate assertions.

This combines [ 247 , Section 4 ] with the comparators of [ 247 , Section 8 ] . Under the hypotheses of Theorem 28.1.2, the same constant \(C_1\), truncated Hamiltonian \(\widetilde H\), ground energy \(e\) and unit ground vector \(\widetilde\Omega \) satisfy all of its support, perturbation, gap and distance conclusions. For every region \(X\) and every positive-mass selection of the actual reduced spectrum of \(\widetilde\Omega \), this same vector also supplies the sequence and all the conclusions of Theorem 28.1.3, with \(H=\widetilde H\) and \(E=e\).

Each truncated term \(\widetilde h_i\) acts on its designated support \(\widetilde X_i\). Every support in \(\mathcal C_X\) is a ball of radius exactly \(r_0\), \(|\mathcal C_X|\le 2\mu K_b(r_0+1)^2b_X\), \(\log d_i\le K_b(r_0+1)^2\log q\) for \(i\in \mathcal C_X\), and \(\mathcal B_X\le C(1+b_X)(r_0+1)^6\). If moreover \(r_0=\lceil C_1(\log n)^2\rceil \), \(n\ge 2\), \(D\ge 1\) and \(b_X\le C_2nD\), then \(\mathcal B_X\le C'nD(\log n)^{12}\). If \(S_0\) is empty every constraint is retained and no label crosses a cut contained in \(S_0\); constraints whose anchors cannot reach \(S_0\) are always retained exactly.

Let \(V\) be the vertex set of a finite graph, let \(I\) be a finite label set with anchors \(a_i\in V\), and let \(q,r_0\in \mathbb N\) and \(S_0\subseteq V\). Use the designated supports \(\widetilde X_i\) of Theorem 28.1.6. For every cut \(X\subseteq V\), let \(\mathcal C_X\) consist of the original labels whose designated support meets both \(X\) and \(V\setminus X\). Then the geometric budget is exactly the supported-family budget in [ 247 , Lemma 3.1 ] :

Here \(\log 0=0\) when \(q=0\). Also, \(\mathcal C_{V\setminus X}=\mathcal C_X\), \(\mathcal B_{V\setminus X}=\mathcal B_X\ge 1\), and \(\mathcal B_\varnothing =\mathcal B_V=1\). No containment \(X\subseteq S_0\) is required. Repeated designated supports and zero terms keep their original labels.

Graph balls of radius \(d\) in \(\Lambda \) have at most \(2(d+1)^2\) sites, at most \(\mu _R\) supports contain a given site (Theorem 27.117.6), at most \(\mu _R\) labels share an anchor, and the pairs \((u,v)\) of adjacent sites with \(u\in X\) and \(v\notin X\) number at most \(|\partial _\Lambda X|\). The sphere bound, the diameter and size bounds for supports and the site budget are Lemma 36.7.1.

Fix an integer \(p\ge 1\), put \(\alpha =p/(p+1)\), and fix \(R\ge 0\), \(J\ge 0\) and \(\Delta {\gt}0\). There are \(C\ge 0\) and \(c{\gt}0\), depending only on \(p,R,J,\Delta \), such that for every \(q\ge 1\), every finite domain, interaction, choice of anchors and gapped ground vector as above, the constraints \(k_i=|M_i|/c_*\) satisfy \(c_*{\gt}0\),

so \(H_F=\sum _ik_i\) satisfies \(H_F\ge g(I-P_\Omega )\) with \(g=\Delta /c_*\) and \(H_F\Omega =0\). For every integer \(l\ge 0\) the operator \(k_{i,l}=E_{i,l}(k_i)\) lies in \([0,I]\) and satisfies \(\| k_i-k_{i,l}\| \le Ce^{-cl^\alpha }\). Each \(k_i\) acts on the connected component of \(a_i\). This is Proposition 4.3 of [ 247 ] .

With \(p,R,J,\Delta \) as in Theorem 28.2.2 there are \(C\ge 0\) and \(c{\gt}0\), depending only on \(p,R,J,\Delta \), such that the roots \(G_i\), \(K_i\), \(G_{i,l}\), \(K_{i,l}\) and the channels \(\mathcal E_i\), \(\mathcal E_{i,l}\) of Lemma 28.3, built from the constraints of Theorem 28.2.2, satisfy every conclusion of that lemma with \(\| G_i-G_{i,l}\| +\| K_i-K_{i,l}\| \le Ce^{-cl^\alpha }\) and \(\| (\mathcal E_i-\mathcal E_{i,l})\otimes \mathrm{id}\| \le Ce^{-cl^\alpha }\) for every auxiliary system. This is Lemma 4.4 of [ 247 ] .

Take \(p=1\) in Theorem 28.2.2, so \(\alpha =1/2\), and fix \(C_0\ge 0\). There is \(C_1{\gt}0\), depending only on \(R,J,\Delta ,C_0\), such that for every lattice instance as above, every real \(n\ge 2\) and every set \(S_0\) of sites with \(|S_0|\le C_0n^2\), the truncation with \(r_0=\lceil C_1(\log n)^2\rceil \), \(g=\Delta /c_*\) and \(\epsilon _n=\min \{ n^{-1000},g/4\} \) has all of the following properties at once: each truncated term is a positive contraction acting on its designated support, \(\| \widetilde H-H_F\| \le \epsilon _n\), and \(\widetilde H\) has a unit ground vector \(\widetilde\Omega \) with ground energy \(e_0\in [0,\epsilon _n]\) and gap at least \(g/2\), the eigenspace of \(\widetilde H\) at \(e_0\) is spanned by \(\widetilde\Omega \), and after a choice of phase \(\| \widetilde\Omega -\Omega \| \le 2(\epsilon _n/g)^{1/2}\) and \(\frac12\| |\widetilde\Omega \rangle \langle \widetilde\Omega |-P_\Omega \| _1 \le (2\epsilon _n/g)^{1/2}\). This is the truncation part of Proposition 4.5 of [ 247 ] , which assumes \(S_0\) nonempty and \(C_0{\gt}0\).

For \(q\ge 1\) there is \(C\ge 0\), depending only on \(q\) and \(R\), such that for every finite domain, every choice of anchors, every \(S_0\), every \(r_0\) and every \(X\subseteq S_0\), with \(b_X=|\partial _\Lambda X|\), every designated support meeting both \(X\) and its complement is the ball of radius \(r_0\) about its anchor and

For \(C_1,C_2\ge 0\) there is \(C\), depending only on \(q,R,C_1,C_2\), such that \(r_0=\lceil C_1(\log n)^2\rceil \), \(n\ge 2\), \(D\ge 1\) and \(b_X\le C_2nD\) imply \(\mathcal B_X\le CnD(\log n)^{12}\). This is the crossing count and cut budget of Proposition 4.5 of [ 247 ] .

A blank or already assigned fill slot makes no physical move but increases its receiving side’s consumed count by one. Assignments never change sides, and the middle only decreases. For a selected set \(B\), the removed middle is exactly \(B\cap Y\). If all rows have at most \(n\) sites, a site remaining in the middle after \(k\) fills has oriented depth at least the corresponding nominal front.

For \(m,M{\gt}0\), the history weights are nonnegative and sum to one. Extending a history multiplies its weight by \((2M)^{-K}\). Each band’s offset has marginal probability \(1/m\), even after subsequent charges; each side–slot pair at the next charge has conditional probability \(1/(2M)\). Charges leave the nominal fronts unchanged. At every old charge history, each unassigned site lies on or beyond both oriented nominal fronts, provided its row has at most \(n\) sites.

For positive \(d\), the ambient depth row is \(T_d\setminus T_{d-1}\); its physical part has no larger cardinality. Graph distance bounds depth variation: \(|d(x)-d(y)|\le d_\Lambda (x,y)\). Let \(Z\) be the actual endpoints of edges crossing \(A\). If \(\operatorname {dist}_\infty (T,Z){\gt}2L+10r_0\), every graph \(r_0\)-ball meeting \(A\cap T_L\) is wholly contained in \(A\).

Suppose \(n,m\ge 2\), \(D\ge 1\), \(r_0\le D\), \(4D\le m\), \(8Km\le L\), and at most \(nm\) fills are consumed. Assume \(|T_d\setminus T_{d-1}|\le n\) for \(1\le d\le L\), and at most \(\mu \) labels have any fixed anchor. Each charge interval \([j-r_0,j+D]\) visits only unsigned depths in \([1,L]\). Its number of labels is at most \((D+r_0+1)n\mu \le 3\mu nD\). Thus \(C_1\ge 3\mu \) permits padding every list to \(M=\lceil C_1nD\rceil \), with a unique slot for every label, including labels with equal anchors. If \(C_1{\gt}0\), then \(M\le (C_1+1)nD\) and \(1/(2M)\ge [2(C_1+1)]^{-1}/(nD)\).

Retain the capacity and clearance assumptions, and suppose \(2r_0\le D\). An old charge history is good when every assigned site of \(A\) has oriented depth at most \(j+D/2\) on its receiving side. For every labelled graph \(r_0\)-ball meeting that side and the actual middle, the anchor lies in \([j-r_0,j+D]\). Its labelled slot is unique. Choosing this side and slot has probability exactly \(1/(2M)\), at least \(c/(nD)\) for \(c=1/[2(C_1+1)]{\gt}0\), and the extended physical history removes exactly the ball’s old \(Y\)-part from \(Y\).

Set \(S_0=A\cap T_L\) and retain the good-history sampling assumptions. Every truncated designated support meeting a receiving side and the old middle is a graph \(r_0\)-ball. Its labelled side–slot event has probability exactly \(1/(2M)\ge c/(nD)\), and the actual extended history removes exactly the designated support’s old unassigned part.

Under the good-history sampling assumptions, let \(F_{g,s}(B)\ge 0\) be a nonnegative cost of a finite set of physical sites moved toward side \(s\) in band \(g\). Sum \(F_{g,s}(B_i\cap Y_g)\) over every splitting incidence \((g,s,i)\), where \(s\) is the receiving side and \(B_i\) is the labelled charge ball. If \(\Delta _g(c)\) is the actual set removed from \(Y_g\) by the simultaneous charge choice \(c\) and \(s_g(c)\) its selected side, then

For \(C_1{\gt}0\), the left coefficient may be replaced by \(1/[2(C_1+1)nD]\). Labels sharing an anchor retain separate summands.

The expected-cost lower bound also holds with the sum restricted to splitting incidences of the truncated designated supports, using their actual old middle parts. The cost may depend on the band and receiving side, and repeated labels remain distinct.

For every finite actual history and every assigned site of \(A\), its oriented depth is at most \(j+D+r_0\), where \(j\) is its side’s current nominal front. This holds both for completed histories and immediately before a charge. Assume the ambient row bound through \(L\), \(8Km\le L\), and the stated clearance of \(T\) from the physical cut endpoints. If a radius-\(r_0\) ball meets a receiving side and the middle, its anchor’s oriented depth \(d_a\) satisfies

No good-history assumption is imposed. The exterior \(\Lambda \setminus A\) is not subject to the assigned-site lead bound.

At fixed band, side and front time, at most \(a+b+1\) offsets place a fixed depth in \([j-a,j+b]\). Every history event contained in that interval event therefore has probability at most \((a+b+1)/m\), regardless of its dependence on later choices. If \(a+b\le m\), the centers’ spacing \(8m\) allows at most one band on each side to reach the fixed depth, even across different offsets. The union over both sides and all bands has probability at most \(2(a+b+1)/m\). Common convex mixtures preserve the bound.

Suppose \(n,m,D{\gt}0\), \(C_1{\gt}0\), \(r_0\le D\), \(4D\le m\), \(8Km\le L\), and the actual ambient row and cut-clearance hypotheses hold. For every label, the total classical weight of designated receiving–middle incidences in any band is at most

This bound holds for the old and new distributions of every fill or charge round, and for every common mixture parameter \(p\in [0,1]\).

Every initial side assignment survives all subsequent fills and charges. For bands \(g{\lt}g'\), and for arbitrary independent histories and times, \(P_{0,g}\cup Y_g\subseteq P_{0,g'}\). The inclusion holds both after a completed fill–charge pair and immediately before a charge, including comparisons between these two kinds of status.

Suppose \(n{\gt}0\), \(1\le D\), \(r_0\le D\), \(4D\le m\), and at most \(nm\) fills have occurred. At the offsets of one history, any physical depth belongs to at most one of the band–side intervals \([j-r_0,j+D+2r_0]\), interpreted with the side’s orientation. In comparing opposite sides of the same band, the offsets agree.

Assume the ambient row bound, physical cut clearance, \(8Km\le L\), \(n{\gt}0\), \(1\le D\), \(r_0\le D\), and \(4D\le m\). Every designated receiving–middle incidence is a terminal split with a unique band and side. For a completed history of \(k\) pairs require \(k\le nm\); for its pre-charge status require \(k+1\le nm\).

Suppose \(n,m\ge 2\), \(1\le D\), \(r_0\le D\), \(4(D+2r_0)\le m\), \(8Km\le L\), and the ambient row and cut-clearance bounds hold. If at most \(\mu \) interaction labels have any fixed anchor, then

This holds for completed histories with \(k\le nm\) and for their pre-charge statuses with \(k+1\le nm\). Distinct labels with the same anchor are counted separately.

The augmented regions partition all physical and auxiliary factors. Every charge transfers a subset of the middle, and its new partition equals the actual partition of the extended physical history. If \(P_0\cup Y_0\subseteq P'_0\), then \(P\cup Y\subseteq P'\).

Suppose every factor has positive dimension, \(t\ge 0\), and \((P,Y,F)\) and \((P',Y',F')\) are partitions satisfying \(P\cup Y\subseteq P'\). For every replica count, their band metrics

commute on the symmetric replica space. Their extensions by the identity on its orthogonal complement commute globally. Consequently, actual scanner bands satisfy cross-band commutation for arbitrary independent histories, times, and completed or pre-charge statuses, including every pair of terminal leaves of a charge round.

Each charge leaf has weight \((2M)^{-K}\) and each history after \(k\) charges has weight \(m^{-K}(2M)^{-Kk}\). Interpolating a charge at parameter \(p\) gives terminal weights \((1-p)w_h\) and \(pw_hq_c\). For positive definite matrix inputs, the old endpoint is the previous history tree evaluated at the old inputs; the new endpoint is exactly the recursively substituted next tree evaluated at its new inputs.

Let \(\Lambda \subset \mathbb Z^2\) be finite and let \(B,C\subseteq \Lambda \). An unordered pair belongs to \(\partial _\Lambda B\) exactly when its sites are adjacent in the induced graph and exactly one belongs to \(B\). Complements are relative to \(\Lambda \), and

Consequently the boundary cardinalities of \(B\cup C\) and \(B\setminus C\) are at most \(|\partial _\Lambda B|+|\partial _\Lambda C|\). If \(R\subseteq \Lambda \) contains at least one site of each edge of \(\partial _\Lambda B\), then \(|\partial _\Lambda B|\le 4|R|\). In particular,

The last estimate is the changed-site bound in the proof of [ 247 , Lemma 9.1(2) ] .

Let \(\Lambda ,T\subset \mathbb Z^2\) be finite, let \(A\subseteq \Lambda \), and let \(Z\) consist of both sites of every edge of \(\partial _\Lambda A\). Fix integers \(L\ge 1\) and \(n,r_0\ge 0\), and suppose \(\| t-z\| _\infty {\gt}2L+10r_0\) for every \(t\in T\) and \(z\in Z\). Every ambient dilation \(T_j\) with \(0\le j\le L\) is disjoint from \(Z\). Suppose further that \(T\) is nonempty and that \(|T_d\setminus T_{d-1}|\le n\) for each integer \(1\le d\le L\). Set \(\delta (x)=\min _{t\in T}\| x-t\| _\infty \). For every integer \(j\le L\), define \(P_j=\{ x\in A:\delta (x)\le j\} \) and \(Q_j=\{ x\in A:0{\lt}\delta (x)\le j\} \). Then

The set identity holds for every integer-valued function \(\delta \). These are the full-row edge estimates used in [ 247 , Lemma 9.1(2) ] .

Let \(\Lambda ,T\subset \mathbb Z^2\) be finite, with \(T\) nonempty, and let \(A\subseteq \Lambda \). Give the collar scan the ambient depth \(d(x)=\min _{t\in T}\| x-t\| _\infty \), and put \(X=A\cap T\). For any band and any finite history, let \((P_0,Y,V)\) be its physical partition either after the completed fill–charge pairs or after the next prescribed fill and before its charge. With \(U=P_0\cap \{ d{\gt}0\} \) and \(UY=(P_0\cup Y)\cap \{ d{\gt}0\} \), one has

These are the moving-region identities of [ 247 , Section 9 ] .

Let \(h\) be a finite collar history and let \(g\) be one of its bands. Consider either the physical partition after \(h\), or the partition after its next prescribed fill and before the next charge. Write \((P_0,Y_0,V_0)\) for its physical near, middle and far regions. Set \(X=A\cap T\) and \(U=P_0\cap \{ x\in \Lambda :d(x){\gt}0\} \). Let \(\iota \) include the physical sites into the domain obtained by adjoining two distinct auxiliary sites \(C\) and \(R\). The augmented partition is

The construction is that of [ 247 , Section 9, moving partitions ] .

Let \(T\subset \mathbb Z^2\) be finite and nonempty, and let \(A\) be a finite physical color. For every integer \(j\), the positive-depth prefix satisfies \(Q_j=A\cap (T_{\max \{ j,0\} }\setminus T)\). These are the collars of [ 247 , Section 9 ] .

Suppose \(n{\gt}0\) and \(8Km\le L\). Fix \(0\le g{\lt}K\), \(0\le r{\lt}m\) and \(0\le k\le nm\). Let \(j_N,j_F\) be the near and oriented far nominal fronts after \(k\) fills, and put \(v=8gm+5m+r\). Then

Let \(S_0=\{ x\in A:d(x)\le L\} \) be the original compact truncation set. For every integer threshold \(j\le L\), the positive-depth prefix satisfies \(Q_j\subseteq S_0\).

Under the preceding scale conditions, actual row padding bounds and graph-ball depth variation give

These inclusions hold for completed and pre-charge histories within their respective horizons, without a good-history hypothesis.

Let \(T\) be nonempty, use its actual ambient depth and the induced physical lattice graph, and assume each ambient layer \(T_d\setminus T_{d-1}\) has at most \(n\) sites for \(1\le d\le L\). Under the preceding scale and horizon conditions, actual statuses satisfy

In addition to the physical prefix-approximation hypotheses, suppose every endpoint of an actual cut edge of \(A\) has ambient distance greater than \(2L+10r_0\) from \(T\). Count each undirected physical edge once. Every completed status with \(k\le nm\) and every pre-charge status with \(k+1\le nm\) obeys

Let \((P,Y,F)\) partition all tensor factors, let \(x\subseteq Y\), and let \(\theta \) be a pure vector. With regional entropies of its reduced pure-state matrices, a move to \(P\) has entropy

A move to \(F\) has the same formula with \(P\) and \(F\) exchanged. Both empty moves have zero entropy. These balanced identities and inequalities require no normalization of \(\theta \).

Every trial cost is nonnegative. If \(B\subseteq Y_{h,g}\), its moved set is exactly \(B\). Write \(\Delta _g(c)\) for the actual removed middle set and \(s_g(c)\) for the selected side of a simultaneous charge choice. Then

The old partition and the vector \(\theta \) are the same for every choice.

Under the physical good-history sampling hypotheses of Theorem 28.3.10, let \(D_i\) be the actual truncated designated supports and let \(c=1/[2(C_1+1)]{\gt}0\). For every pure vector \(\theta \) on the physical and auxiliary factors,

Distinct interaction labels remain distinct summands even when their anchors coincide. The classical charge weights are \(q_{c'}=(1/(2M))^K\); they sum to one when \(M{\gt}0\). The inequality is pointwise before coherent-state and Fourier integration.

If \(m,M{\gt}0\), every history and conditional-choice leaf has positive weight. All old regions partition the physical and auxiliary factors, and every actual move transfers a subset of the old middle.

Consider an actual completed status after \(q\) fills, or the status after its next fill and before the corresponding charge. The physical near side remains in \(A\cap T_L\) whenever \(8Km\le L\). Suppose depth is ambient sup-norm distance to the nonempty target, supports use the induced graph of the finite domain, and the target has clearance greater than \(2L+10r_0\) from the cut endpoints. If the number of consumed fills is at most \(nm\), \(n{\gt}0\), \(r_0\le D\), \(D\ge 1\), and \(4D\le m\), no designated support meets both physical receiving sides. No depth-row cardinality or good-history condition is needed.

Under the hypotheses of Theorem 28.3.47 and the ambient row bound \(|T_d\setminus T_{d-1}|\le n\) for \(1\le d\le L\), every designated support in every completed or pre-charge status has one of the following forms at each band: it is contained in one part, or it meets the middle and exactly one receiving side and is contained in one part of every other band.

Under the hypotheses of Theorem 28.3.48, adjoin \(C\) to the near side and \(R\) to the far side, leaving every designated support on physical tensor factors only. At every old and new terminal leaf of an actual charge round, each designated support lies in one part of every band except possibly one; in that band it meets the middle and exactly one receiving side. This holds for any supplied finite history and conditional-choice mean trees.

Suppose the physically embedded designated supports are compatible with the terminal partitions. At each old history and each pure vector, the sum of the receiving-side incidence costs equals the sum of the terminal split entropies over the interaction labels:

Distinct labels are retained even when their supports coincide.

Continuous symbols are integrable. Finite continuous sums and scalar multiplication pass through the same old-state integral. For admissible transport data with positive factor dimensions, nonnegative metric parameter and commuting band metrics, \(f\le g\) implies \(J_h(f)\le J_h(g)\); nonnegative continuous symbols have nonnegative integrals. The conditional entropy symbol \(G_h\) is continuous and nonnegative, and the entropy gain is

If the common prevector is nonzero and symmetric and \(0{\lt}p{\lt}1\), then \(J_h(1)=1/2\). The entropy-gain formula uses the history weight \(w_h\), without the old terminal edge factor \(1-p\).

For admissible transport data with positive factor dimensions, nonnegative metric parameter and cross-band commutation, fix a continuous real symbol \(F\), a history \(h\), a replica count \(k\ge 0\) and a supplied prevector. The function \(p\mapsto J_h(F)\) is almost everywhere measurable and integrable on both \((0,1)\) and \([0,1]\). Writing \(Q_F=(A_F+A_F^\dagger )/2\) for the Hermitian part of the coherent average \(A_F\), one has \(|J_h(F)|\le \| Q_F\| /2\) in the interior. Fixed finite weighted sums selected by a fixed history predicate retain this regularity, including the actual good-history charge defect. No symmetry or nonzero-prevector hypothesis is needed for these assertions.

Fix a collar scan with positive offset, charge-slot and auxiliary counts \(n,m,M\), positive finite factor dimensions, energy terms, a replica count \(k\ge 0\), a supplied prevector, and \(a\ge 0\). For \(0\le s{\lt}nm\), let \(\mathcal Q_s(p)\) be its actual good-history charge defect with metric parameter \(a/2\). Suppose \(0\le p_0{\lt}p_1\le 1\), \(\rho {\gt}0\), and \(B\in \mathbb R\) satisfy

Then some \(0\le s{\lt}nm\) and \(p\in [p_0,p_1]\) satisfy \(\mathcal Q_s(p)\le B/(\rho nm(p_1-p_0))\). This is the finite averaging step in the scanner argument of [ 247 ] , applied to the actual charge defects.

Assume the physical sampling hypotheses of Theorem 28.3.44, positive offset and charge-slot counts, positive physical and auxiliary factor dimensions, and a nonnegative metric parameter. Give each interaction its actual physically embedded designated support. Put

Then, for every good old history and every coherent vector,

and for every common prevector and interpolation parameter,

These inequalities establish entropy sampling for one actual charge round. They do not construct the common prevector or the remaining comparator and scanner estimates.

With positive offset and charge capacities, the actual fill tree has positive history weights and conditional probability one. Every move lies in the old middle, and its application gives exactly the scheduled physical fill. An already assigned singleton transfers the empty set. For positive local dimensions and \(t\geq 0\), terminal band metrics from distinct bands commute on the symmetric replica subspace, across independent histories and either old or new status.

Assume positive offset and charge capacities, positive local dimensions, and \(t\geq 0\). The ordered product of band metrics at every completed or pre-charge history is positive definite. The fill root at parameter zero is the existing history tree evaluated at its completed metrics; at parameter one it is the same tree evaluated at its pre-charge metrics. The latter is exactly the following charge root at parameter zero. The charge root at parameter one equals the next fill root at parameter zero, with the recursively substituted history tree.

Under the geometric hypotheses of Theorem 28.3.48, with \(k+1\leq nm\), the physical designated supports satisfy the transport compatibility condition at both terminal statuses of a deterministic fill. The history tree may be any supplied finite tree.

In an arbitrary finite induced lattice domain, a graph ball of radius \(r_0\) has at most \((2r_0+1)^2\) sites. Each actual fill transfers at most one physical site, and each actual charge transfers at most \((2r_0+1)^2\) physical sites. If \(8Km\le L\), a designated support meeting both the middle and a receiving side of an actual completed or pre-charge status has the same bound.

Assume \(8Km\le L\). Every designated support not contained in one physical part of an actual completed or pre-charge status equals its anchor’s radius-\(r_0\) graph ball. No row bound, clearance, goodness, or restriction on the number of consumed fills is required.

A physical region \(B\) has dimension \(q^{|B|}\) and \(\log (e\dim B)=1+|B|\log q\). Thus \(\ell _q(r_0)\ge 1\) bounds the logarithmic dimension of every actual charge transfer and, if \(8Km\le L\), every designated support not contained in one actual physical part. A fill has the sharper bound \(1+\log q\). Embedding a physical region among additional factors of arbitrary dimensions does not change its dimension.

Fix \(q\ge 1\) and \(C_r\ge 0\). For \(r_0=\lceil C_r\log ^2 n\rceil \) and \(\log n\ge 1\), \(\ell _q(r_0)\le [1+(2C_r+3)^2\log q]\log ^4 n\). Consequently a positive coefficient and a threshold for this bound may be chosen before the finite domain, labels, scanner, or history.

Fix \(q\ge 1\), \(C_r\ge 0\), and real numbers \(c_{\mathrm{ent}},e_{\mathrm{ent}},c_{\mathrm{en}},e_{\mathrm{en}}\). There are \(C\ge 1\) and \(C_\ell \ge 0\) such that, for every \(n\) with \(\log n\ge 1\) and every \(1\le \ell \le \ell _q(\lceil C_r\log ^2 n\rceil )\),

If \(q\ge 1\) and \(r\ge 1\), then \(\ell _q(r)\le (1+9\log q)r^2\). Consequently, for \(a\ge 0\) and any real \(c_0\), \(ar^2\le c_0/(1+9\log q)\) implies \(a\ell _q(r)\le c_0\). If \(c_0{\gt}0\), the radius-smallness threshold \(c_0/(1+9\log q)\) is positive and depends only on \(q\) and \(c_0\).

Give every physical factor dimension \(q\ge 1\), with arbitrary sizes for \(C\) and \(R\). In every actual fill or charge round, each moved subsystem has logarithmic dimension at most \(\ell _q(r_0)\). If \(8Km\le L\), the same bound holds for a physical designated support whose split-leaf set is nonempty. No bound is imposed on a support contained in one part at every terminal leaf.

Let \(H\) be an arbitrary matrix on the physical configuration space. Write \(\overline H_k=k^{-1}\sum _{j=0}^{k-1}H_j\) and define \(\overline H_0=0\). Let \(\widetilde H=H\otimes I_C\otimes I_R\) be expressed on \(V\sqcup \{ C,R\} \). Then

The bijection \(e\) preserves every physical coordinate and sends the first and second auxiliary coordinates to \(C\) and \(R\), respectively. This is the identity extension used in [ 247 , Proposition 7.4 and Section 8 ] .

Let \(V\) be a finite set of physical sites, of common dimension \(q\), and let both auxiliary factors have dimension \(d\). Let \(H\) be an arbitrary physical matrix. In the successively augmented coordinates, let \(\mathsf H_k\) be the physical replica mean with both auxiliary identities, and in the coordinates \(V\sqcup \{ C,R\} \) let \(\overline{\widetilde H}_k\) be the replica mean of \(\widetilde H=H\otimes I_C\otimes I_R\). For the bijection \(e_k\) of Definition 28.3.69,

Both means are defined to be zero at \(k=0\). The identity includes zero dimensions and empty physical site sets. This is the original energy expectation used in [ 247 , Section 8, defect-mass comparison ] .

Use the spaces, physical matrix and coordinate bijection of Theorem 28.3.71. For every vector \(w\) in the scanner configuration space, put \(w_O=w\circ e_k\). Then

The equality is complex-valued. It requires neither normalization nor a nonzero vector, and also holds at zero replicas.

Suppose \(q\ge 1\), the scanner graph is the induced domain graph, and each anchor lies in its labelled interaction support. The augmented terms are positive contractions supported on their designated physical sets. Their sum is \((\sum _i\widetilde h_i)\otimes I_C\otimes I_R\), and their replica energy is the copy mean of this sum. The statement includes components disjoint from \(S_0\) and the convention \(S_0=\varnothing \), which retains every constraint.

Let \(q\ge 1\), let \(H=\sum _i h_i\) be a local Hamiltonian of range \(R\) and term norm bound \(J\in \mathbb R\), and take \(\| \Omega \| =1\) and \(\Delta {\gt}0\). Suppose the scanner graph is the induced domain graph and each anchor lies in its labelled interaction support. For any truncation depth \(L\in \mathbb N\), let \(\widetilde h_i\) be the physical terms of Definition 28.3.68, centered using this same \(\Omega \). Put

Assume a real \(e\) and a unit vector \(\widetilde\Omega \) satisfy

For every cut \(X\subseteq \Lambda \), put \(\rho _X=\operatorname {Tr}_{\Lambda \setminus X} |\widetilde\Omega \rangle \langle \widetilde\Omega |\) and \(s_X=S_{\widetilde\Omega }(X)\). Let \(K_X\) take the value \(-\log p_j\) with probability \(p_j\) for the positive eigenvalues \(p_j\) of \(\rho _X\), and let \(\mathcal B_X\) be the exact budget of the designated supports for \(S_0=\{ x\in A:d(x)\le L\} \). For every real \(u\) with \(|u|\le 1/(32\sqrt{(1+\vartheta )\mathcal B_X})\),

For every real \(w\),

The same \(e\) and \(\widetilde\Omega \) are used for every cut.

Fix \(R\in \mathbb N\), \(J\ge 0\), \(\Delta {\gt}0\) and \(C_0\ge 0\). There is \(C_{\mathrm{tr}}{\gt}0\), depending only on \(R,J,\Delta ,C_0\), such that the following holds for every \(q\ge 1\), finite induced lattice domain \(\Lambda \), and local Hamiltonian \(H=\sum _i h_i\) of range \(R\) and term norm bound \(J\). Let \(\Omega \) be a unit vector with \(H\Omega =E_0\Omega \) and \(H-E_0I\ge \Delta (I-|\Omega \rangle \langle \Omega |)\). Take any scan at integer scale \(n\ge 2\), with the induced domain graph and every anchor in its labelled interaction support. For any \(L\in \mathbb N\), suppose

Form the physical truncated terms of Definition 28.3.68, centered at \(\Omega \), and put \(\widetilde H=\sum _i\widetilde h_i\) and \(g=\Delta /c_*\) and \(\epsilon _n=\min \{ n^{-1000},g/4\} \). There exist a real \(e\) and a unit vector \(\widetilde\Omega \) satisfying (75), with \(0\le e\le \epsilon _n\). For some real \(\theta \), the same vector satisfies

For this same vector, both marginal inequalities of Theorem 28.3.74 hold for every cut \(X\subseteq \Lambda \), with \(\vartheta =2/g\), the literal designated support budget \(\mathcal B_X\), every real \(w\), and every real \(u\) with \(|u|\le 1/(32\sqrt{(1+2/g)\mathcal B_X})\). The constant \(C_{\mathrm{tr}}\) is chosen independently of the scan’s charge-slot constant \(C_1\), and before \(q\), \(\Lambda \), \(H\), \(\Omega \) and the scan. Empty domains and empty \(S_0\) are permitted.

Fix \(q\ge 1\), \(R\in \mathbb N\), \(J\ge 0\), \(\Delta {\gt}0\) and \(C_T\ge 0\). There exist \(C_{\mathrm{tr}}{\gt}0\), \(C_B{\gt}0\) and \(N\ge 2\), chosen before the domain, Hamiltonian, ground vector, cut, scan and source exponents. Consider any instance of Theorem 28.3.75 with \(n\ge N\), except that the compact-volume bound is replaced by the following source geometry. Let \(T\) be a nonempty ambient target, \(d(x)=\operatorname {dist}_\infty (x,T)\), \(0{\lt}\ell _T{\lt}1\), \(L=\lfloor n^{1-\ell _T}\rfloor \), and \(r_0=\lceil C_{\mathrm{tr}}(\log n)^2\rceil \). Suppose \(|T|\le C_Tn^2\), each positive ambient dilation layer through depth \(L\) has at most \(n\) sites, and \(\operatorname {dist}_\infty (T,Z){\gt}2L+10r_0\), where \(Z\) is the set of endpoints of the actual cut edges. Set \(X=A\cap T\) and \(S_0=A\cap T_L\). Then

One pair \((e,\widetilde\Omega )\) satisfies every energy, phase-distance, trace-distance, moment and all-cut tail conclusion of Theorem 28.3.75. For the target marginal of this same vector, write \(p_i\) for its eigenvalues, \(s_X=S_{\widetilde\Omega }(X)\), \(w=n^{3/5}\) and \(\delta =n^{-100}\). Its literal labelled cut budget and canonical typical set satisfy

In particular, \(E\) is nonempty. Every \(i\in E\) obeys \(\mathrm e^{-s_X-w}\le p_i\le \mathrm e^{-s_X+w}\).

The following assertion holds separately for an actual fill round and for an actual charge round. In each case there exist universal real constants \(c_0,C_{\mathrm{ent}},e_{\mathrm{ent}},C_{\mathrm{en}},e_{\mathrm{en}}\), with \(c_0{\gt}0\), chosen before all the data below.

Let \(\Lambda ,T\subset \mathbb Z^2\) be finite with \(T\ne \varnothing \), let \(q,R_{\mathrm{int}}\in \mathbb N\) with \(q\ge 1\), and let \(J\in \mathbb R\). Take a local Hamiltonian \(H=\sum _i h_i\) of range \(R_{\mathrm{int}}\) and term norm bound \(J\) on \((\mathbb C^q)^{\otimes \Lambda }\), a vector \(\Omega \) with \(\| \Omega \| =1\), and \(\Delta {\gt}0\). The labels \(i\) are the admissible interaction supports, with a fixed linear order. Let the scanner have color \(A\subseteq \Lambda \), parameters \(n,m,K,D,r_0\in \mathbb N\) and \(C_1\in \mathbb R\), the induced nearest-neighbor graph on \(\Lambda \), depth \(d(x)=\min _{t\in T}\| x-t\| _\infty \), and anchors \(x_i\in i\). Fix a stage \(s\in \mathbb N\) and a truncation depth \(L\in \mathbb N\) satisfying

Write \(T_j=T+[-j,j]^2_{\mathbb Z}\) and let \(Z\) consist of both endpoints of every edge crossing \(A\) in \(\Lambda \). For every integer \(1\le j\le L\) assume \(|T_j\setminus T_{j-1}|\le n\), and for every \(t\in T\) and \(z\in Z\) assume \(2L+10r_0{\lt}\| t-z\| _\infty \). Choose arbitrary independent integers \(d_C,d_R\ge 1\) for the auxiliary factors and put

Let \(a{\gt}0\) satisfy \(a\ell \le c_0\).

Use the actual recursive history tree at stage \(s\). For the fill assertion use its scheduled moves and singleton conditional choices; for the charge assertion use its actual pre-charge partitions, selected-side moves, and uniform side–slot choice trees. In both cases set \(t=a/2\) in the band metrics. Let \(\widetilde h_i\) and \(\widetilde X_i\) be exactly the truncated terms and designated supports of Definition 28.3.68 for \(H,\Omega ,\Delta ,L\). The energy terms and their replica sum are

Here the subscript \(b\) means insertion in copy \(b\), and \(\overline H_0=0\).

There exist sequences \(\beta ,\varepsilon :\mathbb N\to \mathbb R\), depending only on the fixed data above, such that \(\beta _r=O(\log (r+1))\) and \(\varepsilon _r\to 0\) as \(r\to \infty \), with the following properties for every \(r\in \mathbb N\) and every supplied nonzero symmetric vector \(\mathrm{pre}\in \operatorname {Sym}^r(\mathcal V)\). Write \(\mathcal M_r(p)\) for the actual interpolating root and set

Here \(N_{r,\mathrm{pre}}(p)\) is the nonnegative square root. Use the history weights \(w_h\), conditional weights \(q_{c\mid h}\), terminal weights \(\pi _j(p)\), transported states \(\sigma _{j,u}(p)\), coherent measures \(\mu _\sigma \), and Fourier weight \(m_{1/4}\) of [ 247 , Proposition 7.4 ] for this actual round and the vector in (91). Let \(C_h=A_h^{-1/2}B_hA_h^{-1/2}\), where \(A_h\) is the old metric and \(B_h\) the conditional choice-tree root. Let \(\mathcal J_i\) be the terminal leaves splitting \(D_i\), and set

Here \(\eta _{h,c,g}\) is the entropy of the actual move and \(\eta _{i,j}\) is the entropy of the designated support’s unique split. Then the following four conclusions hold.

1. For every \(0{\lt}p{\lt}1\), the function \(-\log N_{r,\mathrm{pre}}(p)^2\) is differentiable at \(p\), with

   \begin{align} -\frac{d}{dp}\log N_{r,\mathrm{pre}}(p)^2 & =\mathcal L_{r,\mathrm{pre}}(p). \end{align}

2. For every \(0{\lt}p{\lt}1\),

   \begin{align} ra\mathcal G_{r,\mathrm{pre}}(p) -C_{\mathrm{ent}}raK a^{1/4}\ell ^{e_{\mathrm{ent}}} -\beta _r & \le \mathcal L_{r,\mathrm{pre}}(p). \end{align}

3. For every \(0{\lt}p{\lt}1\) and \(E_0\in \mathbb R\) satisfying \(\overline H_r\mathrm{pre}=E_0\mathrm{pre}\),

   \begin{align} \operatorname {Re}\langle v_{r,\mathrm{pre}}(p), \overline H_r v_{r,\mathrm{pre}}(p)\rangle & \le 2E_0+C_{\mathrm{en}}a^2\ell ^{e_{\mathrm{en}}} \mathcal E_{r,\mathrm{pre}}(p)+\varepsilon _r. \end{align}

4. For every \(0\le p_0\le p_1\le 1\),

   \begin{align} \int _{p_0}^{p_1}\left( ra\mathcal G_{r,\mathrm{pre}}(p) -C_{\mathrm{ent}}raK a^{1/4}\ell ^{e_{\mathrm{ent}}} -\beta _r\right)\, dp & \le \log N_{r,\mathrm{pre}}(p_0)^2 -\log N_{r,\mathrm{pre}}(p_1)^2. \end{align}

The sequences are chosen before \(r,\mathrm{pre},p,p_0,p_1,E_0\). The eigenvector equation is required only in the third conclusion.

This is a one-round consequence of the sampling estimate and [ 247 , Proposition 7.4 ] . Under the joint geometric and Hamiltonian hypotheses of Theorems 28.3.55 and 28.3.77, consider one actual charge round and its literal good-history defect \(\mathcal Q(p)\). There are universal real constants \(c_0,C_{\mathrm{ent}},e_{\mathrm{ent}}\), with \(c_0{\gt}0\), such that for \(a{\gt}0\) and \(a\ell \le c_0\) there is a sequence \(\beta _r=O(\log (r+1))\) satisfying the following assertion. The constants are chosen before the geometry and Hamiltonian; the error sequence is chosen before the replica count and supplied vector. For every \(r\ge 0\), every nonzero symmetric supplied vector, and every \(0\le p_0\le p_1\le 1\), put \(\alpha =1/(2(C_1+1)nD)\). Here \(N\) and \(\ell \) are as in Theorem 28.3.77. Then

The assertions for the actual fill and charge rounds need no further hypothesis. For general finite transport data, the identities below are asserted for admissible data with cross-band commutation; the actual partitions and moves satisfy both conditions. Each measure is finite for every fixed real \(p\), every replica count, and every supplied vector. If \(a\ge 0\), every continuous real function \(f\) on the ambient one-copy vector space satisfies

If \(a\ge 0\) and \(\psi _k\) is nonzero and symmetric, each interior leaf has mass \(1/2\). The old leaf has zero measure at \(p=1\), and the new leaf has zero measure at \(p=0\). Under \(a\ge 0\) and the same nonzero symmetric-vector hypothesis, the complementary old and new leaves at \(p=0\) and \(p=1\) have mass \(1/2\). These statements include \(k=0\) for a supplied nonzero scalar vector.

The assertions for the actual fill and charge rounds need no further hypothesis. For general finite transport data, the identities below are asserted for admissible data with cross-band commutation; the actual partitions and moves satisfy both conditions. For \(a\ge 0\) and every real \(p\), the choice gain is the transport entropy gain, with history factor \(w_h\), and the charge defect is the actual good-history old-state split-entropy defect. On \(0\le p\le 1\), the round’s energy sum is exactly

with \(\pi _{h,\mathrm{old}}=(1-p)w_h\) and \(\pi _{h,c,\mathrm{new}}=pw_hq_{c\mid h}\). For the actual augmented truncated terms, the mean energy is the expectation of their actual replica Hamiltonian in the same filtered vector.

Give every local factor a positive dimension \(n_v\). For a partition \((P,Y,F)\) and a designated support \(B\), let \(x=B\cap Y\). The split entropy is nonnegative. On the unit sphere, either choice of receiving side satisfies

For every finite transport, the split entropy at each leaf is measurable on the unit sphere; for admissible transport it is also nonnegative. If the designated supports are compatible with the leaf partitions and \(\log (\mathrm e\prod _{v\in B_i}n_v)\le \ell \) whenever \(B_i\) has a split leaf, with \(\ell \ge 0\), then \(0\le \eta _{i,j}(\theta )\le 2\ell \) at every leaf. No dimension bound is required for a support with no split leaf.

Let \(w_h\ge 0\) and \(q_{c\mid h}\ge 0\) be finite probability weights. Fix finite old and new leaf measures of mass \(1/2\), nonnegative split entropies bounded by \(B\ge 0\), and a measurable old split entropy. Suppose at most \(N\ge 0\) labels split at each leaf and each label’s terminal split weight is at most \(d\ge 0\). For \(0\le p\le 1\), put

The energy sum with terminal weights \((1-p)w_h\) and \(pw_hq_{c\mid h}\) satisfies

The good-old contribution before multiplication by \(1-p\) is at most \(N^{7/8}\mathcal Q^{1/8}\), including when \(N=0\) or the total good weight is zero.

Fix a finite physical domain \(\Lambda \), a nonempty target \(T\), physical dimension \(q\ge 1\), and independently chosen positive auxiliary dimensions. Let the scan use the induced domain graph and ambient sup-norm distance to \(T\) as depth. Fix a local Hamiltonian, any supplied vector \(\Omega \) and real \(\Delta \), and form their designated truncated supports and augmented terms by the formulas of Definition 28.3.68. Suppose

Every ambient row \(T_d\setminus T_{d-1}\) with \(1\le d\le L\) has at most \(n\) sites, at most \(\mathfrak m\) interaction labels have any fixed anchor, and every \(t\in T\) and physical cut boundary endpoint \(z\) satisfy \(d_\infty (t,z){\gt}2L+10r_0\). Let \(a\ge 0\), \(k\in \mathbb N\), and \(\psi _k\ne 0\) be a symmetric supplied replica vector. At \(0{\lt}p{\lt}1\), use the actual charge round at stage \(s\) with metric parameter \(a/2\) and this same vector. Set

With \(\mathcal Q\) the actual good-old charge defect and \(\mathcal E\) the literal energy error in (107),

The vector need not have norm one. The natural numbers \(K\), \(\mathfrak m\) and \(k\) may vanish. The constant \(\ell \) is independent of both auxiliary dimensions.

Fix admissible finite transport data, a nonnegative metric parameter \(t\), a natural replica count \(k\) with cross-band commutation, and the supplied vector \(\psi _k\). For a fixed good-history predicate and the original history weights, the literal round’s defect \(\mathcal Q_k\) is almost-everywhere measurable on \((0,1)\) and \([0,1]\), and integrable on \([0,1]\). This applies to both actual fill and charge rounds with \(t=a/2\) and \(a\ge 0\), for the same supplied family of vectors. No symmetry, nonzero-vector or positive-copy hypothesis is needed.

Suppose \(8Km\le L\). If a physical partition preserves every initial far-side assignment of a band, every site outside its far side lies in \(S_0=\{ x\in A:d(x)\le L\} \). In particular, \(V^c\subseteq S_0\) at every completed and pre-charge status, with no time-horizon or row assumption.

Fix \(q\ge 1\), the interaction range \(R\), and \(C_r\ge 0\). There is \(C_B{\gt}0\), chosen before the finite domain and scanner, with the following property. Use the induced lattice graph and anchors in their original admissible supports. Suppose \(n\ge 2\), \(T\ne \varnothing \), depth is ambient sup-norm distance to \(T\), and

Assume \(|T_d\setminus T_{d-1}|\le n\) for \(1\le d\le L\), and every endpoint of a physical cut edge of \(A\) has ambient distance greater than \(2L+10r_0\) from \(T\). One common number \(\mathcal B=C_B nD(\log n)^{12}\ge 1\) bounds the designated-support budget of every region in Definition 28.3.1.2, at every completed status with \(k\le nm\) and every pre-charge status with \(k+1\le nm\).

For \(\vartheta \ge 0\) and \(0{\lt}B\le B'\), the radius \([32\sqrt{(1+\vartheta )B}]^{-1}\) decreases when \(B\) is replaced by \(B'\). Under the physical ground-state hypotheses of Theorem 28.3.74, suppose the designated-support budget of a cut \(Q\) is at most \(\mathcal B\). Put \(C_0=512\mathrm e(2/g)\) and \(c_0=[32\sqrt{1+2/g}]^{-1}\). For every real \(u\) with \(|u|\le c_0/\sqrt{\mathcal B}\),

where \(p_j\) are the eigenvalues of the marginal of the same physical ground vector \(\widetilde\Omega \).

Fix \(R\), \(J\ge 0\), \(\Delta {\gt}0\), and \(C_{\mathrm{card}}\ge 0\). There is \(C_{\mathrm{tr}}{\gt}0\) such that, for each \(q\ge 1\), there is \(C_B{\gt}0\), both chosen before the domain, Hamiltonian and scanner. Assume the original Hamiltonian has the ground vector and gap of Theorem 28.3.75, that \(|S_0|\le C_{\mathrm{card}}n^2\), and that the geometric hypotheses of Theorem 28.3.1.3 hold with \(C_r=C_{\mathrm{tr}}\). There are one pair \((e,\widetilde\Omega )\) and one number \(1\le \mathcal B\le C_BnD(\log n)^{12}\) such that the truncated sum has ground vector \(\widetilde\Omega \), energy \(e\) and projector gap \(g/2\). This pair retains

for a real phase \(\theta \). The positive constants \(C_0=512\mathrm e(2/g)\) and \(c_0=[32\sqrt{1+2/g}]^{-1}\) give (119) for all five physical regions, every completed status with \(k\le nm\), every pre-charge status with \(k+1\le nm\), and every real \(u\) in the same interval \(|u|\le c_0/\sqrt{\mathcal B}\).

Fix \(C_r\ge 0\) and real exponents \(0{\lt}\kappa {\lt}\mu \) and \(\ell \). For all sufficiently large integer \(n\), put \(L=\lfloor n^{1-\ell }\rfloor \), \(m=\lfloor n^\mu \rfloor \), \(D=\lceil n^\kappa \rceil \), and \(K=L/(8m)\) with integer division. Then \(n\ge 2\), \(r_0=\lceil C_r(\log n)^2\rceil \le D\), \(D\ge 1\), \(4D\le m\), and \(8Km\le L\). The threshold may depend on the fixed exponents. The last inequality holds for all natural \(L,m\).

Let \(T\ne \varnothing \) be a finite ambient target and let \(A\) be any subset of the finite physical domain. Suppose \(n,L\in \mathbb {N}\), \(L\le n\), \(C\in \mathbb {R}\), \(|T|\le Cn^2\), and \(|T_d\setminus T_{d-1}|\le n\) for \(1\le d\le L\). Then \(|A\cap T_L|\le (C+1)n^2\).

Fix real exponents \(0{\lt}\mu {\lt}1-\ell \). For all sufficiently large integer \(n\), the exact integer quotient \(K=\lfloor n^{1-\ell }\rfloor /(8\lfloor n^\mu \rfloor )\) is at least one. The threshold may depend on \(\mu \) and \(1-\ell -\mu \).

Fix \(R\), \(J\ge 0\), \(\Delta {\gt}0\), \(C_T\ge 0\), and \(0{\lt}\ell {\lt}1\), \(0{\lt}\kappa {\lt}\mu {\lt}1-\ell \). There are \(C_{\mathrm{tr}}{\gt}0\) and \(N\ge 2\), chosen before the local dimension, such that for every \(q\ge 1\) there is \(C_B{\gt}0\) chosen before the physical domain, Hamiltonian and scanner. Use an original gapped ground vector as in Theorem 28.3.1.5, the induced lattice graph, and anchors in their original supports. For \(n\ge N\), use the literal scales \(L=\lfloor n^{1-\ell }\rfloor \), \(m=\lfloor n^\mu \rfloor \), \(D=\lceil n^\kappa \rceil \), \(K=L/(8m)\) with integer division, and \(r_0=\lceil C_{\mathrm{tr}}(\log n)^2\rceil \). Suppose \(T\ne \varnothing \), depth is ambient sup-norm distance to \(T\), \(|T|\le C_Tn^2\), \(|T_d\setminus T_{d-1}|\le n\) for \(1\le d\le L\), and every endpoint of a physical cut edge of \(A\) has ambient distance greater than \(2L+10r_0\) from \(T\).

Then \(K\ge 1\). There are one pair \((e,\widetilde\Omega )\) and one number \(1\le \mathcal B\le C_BnD(\log n)^{12}\) satisfying all the spectral, energy, phase and trace-distance conclusions of Theorem 28.3.1.5. The positive constants \(C_0=512\mathrm e(2/g)\) and \(c_0=[32\sqrt{1+2/g}]^{-1}\) give (119) for every actual region \(XU,V,Y,U,UY\), every completed status with \(k\le nm\), every pre-charge status with \(k+1\le nm\), and every real \(u\) with \(|u|\le c_0/\sqrt{\mathcal B}\), using that same \(\widetilde\Omega \).

Fix an actual fill or charge round, with positive physical and auxiliary dimensions \(q,d\), scan scale \(m{\gt}0\), padded charge capacity \(M{\gt}0\), and interpolation scale \(a/2\). Let \(H=\sum _i\widetilde H_i\) be the sum of its truncated physical terms, formed from the same reference state. For any supplied replica vector \(\mathrm{pre}_k\) and any real interpolation parameter \(p\), let \(\mathcal M\) be its mean at the parameter clamped to \([0,1]\). Put

with the normalized vector interpreted as zero when its numerator is zero. The mean energy of this round satisfies

where \(\mathsf H_k\) is the physical replica mean of \(H\), extended by both auxiliary identities in the successive physical, \(C\), \(R\) coordinates. This is the replica mean in [ 247 , Section 9, scanner energy output ] . The equality includes \(k=0\). It imposes no eigenvalue equation or additional assumption on the supplied vector.

Use an actual charge round with \(m,M{\gt}0\), physical and auxiliary dimensions \(q,d{\gt}0\), and metric parameter \(a/2\) with \(a\ge 0\). For every integer \(k\ge 0\), every nonzero supplied vector \(\mathrm{pre}_k\), and every real \(p\), the filtered vector \(v\) has norm one. Write \(v_O=v\circ e_k\) for the same vector in the successive physical, \(C\), \(R\) coordinates, and retain the physical Hamiltonian \(H=\sum _i\widetilde H_i(\Omega )\) formed from the original reference vector \(\Omega \). Write \(\mathcal E_k(p)\) for the mean energy of this actual charge round.

Let \(\widetilde\Omega \) be a supplied unit vector, separately from \(\Omega \). Suppose \(g{\gt}0\) and

For \(k\ge 1\), suppose that \(v_O\) is invariant under simultaneous copy permutations and lies in the two auxiliary isotypic spaces labelled by \(\lambda _C,\lambda _R\). Let \(\mathsf D_k=\sum _{j=0}^{k-1}(I-|\widetilde\Omega \rangle \langle \widetilde\Omega |)_j\) be the physical-copy defect count. For \(\tau {\gt}0\), let \(\Pi _{k,\tau }\) be the product of its closed spectral projection \(\mathbf1_{(-\infty ,\tau k]}(\mathsf D_k)\), the two auxiliary label projections, and the projection onto simultaneous copy symmetry, all in the coordinates of \(v_O\). Then

This is the gap estimate of [ 247 , Section 8, defect-mass comparison ] , applied to the vector of the actual charge round.

There exist universal real constants \(c_0,C_{\mathrm{ent}},e_{\mathrm{ent}},C_{\mathrm{en}},e_{\mathrm{en}}\), with \(c_0{\gt}0\), chosen before all the following data.

Let \(\Lambda ,T\subset \mathbb Z^2\) be finite with \(T\ne \varnothing \), \(q,R_{\mathrm{int}}\in \mathbb N\) with \(q\ge 1\), and \(J\in \mathbb R\). Let \(H=\sum _i h_i\) be a local Hamiltonian of range \(R_{\mathrm{int}}\) and term norm bound \(J\) on \((\mathbb C^q)^{\otimes \Lambda }\), with its admissible support labels in a fixed linear order. Fix \(\| \Omega \| =1\) and \(\Delta {\gt}0\). Take a scanner of color \(A\subseteq \Lambda \), with parameters \(n,m,K,D,r_0\in \mathbb N\) and \(C_1\in \mathbb R\), the induced nearest-neighbor graph on \(\Lambda \), depth \(d(x)=\min _{t\in T}\| x-t\| _\infty \), and anchors \(x_i\in i\). For a truncation depth \(L\in \mathbb N\), assume

Put \(T_d=T+[-d,d]^2_{\mathbb Z}\) and let \(Z\) contain both endpoints of every edge crossing \(A\) in \(\Lambda \). Assume \(|T_d\setminus T_{d-1}|\le n\) for every integer \(1\le d\le L\), and \(2L+10r_0{\lt}\| t-z\| _\infty \) for every \(t\in T\) and \(z\in Z\). Fix arbitrary auxiliary dimensions \(d_C,d_R\ge 1\) and put

Fix \(a{\gt}0\) with \(a\ell \le c_0\) and use \(t=a/2\) in every band’s metric. Let \(\mathscr D_j\) be the actual round family of Definition 28.3.1.10.

Use one energy family for all rounds: let \(\widetilde h_i\) and \(\widetilde X_i\) be the actual truncated terms and designated supports of Definition 28.3.68 for \(H,\Omega ,\Delta ,L\), retaining every original support label, and set

The subscript \(b\) denotes insertion in copy \(b\).

There exist \(C_\beta \ge 0\) and sequences \(\beta ,\varepsilon :\mathbb N\to \mathbb R\) such that, for every \(k\in \mathbb N\),

These choices precede the round \(j\), replica count \(k\), supplied nonzero symmetric vector \(\mathrm{pre}\in \operatorname {Sym}^k(\mathcal V)\), interpolation parameters, and energy eigenvalue. For every such \(j,k,\mathrm{pre}\), let \(\mathcal M_{j,k}(p)\) be the actual interpolating root and set

For this round, let \(\mathcal L_{j,k,\mathrm{pre}}\), \(\mathcal G_{j,k,\mathrm{pre}}\), and \(\mathcal E_{j,k,\mathrm{pre}}\) be the derivative, entropy-gain, and energy-error expressions in Theorem 28.3.77, with the actual history and conditional-choice trees of \(\mathscr D_j\). In particular, the energy error retains both split probabilities: writing \(\mathcal J_{j,i}\) for the leaves splitting \(\widetilde X_i\), its expression is

Then for every \(0{\lt}p{\lt}1\),

For every \(E_0\in \mathbb R\) satisfying \(\overline H_k\mathrm{pre}=E_0\mathrm{pre}\), the energy conclusion is

For every \(0\le p_0\le p_1\le 1\),

The common errors may depend on the fixed physical data, auxiliary dimensions, \(a\), and finite family, but are independent of \(j,k,\mathrm{pre},p,p_0,p_1,E_0\). This does not assert a uniform limit when the system or auxiliary dimensions grow with \(k\). At \(k=0\), the energy premise and \(\mathrm{pre}\ne 0\) force \(E_0=0\). No physical prevector or identification with a fixed physical ground energy is asserted.

Fix a scanner with \(m,M{\gt}0\), positive factor dimensions, \(t\ge 0\), and a replica count \(k\ge 0\). Let \(\mathcal M_{j,k}(p)\) be the actual round roots, and let \(P_{s,k}\) be the root at parameter zero of the fill after \(s\) completed charges. Whenever \(j+1{\lt}2nm\), \(\mathcal M_{j,k}(1)=\mathcal M_{j+1,k}(0)\). If \(nm{\gt}0\), the last charge satisfies \(\mathcal M_{2nm-1,k}(1)=P_{nm,k}\).

For any supplied vector \(\mathrm{pre}\) in the replica space, set \(N_{j,k,\mathrm{pre}}(p)^2 =\| \mathcal M_{j,k}(p)^{-1/4}\mathrm{pre}\| ^2\) and \((N^{\mathrm{fill}}_{s,k,\mathrm{pre}})^2 =\| P_{s,k}^{-1/4}\mathrm{pre}\| ^2\). Then

For the zero vector, use the convention \(\log 0=0\). No symmetry assumption is needed. If \(n=0\), the sum is empty and both remaining roots are \(P_{0,k}\). The assertion also holds at \(k=0\).

There exist universal real constants \(c_0,C_{\mathrm{ent}},e_{\mathrm{ent}}\), with \(c_0{\gt}0\), such that the following holds. Assume the geometric, Hamiltonian, dimension, and smallness hypotheses of Theorem 28.3.1.13, in particular \(n{\gt}0\) and \(a\ell \le c_0\). There exist \(C_\beta \ge 0\) and a sequence \(\beta :\mathbb N\to \mathbb R\), chosen before the replica count and supplied vector, with \(0\le \beta _k\le C_\beta \log (k+2)\). For every \(k\ge 0\) and every nonzero symmetric vector \(\mathrm{pre}\in \operatorname {Sym}^k(\mathcal V)\),

Here \(t=a/2\), and the entropy gains are those of Theorem 28.3.1.13. The fixed physical and auxiliary dimensions precede the choice of \(\beta \). No energy eigenvector equation or initial or final norm comparison is assumed. At zero replicas the left side is zero and the total error term is \(2nm\beta _0\).

Let \(b_s\) be the classical weight of bad old histories after \(s\) complete fill–charge pairs. For \(s\le H\) and positive charge capacity, \(b_s\) is at most the weight of full histories of length \(H\) having a bad completed prefix.

Fix \(C_r,C_1{\gt}0\), \(C_T\ge 0\), anchor multiplicity bound \(\mathfrak m\in \mathbb N\) and \(P\in \mathbb R\). Use the final scanner exponents \(\ell =10^{-5}\), \(\kappa =10^{-6}\) and \(\mu =1/4\), and set

There is \(N\), depending only on these fixed quantities, such that for all \(n\ge N\) the following holds for every physical scan. Assume \(T\ne \varnothing \), \(|T|\le C_Tn^2\), the induced domain graph, ambient target-distance depth, ambient row sizes at most \(n\) through depth \(L\), anchor multiplicities at most \(\mathfrak m\), and cut clearance greater than \(2L+10r_0\). Then

The first statement holds for every scan with positive charge capacity, without the geometric assumptions above. The margin \(4(D+2r_0)\le m\), together with \(n\ge 2\), \(r_0\le D\), \(1\le D\) and \(8Km\le L\), holds eventually for any real \(\ell \), any \(C_r\ge 0\) and any fixed \(0{\lt}\kappa {\lt}\mu \), with its threshold allowed to depend on these quantities; only the bound on \(b_s\) uses the listed physical assumptions.

With the scales and physical assumptions of Theorem 28.3.1.16, one threshold \(N\) gives the following estimate simultaneously for every charge \(s+1\le nm\), every positive local and auxiliary dimension, every supplied local Hamiltonian and vector \(\Omega \), every \(\Delta \in \mathbb R\), every \(a\ge 0\), replica count \(k\ge 0\), nonzero symmetric supplied vector \(\psi _k\), and common parameter \(0{\lt}p{\lt}1\):

where \(A=10KnD\mathfrak m\) and \(\ell _q=1+(2r_0+1)^2\log q\). The energy and charge defect use the same actual charge data, supplied vector and metric parameter \(a/2\) as in Theorem 28.3.84. Both history capacities are positive as a consequence of the rounded scales. No selected round, physical prevector identity, or norm comparison is part of this statement.

Fix the physical dimension \(q\ge 1\), \(C_r,C_1{\gt}0\), \(C_T\ge 0\), \(\mathfrak m\in \mathbb N\) and \(P,C_{\mathrm{en}},e_{\mathrm{en}}\in \mathbb R\). There exist \(C\ge 1\), \(C_l\ge 0\) and \(N\in \mathbb N\), depending only on these fixed quantities, with the following property. For every \(n\ge N\), use the literal rounded scales (151) and the geometric assumptions of Theorem 28.3.1.16. For every charge \(s+1\le nm\), every pair of positive auxiliary dimensions, supplied local Hamiltonian, vector \(\Omega \) and real \(\Delta \), every \(W\ge 0\), replica count \(k\ge 0\), nonzero symmetric supplied vector \(\psi _k\) and \(0{\lt}p{\lt}1\), put \(a=W/K\). Then both history capacities are positive and

where \(\ell =10^{-5}\) and \(\ell _q=1+(2r_0+1)^2\log q\). Both the energy and defect use the same supplied vector and metric parameter \(a/2\). No round or small defect is selected. The estimate includes zero total weight, zero anchor multiplicity, \(q=1\) and \(k=0\). For every scan with positive history capacities, every supplied vector, \(a\ge 0\) and \(p\), the literal defect \(\mathcal Q_{s,k}(p)\) is nonnegative.

Let \(\mathscr D\) be finite admissible transport data with positive factor dimensions and \(t\ge 0\). Suppose its bands commute on the symmetric replica subspace at replica count \(k\ge 0\). For every supplied vector \(\mathrm{pre}\) and every real \(p\), its entropy gain satisfies \(\mathcal G_{\mathscr D,k,\mathrm{pre}}(p)\ge 0\). The vector may vanish, and \(p\) may be either endpoint.

Assume the geometric and designated-support hypotheses of Theorem 28.3.55, uniformly over the \(nm\) charge stages, with \(m,M{\gt}0\), positive factor dimensions, \(t\ge 0\), a replica count \(k\ge 0\) and a supplied nonzero vector \(\mathrm{pre}\). Put \(\alpha =[2(C_1+1)nD]^{-1}\). Write \(\mathcal Q_{s,k,\mathrm{pre}}\) for the literal good-old-history defect at charge stage \(s\), and \(\mathcal G_{j,k,\mathrm{pre}}\) for the gain of actual round \(j\). For every \(0\le p_0\le p_1\le 1\),

No symmetry hypothesis on the supplied vector is needed.

There are universal real constants \(c_0,C_{\mathrm{ent}},e_{\mathrm{ent}}\), with \(c_0{\gt}0\), for which the following holds. Assume the joint geometric, Hamiltonian and smallness hypotheses of Theorems 28.3.1.20 and 28.3.1.13; in particular, \(n\ge 2\), \(2r_0\le D\), \(C_1{\gt}0\), \(3\mu \le C_1\), \(a{\gt}0\) and \(a\ell \le c_0\), where \(\mu \) bounds the number of labels at each anchor. Use the actual augmented energy terms, set \(t=a/2\), and put \(\alpha =[2(C_1+1)nD]^{-1}\). The physical and auxiliary dimensions are fixed before choosing \(C_\beta \ge 0\) and a sequence \(\beta :\mathbb N\to \mathbb R\) such that \(0\le \beta _k\le C_\beta \log (k+2)\) for every \(k\ge 0\). The same sequence works for every replica count, supplied vector and retained subinterval. For every \(k\ge 0\), every nonzero symmetric supplied vector \(\mathrm{pre}\), and every \(0\le p_0\le p_1\le 1\),

The endpoint norms are those of Theorem 28.3.1.14. The error is paid for every full round; it has no factor \(p_1-p_0\). Zero replicas, zero bands and equal endpoints are included. Identifying the supplied vector with the physical prevector and bounding the initial and final norms are separate obligations.

Fix an integer \(q\ge 1\), a real number \(g{\gt}0\), and a real constant \(C_{\mathrm{vol}}\). For all sufficiently large integers \(n\), the following assertion holds uniformly over the finite domain and all the data below.

Let \(\Omega ,\Omega _t\) be unit vectors on a finite domain \(\Lambda \), with local dimension \(q\). For a subset \(A\subseteq \Lambda \), an integer-valued depth function \(d\), and a natural number \(L\), put

Suppose \(|R_L|\le C_{\mathrm{vol}}n^2\) and, writing \(\Delta \) for the trace distance between the two pure states, suppose

Let \(C_e,e\in \mathbb R\). If the original vector satisfies

for every natural number \(j\le L\), then the same prefixes satisfy

The threshold depends only on \(q,g,C_{\mathrm{vol}}\); in particular, it precedes the choices of both vectors, the depth function, and \(C_e,e\). No sign condition on \(C_e,e\) is needed for this implication.

Let \(V\) be a set, let \(q,d,k\ge 0\) be integers, and let \(Q\subseteq I\) be finite. For \(\sigma \in S_k\), write \(\rho _Q(\sigma )\) for the permutation that permutes the copies at the sites in \(Q\) and fixes all other coordinates. Then, for every \(x\in \mathcal X_I^k\),

No finiteness assumption on \(V\) is needed.

Suppose now that \(V\) is finite. Put \(N=|\mathcal X_I|=|\mathcal X_J|\), and use the replica weight \(w_{k,N,t}(\lambda )\) for the full one-copy dimension \(N\). For every real \(t\), every \(k\ge 0\) and every \(Q\subseteq I\), the two replica metrics satisfy

The weights on both sides use \(N\), including when a coordinate space is empty. The assertion also includes \(k=0\).

Suppose \(V\) is finite, \(q,d,k\ge 0\) are integers and \(t\in \mathbb {R}\). For any three regions \(P,Y,F\subseteq J\), put \(P_I=a^{-1}(P)\), \(Y_I=a^{-1}(Y)\) and \(F_I=a^{-1}(F)\). Define

Then \(A_J(P,Y,F)=\mathcal U_k A_I(P_I,Y_I,F_I)\mathcal U_k^*\). The regions need not form a partition. Empty coordinate spaces and zero replicas are included; inverses have the usual total matrix convention.

Let \(q,d{\gt}0\), \(t\ge 0\), and let \(D\) be admissible transport data on \(J\) with \(K\) bands. Thus its history and conditional weights are positive, its old regions partition \(J\), and every move transfers a part of the old middle region to one side. Assume cross-band commutation on the symmetric replica space. Fix \(k,N\ge 0\), \(p\in \mathbb {R}\), and a supplied matrix \(Z:\mathbb {C}^N\to \mathbb {C}^{\mathcal X_J^k}\) satisfying

where \(\Pi _k\) is the symmetric projection. Let \(v\in \mathbb {C}^{\mathcal X_J^k}\) satisfy \(\Pi _kv=v\).

Let \(T\) be the original transport tree at the parameter \(p\) clamped to \([0,1]\). For each of its terminal labels \(j\) and bands \(g\), pull back the actual partition \((P_{j,g},Y_{j,g},F_{j,g})\) through \(a\), and let \(A^I_{j,g}\) be its band matrix in the original coordinates. Put \(Z_I=\mathcal U_k^*Z\), \(v_I=\mathcal U_k^*v\), and

The product in (167) has the original band order. Write \(L(p)\) for the actual transport root. Then \(M\) is positive definite, and for every \(r\in \mathbb {R}\),

A zero denominator gives the zero vector. No nonzero-vector premise is needed for these equalities.

Suppose the target is nonempty, \(8Km\le L\), and the actual cut endpoints have ambient distance greater than \(2L+10r\) from the target. Every assigned site of \(A\) has a chronological selected-charge path rooted at an initially assigned site or a scheduled fill slot. Its charge times are strictly increasing. For a nonempty path, its endpoint lies in its last charge ball, and successive anchors are at graph distance at most \(2r\). Initial assignments persist. A bad endpoint was initially middle and lies in \(A\cap T_L\). A scheduled fill cannot create a bad lead.

The fronts are monotone. For \(n{\gt}0\) and pair times \(a\le b\),

If an assigned endpoint has lead greater than \(D/2\), its ancestry is nonempty. For its length \(s\) and every recorded charge time \(t\),

Assume \(M{\gt}0\) and fix all initial offsets. Requiring a specified side and label in one band at each of \(s\) distinct times has probability at most \((2M)^{-s}\). The same upper bound holds after imposing arbitrary additional conditions. For a finite family of such prescriptions covering an event, the event’s probability is at most the sum of these bounds.

Let \(B\) bound the number of labels whose anchors lie within graph distance \(2r\) of any site. The number of reversed label paths of length \(s\) from a fixed endpoint is at most \(B^s\). In an induced square-lattice domain with anchor multiplicity at most \(\mu \), one may take

A terminal window specified by \(k-(t+1)\le W\) and \(0\le t{\lt}k\) contains at most \(W+1\) indices. Choosing \(s\) of its \(v\) indices and a reversed label path from a finite family of size \(b\) gives at most \(\binom vs b\) chronologically labelled paths. Their times are distinct. For \(n,r,s{\gt}0\) and \(W=4n(rs+1)\), one has \(v\le 9nrs\).

For a fixed endpoint and pair time \(k\), all bad ancestries lie in the finite union of the timed spatial path families with \(0\le s\le k\) and \(D{\lt}4rs\). Every path in this family has valid distinct times. For every \(q\ge 0\), its sum of weights \(q^s\) is at most

For \(s{\gt}0\), \(a\ge 0\), and \(v\le Cs\),

If every positive ambient layer through \(L\) has at most \(n\) sites, then

Restricting a full history to its first \(j\) pairs preserves its offsets and earlier charge coordinates. For \(j\le N\) and \(M{\gt}0\), every real-valued function of this prefix has the same expectation under length-\(N\) history weights as under length-\(j\) history weights.

Fix the offsets, a band, a side, and an endpoint in \(A\). Under the actual cut-clearance hypotheses and \(n,M{\gt}0\), the probability that the endpoint is assigned with lead greater than \(D/2\) is at most

For \(m{\gt}0\), the total weight of bad completed histories at time \(k\) is at most \(2K|A\cap T_L|R_k\). The same bound holds for old charge histories after their preceding fill.

Under the source ambient layer bound, the probability that some completed prefix of one length-\(N\) actual history is bad is at most

Goodness of these completed prefixes also guarantees goodness after every intervening deterministic fill.

If additionally \(r{\gt}0\), set

The probability of a bad prefix is at most

Fix \(0{\lt}\ell \), \(0{\lt}\kappa {\lt}\mu {\lt}1-\ell \), \(C_r,C_1{\gt}0\), \(C_T\ge 0\), a finite anchor multiplicity bound \(b\), and a real exponent \(p\). Set

For all sufficiently large \(n\), uniformly over finite lattice domains, nonempty targets with \(|T|\le C_Tn^2\), and actual scanners satisfying the ambient row bound, the cut clearance above, and anchor multiplicity at most \(b\), the probability that any prefix through \(nm\) fill–charge pairs has bad lead on either side in any band is at most \(n^{-p}\). In particular \(p=200\) gives the bad-history exponent in Lemma 9.1.

Fix scanner exponents satisfying the preceding conditions and put \(K_n=\lfloor \lfloor n^{1-\ell }\rfloor /(8\lfloor n^\mu \rfloor )\rfloor \). For every integer \(B\ge 0\) there is a threshold \(N\), depending only on the exponents and \(B\), such that \(B\le K_n\) whenever \(n\ge N\). These are the rounded scales of [ 247 , Section 9 ] .

There are constants, independent of \(W\) and depending only on the exponents and on the constants of the inputs, such that for all sufficiently large \(n\) and every scan at scale \(n\) satisfying the conclusions of Lemmas 9.1, 2.1, 2.3 and Propositions 7.4, 8.1, the comparator budgets \(B_{\mathrm{sh}}\le C(nL^e+n)\), \(B_{\mathrm{exc}}\le CnD\) and \(\mathcal B\le CnD(\log n)^{C}\) hold, and there is a remainder \(\rho _k\to 0\), uniform over the selected round and parameter, such that for every \(k\ge 1\) some charge round and some \(p_k\in [\epsilon /2,\epsilon ]\) satisfy \(\mathcal Q(p_k)/(KnD)\le \delta _n+\rho _k\) and

Let \(C_e\) be the original entropy-input constant, and write \(\widetilde S_X\) and \(\widetilde S_Q(j)\) for the target and shell entropy values of the truncated state recorded in the same input data. The threshold can also be chosen so that, for every natural shell index \(j\le L\),