Dependencies

Fix the original region \(A\), its depth function \(d\) and the collar depth \(L\). Suppose \(n{\gt}0\) and \(8Km\le L\). The nested family \(Q_0,\ldots ,Q_L\), where \(Q_j=\{ x\in A:0{\lt}d(x)\le j\} \), then contains both \(Q_{j_N-1}\) and \(Q_{-j_F}\) for every band, offset and round within the fill horizon. This one family is fixed before all such choices.

The configuration space of \(\Lambda \) is the product of those of \(X\), \(Y\) and \(Z'\). In these coordinates, with the factor \(X\) written in the eigenbasis \(\{ |x\rangle \} \) of \(\rho _X\), the vector \(\Omega \) and the projection \(Q\) are the inputs of \(R_0\), \(B_0\), \(\phi \) and \(\eta \).

Let \(\Omega \) be a vector on a finite induced lattice domain \(\Lambda \), let \(A\subseteq \Lambda \), and let \(w\) be real. Write \(\rho _A=\sum _jp_j|e_j\rangle \langle e_j|\) for an eigendecomposition of the reduced state. The typical projection of \(A\) at width \(w\) is

For a real sequence \((\varepsilon _l)_{l\ge 0}\), define

Here the square root of a negative real number is interpreted as zero. Every localization-error inequality below forces its error bound to be nonnegative at the radius in question.

For a nonempty word \(w=(i_0,\ldots ,i_{m-1})\), define

For the empty word \(\varnothing \), take \(\mathcal E_{\varnothing }=\mathrm{id}\). Thus the first event acts first. Given an initial operator \(B_0\), write \(B_w=\mathcal E_w(B_0)\).

For an operator \(B\) and a word \(w\), put \(D_P(w;B)=\| \mathcal E_w(B)-\mathcal E_{w|_P}(B)\| \).

For real parameters \(D,b,\alpha \) and a cutoff \(N\in \mathbb {N}\), define

For arbitrary real exponents, use \(0^0=1\) and \(0^\alpha =0\) when \(\alpha \ne 0\). With \(D=2A\) and \(N=n\), the right side of (277) is \(\sum _{z\in I}\mathsf A^{(n)}_{yz}d_z(B)\).

Let \(V\) be any set and let \(G\) be a simple graph on \(V\). The graph distance \(d_G(x,y)\) is the length of a shortest path when one exists, and is \(\infty \) otherwise. Let \(\iota :\mathbb {N}\cup \{ \infty \} \to [0,\infty ]\) be the canonical inclusion, sending each natural number to the same real number and \(\infty \) to \(\infty \). Define

This is an extended pseudometric on \(V\): it satisfies \(\rho _G(x,x)=0\), \(\rho _G(x,y)=\rho _G(y,x)\), and \(\rho _G(x,z)\le \rho _G(x,y)+\rho _G(y,z)\).

Let \((K_l)_{l\ge 0}\) be any sequence of physical regions and put \(\mathcal E_l=\mathcal E_{K_l}\). Define

Equivalently, take \(\mathcal E_{-1}=\mathrm{id}\) and \(\Delta _l=\mathcal E_l-\mathcal E_{l-1}\), as in [ 247 , Section 10 ] . Distributivity gives \(\Delta _l(B-C)=\Delta _l(B)-\Delta _l(C)\).

Set \(k_K=E_K(k)\), and define the positive square roots

The full and localized channels are

In particular, \(\mathcal E(B-C)=\mathcal E(B)-\mathcal E(C)\) and \(\mathcal E_K(B-C)=\mathcal E_K(B)-\mathcal E_K(C)\) by distributivity. These are the localized channels of [ 247 , Section 10 ] , extended by the identity on the spectator.

For \(m,M{\gt}0\), place the actual old charge partitions and actual selected-side moves on the recursive physical history tree. At each history use the fixed uniform charge-choice tree already used to construct the next history tree. No matrix mean is reassociated.

For a local interaction \(H=\sum _i h_i\), a unit vector \(\Omega \), and \(\Delta {\gt}0\), put \(M_i=\mathcal F_{1,\Delta /2}(H,h_i)-\langle \Omega ,h_i\Omega \rangle I\) and \(k_i=|M_i|/c_*\), where \(c_*=\max \{ 1,J(\| f_{1,\Delta /2}\| _1+1)\} \). Truncate \(k_i\) near \(S_0=\{ x\in A:d(x)\le L\} \) at the variable radius of [ 247 , Proposition 4.5 ] , with base radius \(r_0\). Write the resulting term and designated support as \(\widetilde h_i\) and \(\widetilde X_i\). The augmented term is \(\widetilde h_i\otimes I_C\otimes I_R\), with designated support \(\widetilde X_i\) embedded among physical factors. The dimensions of \(C\) and \(R\) are independently specified.

At fill index \(k\), read the scheduled padded row slot in each band. A blank slot makes no move. If the slot contains \(x\), transfer \(\{ x\} \cap Y\) to the scheduled receiving side. Keep the existing recursive history tree and use a singleton conditional choice tree at every history. The old partition is the completed physical status and the new partition is its following pre-charge status.

For each band \(0\le g{\lt}K\), choose an independent uniform offset \(r_g\in \{ 0,\ldots ,m-1\} \) and set \(u_g=8gm+m+r_g\), \(v_g=8gm+5m+r_g\). Each charge chooses, independently in every band, a uniform side and a uniform slot among \(M\) fixed slots. A history after \(k\) fill–charge pairs consists of these \(K\) offsets and \(kK\) side–slot choices. Its classical weight is \(m^{-K}(2M)^{-Kk}\); a new charge has weight \((2M)^{-K}\). Evaluate each history recursively: first perform the next prescribed fill, then the chosen charge at the resulting front. A nonblank labelled slot selects its graph \(r_0\)-ball; a blank slot makes no move.

The picture shows the nominal partitions of \(A\) in two consecutive bands, ordered by depth, with \(X=A\cap T\) and \(U=P_0\setminus X\); arrows show the fill directions at the fronts, and brackets mark the charge intervals of oriented depth \([j-r_0,j+D]\) on the two sides.

For a fixed scanner with \(m,M{\gt}0\), index its rounds by integers \(0\le j{\lt}2nm\) and put \(s_j=\lfloor j/2\rfloor \). Write \(\mathcal Q=(\{ 0,1\} \times \{ 0,\ldots ,M-1\} )^K\) for the charge choices. The history and conditional-choice sets are

The transport data \(\mathscr D_j\) are the actual fill data at stage \(s_j\) for even \(j\) and the actual charge data at stage \(s_j\) for odd \(j\). Thus round zero is a fill, rounds zero and one both have empty charge history, and rounds two and three both have one completed charge. Every round has exactly \(K\) bands. Empty physical moves remain among these rounds. A property depending on the conditional-choice sets and transport data can be proved separately for the fill and charge cases, identifying their conditional-choice sets with \(\mathcal C_{j,h}\).

Adjoin the fixed factors \(C\) and \(R\) to the physical near and far sides: \(P=C\sqcup P_0\), \(Y=Y_0\), and \(F=R\sqcup V_0\). The target \(X\) remains in \(P_0\); it is not removed from the near region. An enabled charge with selected ball \(B\) transfers exactly \(B\cap Y_0\) to the selected side. A blank slot or a ball not meeting both that side and the middle makes no move. History and conditional-choice mean trees retain their supplied shapes.

Let \(V\) be a finite set of physical sites, each of dimension \(q\), and let the dimensions of \(C\) and \(R\) both be \(d\). Write \(X=\{ 0,\ldots ,q-1\} ^V\) for the physical configuration space. Identify the physical site \(v\) with \(v\) in \(V\sqcup \{ C,R\} \). The two successive adjoining operations place \(C\) at the first added site and \(R\) at the second. Let \(e\) be this identification of one-copy configurations and \(e_k\) its copywise extension. Let \(g_k\) group the configurations in the order \(X^k,C^k,R^k\), where each auxiliary configuration space has \(d\) elements. These are bijections also for zero dimensions and for \(k=0\).

Fix an actual old history \(h\) and a pure vector \(\theta \) on the physical and auxiliary factors. For a physical trial set \(B\), band \(g\) and receiving side \(s\), put \(x=B\cap Y_{h,g}\) and define

Only physical sites are moved; the auxiliary factors remain in their respective receiving sides.

Let \(T\) be finite and nonempty. Write \(d(x)=\operatorname {dist}_\infty (x,T)\) and set

For a physical status \((P_0,Y,V)\), let \(U=P_0\cap \{ d{\gt}0\} \) and \(UY=(P_0\cup Y)\cap \{ d{\gt}0\} \). These prefixes contain whole rows; the partially consumed nominal row contributes to the error estimate.

A label has a receiving–middle incidence if its designated truncated support meets a receiving side and the actual middle of one band. The terminal split event of [ 247 , Section 7, terminal split rule ] additionally requires exactly one receiving side and containment in a single part of every other band, so it is contained in this incidence event. These are the physical statuses \(P_0,Y,V\) underlying \(P=CP_0,Y,F=RV\). No dilution is asserted for arbitrary entropy cuts such as \(U=P_0\setminus X\), whose fixed inner boundary is not a terminal metric split.

For positive offset and slot capacities, choose a fixed binary uniform tree for the finite initial offsets and a fixed binary uniform tree for one simultaneous charge choice. A tree with \(r+1\) leaves is constructed recursively from a first leaf of weight \(1/(r+1)\) and an \(r\)-leaf subtree of total weight \(r/(r+1)\). At every charge round, substitute the charge tree at each existing history leaf and append that charge to its label. Deterministic fills have a singleton conditional choice tree.

Fix one actual fill or charge round, its recursive history and choice trees, dimensions, and a supplied family of replica vectors \(\psi _k\). Let \(\Theta \) be the unit sphere of the one-copy space and set \(t=a/2\). For each terminal leaf \(j\), let \(\mu _{k,p,j}\) be the image on \(\Theta \) of

where \(dU\) is normalized Haar measure and \(\sigma _{k,p,j,u}\) is the transported state of the filtered vector obtained from this same \(\psi _k\). For \(a\ge 0\) the real density is nonnegative, so the maximum leaves it unchanged. The old and new measures are \(\mu _{k,p,(h,\mathrm{old})}\) and \(\mu _{k,p,(h,c,\mathrm{new})}\). History and choice weights are not part of these measures. The round uses the corresponding split entropies, choice entropy, logarithmic root norm and replica-energy expectation.

At history \(h\), let \(\rho _{h,\varnothing }(p,u)\) be its old transported state, let \(\beta (u)\) be the Fourier weight, and let \(\mathcal H_\rho \) denote the coherent-state integral with density \(\rho \). Define

The old state in \(J_h\) is fixed while the conditional choice in the entropy symbol varies.

For physical local dimension \(q\ge 1\), set \(\ell _q(r_0)=1+(2r_0+1)^2\log q\).

A physical partition assigns each site to the near side \(P_0\), the middle \(Y\), or the far side \(V\). For cutoffs \(u{\lt}v\), initially \(P_0=\{ x\in A:d(x)\le u\} \) and \(V=(\Lambda \setminus A)\cup \{ x\in A:d(x){\gt}v\} \). Order each positive-depth row once and pad it to \(n\) slots. Sites outside \(A\) and absent entries are blank. The far side visits rows in decreasing depth, using the same fixed order within each row. Fills alternate near, far, starting near. After \(k\) fills the two consumed slot counts are \(t_P=\lfloor (k+1)/2\rfloor \) and \(t_F=\lfloor k/2\rfloor \); the oriented fronts are \(j_P=u+1+\lfloor t_P/n\rfloor \) and \(j_F=-v+\lfloor t_F/n\rfloor \). A move of a set \(B\) transfers only \(B\cap Y\) to its receiving side. A charge transfers this set only if \(B\) meets both that side and \(Y\).

The physical marginal regions at a status are \(XU,V,Y,U,UY\), where \(XU\) is the near receiving side, \(V\) is the far side, \(Y\) is the middle, and \(U,UY\) are the positive-depth portions of \(XU,XU\cup Y\), respectively. Coincident and empty regions are allowed.

A support splits band \(g\) on one side when it meets \(Y_g\) and that receiving side, misses the opposite side, and is contained in a single physical part of every other band. This is the physical terminal split rule of [ 247 , Section 7 ] .

For a finite induced domain \(\Lambda \) and a cut \(A\subseteq \Lambda \), the boundary endpoints are the integer endpoints of the crossing edges,

For a finite set \(P\subset \mathbb Z^2\) of parent indices, its dyadic child indices are

These are the four children of each parent in a translated dyadic mesh.

For a unit vector \(\Omega \), a real \(E_0\), and \(\Delta {\gt}0\), the eigenvector and full-system projector-gap conditions are

For \(A\subseteq \Lambda \), put

All logarithms are natural, with \(0\log 0=0\). The reduced matrix is Hermitian, including when its spectrum contains zero. No condition is imposed on the gap of a Hamiltonian restricted to a smaller region. The uniform boundary assertion requires a single finite constant for each \(q,R,J,\Delta \), chosen before every domain, Hamiltonian, ground vector, and cut.

Fix an integer \(R\ge 0\) and \(J{\gt}0\). For each nonempty \(X\subseteq \Lambda \) with \(\operatorname {diam}_{d_\Lambda }X\le R\), assign one Hermitian operator \(h_X\) supported on \(X\), with \(\| h_X\| \le J\). The assigned operator may be zero, and acts as the identity on the complementary tensor factors. Define

The norm is the Euclidean operator norm. Boundary edges are unordered and counted once; empty and full cuts have empty boundary.

A local circuit of depth \(T\) on a chain of \(N\) sites is a product of \(T\) layers, each a tensor product of unitaries acting on disjoint pairs of neighbouring sites. A state is prepared in depth \(T\) if it equals a local circuit of depth \(T\) applied to a product state.

More generally, a bond geometry on a finite set of sites \(I\) is a family \((B_b)_{b\in \beta }\) of subsets of \(I\), the bonds, and a layer is a product of unitaries acting on pairwise disjoint bonds. The model of [ 227 ] on a lattice takes for bonds the pairs of nearest-neighbour sites, that is the edges of a graph. Local circuits of depth \(T\) and states prepared in depth \(T\) are defined in the same way for every bond geometry.

The chain is a ring with sites \(1,\dots ,N\): the neighbouring pairs are \(\{ k,k+1\} \) with indices modulo \(N\); for \(N=1\) the only pair is \(\{ 1\} \), and for \(N=2\) both pairs are \(\{ 1,2\} \). The open chain has the pairs \(\{ k,k+1\} \) with \(k{\lt}N\). An operator acts on a set of sites \(X\) if it lies in \(M_{d}(\mathbb {C})^{\otimes X}\otimes \mathbb 1\), the span of the products \(\bigotimes _i m_i\) with \(m_i=\mathbb 1\) for \(i\notin X\).

Acting on a set of sites needs no geometry: for any finite set of sites \(I\) and \(X\subseteq I\), an operator on \((\mathbb {C}^d)^{\otimes I}\) acts on \(X\) if it lies in the span of the products \(\bigotimes _{i\in I}m_i\) with \(m_i=\mathbb 1\) for \(i\notin X\). The chain of \(N\) sites is the case \(I=\{ 1,\dots ,N\} \).

The map \(E_K\) on the operators of \((\mathbb {C}^q)^{\otimes I}\) defined above.

For \(y\in I\), define

This is the on-site unitary supremum of [ 247 ] , with every test unitary acting as the identity on the spectator.

For \(a,b\in J\), let \(B_{ab}\) be the physical matrix whose entries are \((B_{ab})_{\sigma ,\tau }=B_{(\sigma ,a),(\tau ,b)}\). Define \(E_K^{\mathcal K}\) by applying \(E_K\) to each such block:

In particular, when \(q{\gt}0\), set \(Q=I\setminus K\) and write \(u\sqcup \rho \) for the configuration equal to \(u\) on \(K\) and to \(\rho \) on \(Q\). Then

Both spectator indices remain unchanged.

Fix \(C,\lambda \ge 0\). There is a threshold depending only on these constants such that, for every natural number \(n\) above the threshold and every real \(t\le n^{-500}\),

This is the continuity cost of [ 247 , Lemma 2.1 ] at the trace distances arising in [ 247 , Proposition 9.2, proof ] .

Fix \(A,C,\lambda \ge 0\). There is a threshold depending only on these constants such that, for every natural number \(n\) above the threshold and every real \(t\le A n^{-500}\),

Here the real square root is zero on negative arguments. This is the scalar estimate used in the entropy transfer of [ 247 , Proposition 9.2, proof ] .

Let \(G_i=(I-k_i)^{1/2}\), \(K_i=k_i^{1/2}\), \(G_{i,l}=(I-k_{i,l})^{1/2}\) and \(K_{i,l}=k_{i,l}^{1/2}\). These are positive contractions, \(G_i\Omega =\Omega \), \(K_i\Omega =0\), \(G_{i,l}^2+K_{i,l}^2=I\), and \(\| G_i-G_{i,l}\| +\| K_i-K_{i,l}\| \le 2\| k_i-k_{i,l}\| ^{1/2} \le 2C^{1/2}e^{-(c/2)l^\alpha }\). The maps \(\mathcal E_i(B)=G_iBG_i+K_iBK_i\) and \(\mathcal E_{i,l}(B)=G_{i,l}BG_{i,l}+K_{i,l}BK_{i,l}\) are unital and completely positive, contractive after adjoining any auxiliary system, and \(\| (\mathcal E_i-\mathcal E_{i,l})\otimes \mathrm{id}\| \le 4\| k_i-k_{i,l}\| ^{1/2}\) uniformly in the auxiliary dimension. The map \(\mathcal E_{i,l}\) fixes every operator commuting with all operators acting on \(N_l(a_i)\), and \(\mathcal E_{i,l}\otimes \mathrm{id}\) fixes every operator commuting with \(A\otimes I\) for all operators \(A\) acting on \(N_l(a_i)\) (Theorem 28.11.1.4).

Let \(A\) be finite, let \(d\) be any integer-valued depth function, and let \(L\ge 0\) be an integer. For \(Q_j=\{ x\in A:0{\lt}d(x)\le j\} \) and \(j_+=\max (j,0)\), one has \(Q_j=Q_{j_+}\). Consequently, every integer cutoff \(j\le L\) selects a member of the fixed nested family

This family depends only on \(A,d,L\) and may contain repeated regions.

Write \(b\) for the fixed anchor multiplicity bound. For \(n,D,r{\gt}0\) and \(M\ge C_1nD\), where \(C_1{\gt}0\), the actual ratio satisfies

If \(n\ge 1\), \(a\ge 0\), \(4rJ\le D\), and \(\rho \le n^{-a}\), then

If moreover \(K,L,m\le n\), \(|T|\le C_Tn^2\), \(C_T\ge 0\), and \(N=nm\), then

For \(n\ge 8(C_T+1)\) and \(p+8\le aJ\), this prefactor times \(n^{-aJ}\) is at most \(n^{-p}\).

If \(R_a\subseteq E\subseteq R_b\), every row of depths \(a+1\) through \(b\) has at most \(n\) sites, and \(c\in \{ a,b\} \), then for any fixed site predicate \(Z\),

The preceding sandwiches have \(w+1=D+r_0+1\le 3D\) uncertainty rows. They give the same \(3nD\) estimate before and after removing the target.

Both \(R_j\) and \(Q_j\) are monotone in the integer cutoff \(j\). Suppose \(n{\gt}0\), \(D\ge 1\), \(r_0\le D\), \(4D\le m\), and \(8Km\le L\). After \(k\le nm\) consumed fills, put \(w=D+r_0\) and let \(j_N,j_F\) be the near and oriented far nominal fronts. Then

The same assertions apply to a pre-charge status using \(k+1\le nm\). All comparison prefixes therefore belong to one nested family, including comparisons from unrelated histories and bands.

Fix \(C_r\ge 0\) and \(r(n)=\lceil C_r(\log n)^2\rceil \). For every \(\varepsilon {\gt}0\), one has \(r(n)=o(n^\varepsilon )\). Consequently, for every fixed real \(A\), both \(Ar(n)\le n^\varepsilon \) and \(Ar(n)^3\le n^\varepsilon \) hold for sufficiently large \(n\). If \(C_r{\gt}0\), then \(r(n)\ge 1\) for sufficiently large \(n\).

If the physical middle has positive depth, then \(Y=UY\setminus U\). For every physical status, \(V=\Lambda \setminus (P_0\cup Y)\).

Fix exponents \(0{\lt}\ell {\lt}1\), \(0{\lt}\kappa {\lt}\mu {\lt}1-\ell \), \(\nu {\gt}0\) and \(0{\lt}e{\lt}1\) with \(\kappa \le e\), and put \(L=\lfloor n^{1-\ell }\rfloor \), \(m=\lfloor n^\mu \rfloor \), \(K=\lfloor L/(8m)\rfloor \), \(D=\lceil n^\kappa \rceil \) and \(\epsilon =n^{-\nu }\), the scales of [ 247 , Section 9 ] . There is a constant \(C_1\ge 1\), depending only on the exponents, such that for all sufficiently large integers \(n\): \(n\ge 2\), \(K\ge 1\), \(m\ge 1\), \(D\ge 1\), \(L\le n\), \(0{\lt}\epsilon {\lt}1\), and

The expectation \(E_K\) onto a set of sites \(K\) preserves adjoints and the order, maps \([0,I]\) into \([0,I]\), and fixes every operator commuting with all operators acting outside \(K\); such an operator acts on \(K\). In particular \(E_K\) fixes \(|Y|\) for every Hermitian \(Y\) acting on \(K\), and \(Y^{1/2}\) acts on \(K\) for every \(Y\ge 0\) acting on \(K\).

If \(\| k_i-k_{i,l}\| \le Ce^{-cl^\alpha }\) for all \(l\), then \(\| \sum _i\widetilde h_i-\sum _ik_i\| \le C'|S_0|e^{-(c/2)r_0^\alpha }\) with \(C'\) depending only on \(C,c,\alpha ,K_b,\mu \).

Adjacent sites of \(\Lambda \) have ambient displacement one, and at most \(2(d+1)^2\) sites lie at distance exactly \(d\) from any site. Every original support has \(d_\Lambda \)-diameter at most \(R\) and at most \(v_R\) sites, and for every site \(x\),

The boundary consists exactly of edges \(\{ p,q\} \) with \(p\in S\), \(q\notin S\), and \(q\in \mathcal N(p)\). The inside–outside orientation is unique, and mapping back from the finite enclosure preserves the number of unordered edges. Each point has at most four neighbors; a nearest-neighbor step from \(S_r\) lies in \(S_{r+1}\). The empty set has empty boundary.

The neighbor relation is symmetric. If a finite set \(R\) meets every edge of \(\partial S\), then \(|\partial S|\le 4|R|\). In particular,

Let \(H=\sum _jh_j\) with \(h_j\in \mathcal A_{T_j}\), and let \(C\) be a set of sites such that each \(T_j\) is contained in \(C\) or disjoint from \(C\). If \(A\in \mathcal A_C\) and \(B\in \mathcal A_D\) with \(C\cap D=\varnothing \), then \([\tau _t(A),B]=0\) for every \(t\).

Let \(q\geq 1\), and for \(p=(a,b)\) let \(W(p)=X^aZ^b\) be the Weyl operators on \(\mathbb {C}^q\). Then \(E_K(B)\) is the uniform average of \(W_pBW_p^\dagger \) over all products \(W_p=\bigotimes _{x\notin K}W(p_x)\) of Weyl operators outside \(K\).

For \(q\geq 1\), the map \(E_K\) takes values in \(\mathcal A_K\), fixes every operator in \(\mathcal A_K\), is unital, completely positive and trace preserving, and is a contraction in operator norm.

For every \(i\in F\), let \(k_i\) be a positive contraction supported on \(C(a_i)\) and put \(\mathcal E_i=\mathcal E_{k_i}\). Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\). Take \(A=2+8\sqrt C\exp (c/2)\) and use the kernel (279) with \(D=2A\) and \(b=c/2\). Fix \(N\in \mathbb {N}\) with \(N\ge R_{a_i}\) for every \(i\in F\). Suppose that \(\| k_i-E_{N_l(a_i)}(k_i)\| \le C\exp (-cl^\alpha )\) for every \(i\in F\) and every integer \(l\) with \(0\le l\le N\). Then, for every \(y\in I\) and every operator \(B\),

The spectator is any finite-dimensional space, and \(B\) is any complex operator on the physical space together with that spectator.

For every choice of real constants \(C,K,\mu \ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\), there is a number \(R\ge 0\) with the following property. Choose any finite site set \(I\), finite label set \(F\), finite spectator index set \(J\), and local dimension \(q\in \mathbb {N}\) with \(q{\gt}0\). Let \(G\) be any simple graph on \(I\) and \((a_i)_{i\in F}\) any anchor family. For each \(i\in F\), let \(k_i\) be a positive contraction on \((\mathbb {C}^q)^{\otimes I}\) supported on \(C(a_i)\), and put \(\mathcal E_i=\mathcal E_{k_i}\). Assume the ball bound (281) for every site and radius, and the anchor-fiber bound (282) for every site. Set \(A=2+8\sqrt C\exp (c/2)\), and use (279) with \(D=2A\) and \(b=c/2\).

For every cutoff \(N\in \mathbb {N}\) satisfying \(N\ge R_{a_i}\) for all \(i\in F\), assume that \(\| k_i-E_{N_l(a_i)}(k_i)\| \le C\exp (-cl^\alpha )\) for all \(i\in F\) and all integers \(l\) with \(0\le l\le N\). Then the following two conclusions hold. Every \(y\in I\) satisfies

For every \(y\in I\) and every complex operator \(B\) on \((\mathbb {C}^q)^{\otimes I}\otimes \mathbb {C}^J\),

The constant \(R\) depends only on \(C,K,\mu ,c,\alpha \). It is chosen before \(I,F,J,q,G\), the anchors, the effects \(k_i\), the cutoff, and the input operator.

Suppose \(1-\operatorname {tr}(\rho _AP_A^w)\le \delta \) for \(A=X,Y,S_0\), and put \(\mathcal I=I_\Omega (X:Z')\) and \(M_0=\exp \bigl((\mathcal I+3w)/2\bigr)\). Then

the last for every physical operator \(L'\).

If \(1-\operatorname {tr}(\rho _AP_A^w)\le \delta \) for \(A=X,Y,S_0\), then each of \(P_X\), \(T_Y\) and \(Q\) moves \(\Omega \) by at most \(\sqrt\delta \).

Fix \(q\ge 1\), a range \(R\), a strength \(J\ge 0\), a gap \(\Delta {\gt}0\), \(C_0\ge 0\) and an integer \(K\ge 0\). There is a threshold \(N\), depending only on these data, such that the following holds for every \(n\ge N\), every range-\(R\) interaction of strength \(J\) on a finite induced lattice domain with a unit ground vector \(\Omega \) of full-system gap \(\Delta \), and every disjoint \(X,Y\) such that \(X\), \(Y\) and \(S_0\) have at most \(C_0n\) crossing edges each. With \(w=n^{3/5}\), the estimates of Theorem 28.20.5 hold with \(3\sqrt\delta =3n^{-K}\).

For every vector \(\Omega \) and disjoint \(X,Y\),

The projection \(Q\) commutes with \(\rho _{S_0}\), and

Fix \(q\ge 1\), a range \(R\), a strength \(J\ge 0\), a gap \(\Delta {\gt}0\) and \(C_0\ge 0\). There are constants \(C,c{\gt}0\) with the following property. Let \(n\ge 1\), let \(H\) be a range-\(R\) interaction of strength \(J\) on a finite induced lattice domain \(\Lambda \), and let \(\Omega \) be a unit ground vector with full-system gap \(\Delta \). Then every cut \(A\subseteq \Lambda \) with \(|\partial _\Lambda A|\le C_0n\) satisfies

Let \(\Omega \) be a unit vector, \(A\subseteq \Lambda \) and \(w\) real. Then \(P_A^w\) is an orthogonal projection commuting with \(\rho _A\). Every selected eigenvalue satisfies \(p_j{\gt}0\) and \(\mathrm e^{-S_\Omega (A)-w}\le p_j\le \mathrm e^{-S_\Omega (A)+w}\). With \(K\) as in Theorem 27.118.3,

Fix \(q\), \(R\), \(J\), \(\Delta \) and \(C_0\) as in Theorem 28.19.3, and an integer \(K\ge 0\). There is a threshold \(N\), depending only on these data, such that the following holds for every \(n\ge N\), every such \(\Lambda \), \(H\) and \(\Omega \), and every three cuts \(X\), \(Y\), \(S_0\) with at most \(C_0n\) crossing edges each. Put \(w=n^{3/5}\) and \(P_X=P_X^w\), \(T_Y=P_Y^w\), \(Q_{S_0}=P_{S_0}^w\). Each of these is an orthogonal projection commuting with its reduced state, its selected eigenvalues are positive and lie in \([\mathrm e^{-S_\Omega (A)-w},\mathrm e^{-S_\Omega (A)+w}]\) for its cut \(A\), and each fails with probability at most

This is the construction of [ 247 , Section 10 ] preceding the Schmidt expansion of \(\Omega \).

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.

For either \(b\in \{ \mathrm{false},\mathrm{true}\} \), let \(\Pi _{\{ b\} ,\lambda }\) denote the central label projection for the copy action at the single auxiliary site \(b\). Then

The label is the same supplied \(\lambda \) at both sites, including the zero vector and zero-dimensional or zero-copy cases.

Let \(P_{\mathrm{sym}}\) average the simultaneous permutations of the \(k\) copies of \(\mathcal X_J\). Then

No normalization of \(\Omega \) or positivity of \(q,d,k\) is required.

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 \(k=0\).

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 (173) 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.

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.

Let \(0\le k\le \mathbb 1\), let \((K_l)_{l\ge 0}\) be increasing sets of sites, let \(C\ge 0\), \(c{\gt}0\) and \(0{\lt}\alpha \le 1\), and suppose \(\| k-E_{K_l}k\| \le Ce^{-cl^\alpha }\) for \(0\le l\le n\). With \(A=2+8\sqrt Ce^{c/2}\), every \(B\) satisfies

For graph balls \(K_l=N_l(a)\), if \(k\) acts on the component of \(a\) and \(n\) is at least the largest distance from \(a\) in that component, the left side is \(\| \mathcal E(B)-B\| \) for the full channel of \(k\).

Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\), and take \(A,b\) from (265). Fix \(n\in \mathbb {N}\) and \(y\in I\). Suppose that \(\| k-E_{K_l}(k)\| \le C\exp (-cl^\alpha )\) for every integer \(l\) with \(0\le l\le n\). Then

The estimate holds uniformly over all finite spectators and all complex operators \(B\) under the stated finite set of error bounds.

Fix \(n\in \mathbb {N}\), \(y\in I\), and a real sequence \((\varepsilon _l)\). Suppose that \(\| k-E_{K_l}(k)\| \le \varepsilon _l\) for every integer \(l\) with \(0\le l\le n\). Then

Every regional sum is evaluated on the original operator \(B\). This is the finite-radius one-event estimate underlying [ 247 , Section 10 ] .

Under the effect, component support, tail and cutoff hypotheses of Theorem 28.14.8, put \(\beta =c/(4\cdot 2^\alpha )\) and use the actual kernel \(\mathsf A^{(N)}\) with \(D=2(2+8\sqrt C e^{c/2})\) and \(b=c/2\). Suppose its weighted rows satisfy (322) for some \(v\ge 0\). For every initial operator \(B\) whose physical blocks are supported on \(S\), all conclusions of Theorem 28.15.1 hold.

Suppose \(q{\gt}0\), let \(G\) be a simple graph on the finite physical site set \(I\), and let \(A_{yz}\ge 0\) be a real kernel. Assume that, for every operator \(W\) and every \(y\in I\),

Suppose \(A_{yz}=0\) when \(\rho _G(y,z)=\infty \), and suppose \(\beta \ge 0\), \(0{\lt}\alpha \le 1\), and \(v\ge 0\) satisfy

for every \(y\in I\). Let \(S\subseteq I\), and suppose every physical matrix block of \(B\) is supported on \(S\). Then \(w\mapsto d_y(B_w)\) is integrable for every \(t\ge 0\) and every \(y\in I\), with

If \(d_G(y,S)=\infty \), then \(d_y(B_w)=0\) for every word \(w\). The input \(B\) may be non-Hermitian and the spectator space may have dimension zero.

Under the hypotheses of Theorem 28.15.3, sample a word \(w\) from the full event alphabet \(F\), and retain only the events satisfying \(P\). Then \(w\mapsto d_y(\mathcal E_{w|_P}(B))\) is integrable under \(\nu _{F,t}\), and the same constants \(c{\gt}0\), \(v\ge 0\) satisfy

Here \(\alpha =p/(p+1)\), \(\beta =c/(4\cdot 2^\alpha )\), and \(H_S\) is (320). If the component of \(y\) does not meet \(S\), the filtered observable has zero oscillation at \(y\) for every full word. The constants precede the choice of domain, local dimension, Hamiltonian, retained predicate, spectator and supported operator \(B\).

Fix \(p\ge 1\), interaction radius \(R\in \mathbb {N}\), interaction bound \(J\ge 0\), and gap \(\Delta {\gt}0\). There are \(c{\gt}0\) and \(v\ge 0\) depending only on \(p,R,J,\Delta \) with the following property. Take any local dimension \(q{\gt}0\), finite lattice domain \(\Lambda \), Hamiltonian, anchors and gapped ground vector satisfying the hypotheses of Theorem 28.13.8.2. Retain any subfamily \(F_P\) of its event labels, and use its actual positive constraints \(k_i\), formed from the original Hamiltonian. Put \(\alpha =p/(p+1)\) and \(\beta =c/(4\cdot 2^\alpha )\). For every finite spectator, support \(S\subseteq \Lambda \), operator \(B\) whose physical blocks are supported on \(S\), and time \(t\ge 0\), the retained chronological word satisfies

The oscillation is integrable. It vanishes at every word when the component of \(y\) does not meet \(S\). The same \(c,v\) work for all domains, local dimensions, retained families, spectators and initial operators, including empty retained families and empty supports.

Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\). Set \(\varepsilon _l=C\exp (-cl^\alpha )\) and

For every \(l\in \mathbb {N}\),

For every real sequence \((\varepsilon _l)\) and every \(l\ge 0\), the coefficient \(\eta _l\) is nonnegative.

Suppose the regions are increasing: \(K_l\subseteq K_m\) whenever \(l\le m\). If \(B\) commutes with \(\widehat A\) for every \(A\in \mathcal A_{K_l}\), then \(\Delta _l(B)=0\).

Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\), and take \(A,b\) from (265). Fix \(l\in \mathbb {N}\). Suppose that, for every integer \(j\) with \(0\le j\le l\),

Then

Fix \(l\in \mathbb {N}\) and a real sequence \((\varepsilon _j)\). Suppose that, for every integer \(j\) with \(0\le j\le l\),

Then

In particular, if \(\| k-E_{K_j}(k)\| =0\) for every \(0\le j\le L\), then \(\Delta _j(B)=0\) for \(1\le j\le L\). The zeroth shell still has coefficient \(\eta _0=2\).

For every sequence of regions, \(\| \Delta _0(B)\| \le 2\| B\| \). Fix \(l\ge 0\) and real numbers \(\varepsilon _{\rm next}, \varepsilon _{\rm prev}\) such that

Then

Suppose \((K_l)_{l\ge 0}\) is increasing. Then

Fix \(l\ge 0\) and real numbers \(\varepsilon _{\rm next}, \varepsilon _{\rm prev}\) satisfying (246) and (247). Then

These bounds hold for every finite spectator and every complex operator \(B\), without a Hermiticity assumption.

For every operator \(B\), every finite word \(w\), and every label \(i\),

In particular, \(B_{\varnothing }=B_0\) and \(B_{wi}=\mathcal E_i(B_w)\).

Suppose \(0\le k_i\le \mathbb 1\) for every label of a finite set \(F\), and put \(F_P=\{ i\in F:P(i)\} \). Then \(w\mapsto D_P(w;B)\) is integrable under \(\nu _{F,t}\), and its integral is at most

Let \(P\) be a predicate on the event alphabet \(F\), and let \(F_P=\{ i\in F:P(i)\} \). For a word \(w\) over \(F\), let \(\pi _P(w)\) be the word over \(F_P\) obtained by retaining exactly the letters satisfying \(P\), in their original order and with their original multiplicities. Write \(w|_P\) for the same list regarded as a word over \(F\). Then

Let \(F\) be finite and \(t\geq 0\). Let \(\nu _{F,t}\) be the probability measure on finite words over \(F\) assigning to each word \(w\) of length \(m\) the mass

Let \(P\) be a predicate on \(F\), and let \(\Phi \) be a function from operators on \(\mathcal H\otimes \mathcal K\) to a normed real vector space. The function \(w\mapsto \Phi (\mathcal E_{w|_P}(B_0))\) is integrable under \(\nu _{F,t}\) if and only if \(u\mapsto \Phi (\mathcal E^P_u(B_0))\) is integrable under \(\nu _{F_P,t}\), and their Bochner integrals satisfy

In particular, \(\Phi \) may be the identity on matrices equipped with the Euclidean operator norm, or the site oscillation \(B\mapsto d_y(B)\).

Fix \(y\in I\) and \(R\in \mathbb {R}\). If \(u\mapsto d_y(\mathcal E^P_u(B_0))\) is integrable under \(\nu _{F_P,t}\) and its integral is at most \(R\), then \(w\mapsto d_y(\mathcal E_{w|_P}(B_0))\) is integrable under \(\nu _{F,t}\) and

Suppose \(0\le k_i\le \mathbb 1\) and \(k_i\Omega =0\) for every \(i\in F\), where \(\Omega \in \mathcal H\). Put \(G_i=(\mathbb 1-k_i)^{1/2}\), and, for a chronological word \(w=(i_0,\ldots ,i_{m-1})\), put \(R_w=G_{i_{m-1}}\cdots G_{i_0}\), with \(R_{\varnothing }=\mathbb 1\). For every \(\xi \in \mathcal K\) and every operator \(B\) on \(\mathcal H\otimes \mathcal K\), one has

This is the finite composition identity in [ 247 , Section 10 ] . Neither vector need be normalized, and \(B\) need not be Hermitian.

Suppose \(q{\gt}0\) and \(F\) is finite. Let \(G\) be a simple graph on \(I\), and choose anchors \(a_i\in I\) for \(i\in F\). Assume \(0\le k_i\le \mathbb 1\) and that \(k_i\) is supported on the connected component of \(a_i\) for every \(i\). Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\). Fix a cutoff \(N\in \mathbb {N}\) with \(N\ge R_{a_i}\) for every \(i\), where \(R_{a_i}\) is the maximum distance from \(a_i\) within its component. Suppose that, for every \(i\in F\) and every integer \(l\) with \(0\le l\le N\),

Use the kernel (279) with \(D=2(2+8\sqrt C\exp (c/2))\) and \(b=c/2\). Then, for every finite word \(w\), every \(y\in I\), and every initial operator \(B_0\),

Every oscillation on the right is evaluated at the same current observable \(B_w\). Labels with the same anchor remain distinct summands. The estimate holds for arbitrary finite spectators and arbitrary complex initial operators \(B_0\).

Suppose \(0\le k_i\le \mathbb 1\) for every \(i\in F\). Then \(\| \mathcal E_w(B)\| \le \| B\| \) for every finite word \(w\) and every operator \(B\).

Suppose \(0\le k_i\le \mathbb 1\) for every label. Every \(\mathcal E_w\) is linear and contracts the norm of differences, and \(0\le D_P(w;B)\le 2\| B\| \). Appending a label \(i\) to \(w\) raises the defect by at most \(0\) when \(P(i)\) holds, and by at most \(\| (\mathcal E_i-\mathrm{id})(\mathcal E_{w|_P}(B))\| \) otherwise.

Suppose \(0\le k_i\le \mathbb 1\) for every label. With \(w_{{\lt}j}\) the prefix of length \(j\),

Suppose \(0\le k_i\le \mathbb 1\) for every \(i\in F\). For every finite word \(w\), site \(y\in I\), and initial operator \(B_0\), one has \(0\le d_y(B_w)\le 2\| B_0\| \).

Let \(E\) be any label set and let \(f:E\to F\) be any map. Define \(\widetilde k_j=k_{f(j)}\) and \(\widetilde{\mathcal E}_j=\mathcal E_{f(j)}\) for \(j\in E\). For a word \(w\) over \(E\), let \(f(w)\) be the word obtained by applying \(f\) to each label. Then, for every operator \(B\),

The map \(f\) may identify distinct labels.

Under the hypotheses and notation of Theorem 28.14.4, let \(u,w\) be any two finite words. Then

Vector norms are Euclidean and the matrix norm is the induced operator norm. This is the pointwise conversion used in [ 247 , 09-amplification.tex, lines 218–224 ] .

If \(\| k-E_{N_l(a)}k\| \le Ce^{-cl^\alpha }\) for every \(l\), with \(c,\alpha {\gt}0\), then \(k\) acts on the component of \(a\).

In the setting of Theorem 28.17.1, let the alphabet be finite and let \(\mu _t\) be the Poisson-word law at time \(t\ge 0\). Fix a set \(P\) of retained labels, and for a full word \(w\) let \(w_P\) be its subword of retained letters, in their original order. Let \(k_i,k_i'\) be given for \(i\in P\), with roots within \(\varepsilon \) of each other. Then \(w\mapsto \| D_{w_P}-D_{w_P}'\| \) is integrable and

Let \(k_i\) and \(k_i'\) be positive operators on a finite-dimensional space, indexed by a common alphabet. For a word \(w=(i_0,\ldots ,i_{m-1})\) put \(D_w=(I-k_{i_{m-1}})^{1/2}\cdots (I-k_{i_0})^{1/2}\), and define \(D_w'\) in the same way from the \(k_i'\). Then \(\| D_w\| \le 1\). If \(\| (I-k_i)^{1/2}-(I-k_i')^{1/2}\| \le \varepsilon \) for every label \(i\), then \(\| D_w-D_w'\| \le m\varepsilon \).

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 \(C,\mu \ge 0\), \(c{\gt}0\) and \(0{\lt}\alpha \le 1\). There are \(A\ge 0\) and \(v,\gamma {\gt}0\) with the following property. Let \(0\le k_i\le \mathbb 1\) on a finite domain \(\Lambda \), anchored at \(a_i\) with at most \(\mu \) labels per site, satisfy \(\| k_i-E_{N_l(a_i)}k_i\| \le Ce^{-cl^\alpha }\) for all \(l\ge 0\). Let \(B\) act on \(S\) and a finite spectator, and let every label outside \(P\) be at distance more than \(r\ge 0\) from \(S\). Then \(w\mapsto D_P(w;B)\) is integrable and

This is the coupled channel estimate in the proof of the local constraint products lemma.

Under the hypotheses of Theorem 28.18.3.2, with the same constants, suppose also \(k_i\Omega =0\) for a unit vector \(\Omega \). Fix an integer \(r\ge 0\) and retain the labels with \(d_\Lambda (a_i,S)\le r\). On each realization, let \(D\) be the product of all \(G_i\) in occurrence order, later factors on the left, and let \(D_{\rm ret}\) be the product over the retained events. For every unit spectator vector \(\xi \),

This is the omission estimate of the local constraint products lemma. The source takes \(S\) nonempty and \(r\ge 1\); neither is needed.

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.

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.

Let \(V\) be a finite set of physical sites of local dimension \(q\), let \(\Omega \) be a unit vector, and fix \(X\subseteq V\). Select a spectral set \(E\) of the reduced density on \(X\) with mass \(z{\gt}0\), and put \(d=|E|\). Let \(\psi \) be its normalized selected Schmidt vector. Suppose that a supplied label sequence \(\lambda _k\) satisfies

for every sufficiently large \(k\). Form the original physical prevector \(\xi _k\) from \(\Omega \) and these labels, as in Theorem 28.1.3.

Fix a finite index set \(J\), a finite nested list \(\mathcal R\) of physical regions disjoint from \(X\), and \(K\geq 8\). There is a constant \(c\geq 0\) for which the following holds eventually in \(k\). Choose \(Z\) with \(Z^*Z=I\) and \(ZZ^*=P_{\mathrm{sym}}\), physical regions \(U_{jg},Y_{jg}\) for \(j\in J\) and \(0\leq g{\lt}K\), and an iterated weighted geometric mean \(T\). Assume

For \(g{\lt}h\), also assume \(U_{jg}\cup Y_{jg}\subseteq U_{j'h}\). Suppose that \(S_\Omega (D)\leq B_{\mathrm{sh}}\) for every \(D\in \mathcal R\). For each \(Q=U_{jg}\) or \(Q=U_{jg}\cup Y_{jg}\), suppose that some \(D\in \mathcal R\) is nested with \(Q\) and satisfies

At parameter \(t=2/K\), use the full physical and auxiliary dimension in the replica metric \(W\). With auxiliary sites \(C\) and \(R\), put

Complements are taken in \(V\). Then \(\| w_k\| ^2{\gt}0\) and

The constant and eventual threshold precede all the choices of \(Z,U,Y,T,B_{\mathrm{sh}}\) and \(B_{\mathrm{exc}}\). This is the supplied-label application of the rough upper comparison in [ 247 , Section 8, lines 130–169 and 332–381 ] .

Let labels on a graph be anchored at \(a_i\) with at most \(\mu \) labels per site. For every site \(y\) and every \(l\ge 0\), at most \(\mu |N_l(y)|\) labels satisfy \(y\in N_l(a_i)\).

Suppose \(0\le k\le \mathbb 1\) and \(k\in \mathcal A_{C(a)}\). Let \(C\ge 0\), \(c{\gt}0\), \(0{\lt}\alpha \le 1\), and take \(A,b\) from (265). Fix an integer \(n\ge R_a\). Suppose that \(\| k-E_{N_l(a)}(k)\| \le C\exp (-cl^\alpha )\) for every integer \(l\) with \(0\le l\le n\). Then, for every \(y\in I\),

All regional sums use the original operator \(B\).

Let \(C,K,\mu \ge 0\), \(c{\gt}0\) and \(0{\lt}\alpha \le 1\). There are \(D\ge 0\) and \(v,\gamma {\gt}0\) with the following property. Let \(G\) be a graph on a finite set with \(|N_l(x)|\le K(l+1)^2\), and let \(0\le k_i\le \mathbb 1\) be anchored at \(a_i\) with at most \(\mu \) labels per site, acting on the component of \(a_i\), with \(\| k_i-E_{N_l(a_i)}k_i\| \le Ce^{-cl^\alpha }\) for \(l\) up to a cutoff beyond all component distances. Let \(B\) act on \(S\) and a spectator, and let every label outside \(P\) be at distance more than \(r\ge 0\) from \(S\). Then

For each \(i\in F\), let \(k_i\) be a positive contraction supported on \(C(a_i)\), and let \(\mathcal E_i=\mathcal E_{k_i}\) be its full root channel. Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\), and use \(A,b\) from (265). Fix a common cutoff \(n\in \mathbb {N}\) satisfying \(n\ge R_{a_i}\) for every \(i\in F\). Suppose that, for every \(i\in F\) and every integer \(l\) with \(0\le l\le n\),

Then, for every \(y\in I\),

Each channel and every site oscillation on the right act on the same original operator \(B\). The estimate holds for arbitrary finite spectators and arbitrary complex \(B\). This is the finite event-label sum in [ 247 , Section 10 ] .

For the finite graph and label family in Definition 28.13.6.1, suppose \(D\ge 0\). Then \(\mathsf A^{(N)}_{yz}\ge 0\) for every real \(b,\alpha \), every cutoff \(N\in \mathbb {N}\), and all sites \(y,z\).

For every \(x,y\in V\), one has \(\rho _G(x,y)\ne \infty \) if and only if a path in \(G\) connects \(x\) to \(y\). Equivalently, \(\rho _G(x,y)=\infty \) if and only if \(y\) is unreachable from \(x\).

For every \(x,y\in V\),

where \(d_G^0(x,y)\) is the ordinary graph distance for reachable pairs and zero for unreachable pairs.

If \(y,z\) are in different connected components of \(G\), then \(\mathsf A^{(N)}_{yz}=0\) for every cutoff \(N\) and every choice of real \(D,b,\alpha \).

Let \(D,K,\mu \ge 0\) and \(b,\alpha {\gt}0\), and set

Suppose that every site \(x\in I\) and radius \(l\in \mathbb {N}\) satisfy

and that every site \(x\in I\) satisfies

Then, for every cutoff \(N\in \mathbb {N}\) and site \(y\in I\),

For every choice of real constants \(D,K,\mu \ge 0\) and \(b,\alpha {\gt}0\), there exists \(V\ge 0\) with the following property. For every finite site set \(I\), every finite label set \(F\), every simple graph \(G\) on \(I\), and every anchor family \((a_i)_{i\in F}\), assume that (281) holds for all sites and radii and (282) holds for all sites. Then, for every \(N\in \mathbb {N}\) and \(y\in I\),

Thus \(V\) depends only on \(D,K,\mu ,b,\alpha \), and is chosen before the site set, label set, graph, anchors, cutoff, and site \(y\). This is the finite-kernel weighted row estimate of [ 247 , Section 10 ] .

Let \((w_l)_{l\ge 0}\) be any real sequence and let \((v_z)_{z\in I}\) be any family of real numbers. For every \(n\in \mathbb {N}\) and \(y\in I\),

Both the weights \(w_l\) and the site values \(v_z\) may have either sign.

Fix integers \(p\ge 1\) and \(R\ge 0\), and real constants \(J\ge 0\) and \(\Delta {\gt}0\). Put \(\alpha =p/(p+1)\). There exist \(C\ge 0\), \(c{\gt}0\), and \(v\ge 0\), depending only on \(p,R,J,\Delta \), with the following property.

Choose any local dimension \(q\in \mathbb {N}\) with \(q{\gt}0\) and any finite domain \(\Lambda \subseteq \mathbb {Z}^2\), with its induced nearest-neighbour graph \(G_\Lambda \). Let \(\mathcal F_{\Lambda ,R}\) consist of all nonempty supports \(X\subseteq \Lambda \) of diameter at most \(R\) in this graph. For each \(X\in \mathcal F_{\Lambda ,R}\), let \(h_X\) be Hermitian, supported on \(X\), and satisfy \(\| h_X\| \le J\). Put \(H=\sum _{X\in \mathcal F_{\Lambda ,R}}h_X\) and choose an anchor \(a_X\in X\) for every support. Let \(E_0\in \mathbb {R}\) and let \(\Omega \) be a unit vector such that \(H\Omega =E_0\Omega \) and \(H-E_0\mathbb 1\ge \Delta (\mathbb 1-|\Omega \rangle \langle \Omega |)\).

Use the spectral-filter time kernel \(f=f_{p,\Delta /2}\) and the positive constraints of Theorem 28.2.2:

with \(c_*=\max \{ 1,J(\| f\| _1+1)\} \) and \(k_X=c_*^{-1}|M_X|\). Write \(\mathcal E_X=\mathcal E_{k_X}\) for the full root channel.

Choose any predicate \(P\) on \(\mathcal F_{\Lambda ,R}\) and put \(F_P=\{ X\in \mathcal F_{\Lambda ,R}:P(X)\} \). Every retained \(k_X\) uses the original full Hamiltonian \(H\) and ground vector \(\Omega \) in (296). Take the kernel (279) on \(G_\Lambda \), with label set \(F_P\), the restricted anchors, \(D=2(2+8\sqrt C\exp (c/2))\), and \(b=c/2\). Denote this kernel by \(\mathsf A^{(N,P)}\). Its incidence counts are

Choose any finite spectator index set \(T\) and any cutoff \(N\in \mathbb {N}\) such that \(N\ge R_{a_X}\) for every \(X\in F_P\), where \(R_{a_X}\) is the maximum distance from \(a_X\) inside its component in \(G_\Lambda \). Then, for every \(y\in \Lambda \),

For every \(y\in \Lambda \) and every complex operator \(B\) on \((\mathbb {C}^q)^{\otimes \Lambda }\otimes \mathbb {C}^T\),

The constants \(C,c,v\) are chosen before \(q\), the domain, Hamiltonian, anchors, ground vector, retained predicate, spectator, and cutoff. The gap assumption concerns the original full Hamiltonian \(H\). This applies [ 247 , Proposition 4.3 ] to the finite channel estimate in [ 247 , Section 10 ] .

Let \(p\ge 1\), \(R\), \(J\ge 0\) and \(\Delta {\gt}0\), and put \(\alpha =p/(p+1)\). There are \(A\ge 0\) and \(v,\gamma {\gt}0\) such that the conclusion of Theorem 28.18.3.3 holds for the exact positive constraints of every gapped lattice Hamiltonian with ground state \(\Omega \), with any anchoring \(a_i\in X_i\).

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 27.118.6), 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.

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 ] .

Let \(\Lambda \) be a finite induced lattice domain with local dimension \(q\ge 1\), and let \(H\) be a range-\(R\) interaction of strength \(J\ge 0\). Suppose that \(\Omega \) is a unit ground vector of energy \(E_0\) with full-system gap \(\Delta {\gt}0\). Fix an integer \(p\ge 1\), and let \(k_i=|M_i|/c_*\) be its exact positive constraints from Theorem 28.2.2. Put \(g=\Delta /c_*\), \(P_\Omega =|\Omega \rangle \langle \Omega |\) and \(G_i=(I-k_i)^{1/2}\).

For \(t\ge 0\), let \(\mu _t\) be the probability law of independent Poisson counts of mean \(t\) for the interaction labels, followed by a uniformly ordered word with those counts. For a chronological word \(w=(i_0,\ldots ,i_{m-1})\), set \(D_w=G_{i_{m-1}}\cdots G_{i_0}\) and \(D_{\varnothing }=I\). For every finite-dimensional spectator space \(\mathcal K\) and every \(\zeta \in \mathcal H_\Lambda \otimes \mathcal K\), put \(\xi =((I-P_\Omega )\otimes I)\zeta \). The squared norm \(\| (D_w\otimes I)\xi \| ^2\) is integrable and satisfies

This is the fixed-time excited-state estimate in [ 247 , Section 10 ] .

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 ] .

Fix an integer \(p\ge 1\), a range \(R\), a strength \(J\ge 0\) and a gap \(\Delta {\gt}0\), and put \(\alpha =p/(p+1)\). There are constants \(C\ge 0\) and \(c{\gt}0\) with the following property. Let \(\Lambda \) be a finite induced lattice domain with local dimension \(q\ge 1\), let \(H\) be a range-\(R\) interaction of strength \(J\), and let \(\Omega \) be a ground vector with full-system gap \(\Delta \). Let \(k_i\) be the exact positive constraints of Theorem 28.2.2, anchored at \(a_i\in X_i\), and let \(E_{i,r}\) be the conditional expectation onto \(N_r(a_i)\). Fix \(S\subseteq \Lambda \), an integer \(r\ge 0\) and \(t\ge 0\), and retain the labels with \(d_\Lambda (a_i,S)\le r\). For a full Poisson word \(w\), let \(D_{\rm ret}(w)\) be the chronological product of the roots \((I-k_i)^{1/2}\) over its retained events, and let \(L(w)\) be the same product of \((I-E_{i,r}k_i)^{1/2}\). Then \(\| D_{\rm ret}-L\| \) is integrable for the Poisson-word law \(\mu _t\),

and every \(L(w)\) is a contraction supported on \(N_{2r}(S)\). The constants depend only on \(p,R,J,\Delta \). This is the first estimate of [ 247 , Lemma 10.1 ] , together with its contraction and support clause, without the assumptions that \(S\) is nonempty and \(r\ge 1\).

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 \(C_0{\gt}0\).

There is a constant \(C_K\ge 1\) such that all sufficiently large integers \(n\) satisfy \(n\ge 2\), \(K\ge 1\), and \(n^\alpha \le C_KK\).

Use the final parameters of [ 247 , Section 9, lines 839–865 ] :

Their exact margin is

Fix \(C_B\ge 0\), \(c{\gt}0\) and \(t\in \mathbb R\). There is \(N\ge 2\) such that, for every \(n\ge N\), the literal rounded scales in (208) have \(K\ge 1\) and, with \(a=n^\omega /K\), every \(1\le B\le C_BnD(\log n)^{12}\) satisfies \(|ta|\le c/\sqrt B\).

Let \(n\ge 1\), \(C_B\ge 0\) and \(B\in \mathbb R\) satisfy \(B\le C_BnD(\log n)^{12}\). Then

The real square root is zero on negative arguments.

Fix \(\omega \in \mathbb R\), \(C_B,A\ge 0\) and \(c{\gt}0\). Assume the strict margin

There is an integer \(N\ge 2\), chosen before \(n,W,B\), such that every \(n\ge N\) has \(K\ge 1\) and the following property. For every \(0\le W\le n^\omega \) and real \(B\le C_BnD(\log n)^{12}\), put \(a=W/K\). Then \(Aa\sqrt B\le c\). If also \(B\ge 1\), then \(Aa\le c/\sqrt B\).

Let \(K,\mu \ge 0\), \(b,\beta {\gt}0\) and \(0{\lt}\alpha \le 1\). There is \(L\ge 0\) with the following property, with \(\gamma =\min (b,\beta )/2\). Let \(G\) be a graph on a finite set whose balls satisfy \(|N_l(x)|\le K(l+1)^2\), and let labels be anchored at \(a_i\) with at most \(\mu \) labels per site. Put \(p(z)=e^{-\beta d(z,S)^\alpha }\), and \(p(z)=0\) when \(z\) is not connected to \(S\). If every label in a set \(E\) is at distance more than \(r\ge 0\) from \(S\), then for every \(N\),

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 \(F\) be finite, let \(P\) be a predicate on \(F\), and put \(F_P=\{ i\in F:P(i)\} \). Suppose a real function \(H(u,i)\), for words \(u\) over \(F_P\) and \(i\in F\), satisfies \(0\le H(u,i)\le M\) for a real constant \(M\). For \(t\ge 0\), the sum over omitted positions is integrable under \(\nu _{F,t}\) and

Let \(f\) be a real function on finite words over any alphabet. For a word \(w=(i_0,\ldots ,i_{m-1})\), let \(w_{{\lt}j}\) be its first \(j\) letters. Then

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.

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 \(a_i\).

Let \(\Lambda \) be a finite induced lattice domain, \(R\ge 0\), and let each admissible support \(X_i\) of range \(R\) carry an anchor \(a_i\in X_i\). Let \(\mu _R\) be the anchor multiplicity bound of Theorem 28.2.1. For every \(S\subseteq \Lambda \) and \(r\ge 0\), if \(d_\Lambda (a_i,S)\le r\) then \(N_r(a_i)\subseteq N_{2r}(S)\), and at most \(2\mu _R|S|(1+r)^2\) labels satisfy \(d_\Lambda (a_i,S)\le r\).

Let \(F\) be a finite label set, let \(V\) be any set, and let \(a:F\to V\) assign an anchor to every label. For a predicate \(P\) on \(F\), put \(F_P=\{ i\in F:P(i)\} \). For every \(x\in V\),

Distinct retained labels at the same anchor remain distinct.

If \(B\) commutes with every \(\widehat C\) for \(C\in \mathcal A_K\), then \(\mathcal E_K(B)=B\). If an operator \(A\) on \(\mathcal H\otimes \mathcal K\) commutes with every such \(\widehat C\), then

In particular, every physical \(U\in \mathcal A_{I\setminus K}\) satisfies \([\widehat U,\mathcal E_K(B)] =\mathcal E_K([\widehat U,B])\).

The full and localized channels satisfy

Suppose \(\| k-E_K(k)\| \le \varepsilon _K\) for a real number \(\varepsilon _K\). Then

If also \(\| k-E_L(k)\| \le \varepsilon _L\), then

The sets \(K,L\) need not be nested. Both bounds are uniform over all finite spectator spaces.

If \(y\notin K\), then \(d_y(\mathcal E_K(B))\le d_y(B)\).

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.

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.

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.

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 28.3.69. 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 (108),

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 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 (152) 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.

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 28.3.85. 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.

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.

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 28.3.78, 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 28.3.78. Then

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.

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.

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.

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.

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.

Fix a collar scan with positive offset and charge capacities, positive physical and auxiliary dimensions, and \(a{\gt}0\). There exists \(C\ge 0\), chosen before the number \(k\ge 0\) of replicas and the vector \(\xi \), such that \(\| \xi \| \le 1\) implies

Here \(M_{0,k}(0)\) is the initial actual fill root with band exponent \(a/2\). The same vector is used in the filtered norm. The real logarithm has value zero at zero. For a nonzero vector, this is the initial normalization estimate in the scanner argument.

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 28.3.69, 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.

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.

Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(T\) be finite and nonempty, and put \(X=A\cap T\) for a color \(A\subseteq \Lambda \). Use the induced physical lattice graph and ambient distance from \(T\). Fix a collar scan with \(K\) bands, \(n,m,M{\gt}0\), and scales satisfying \(r_0\le D\), \(1\le D\), \(4D\le m\), and \(8Km\le L\). Suppose every ambient layer \(T_j\setminus T_{j-1}\) has at most \(n\) sites for \(1\le j\le L\). Fix \(q\ge 1\), a unit physical vector \(\widetilde\Omega \), and \(a{\gt}0\). Assume

where \(Q_j=A\cap (T_j\setminus T)\). Let \(E\) be a set of Schmidt indices at \(X\), with mass \(z{\gt}0\) and cardinality \(d\). Let \(\psi \) be the normalized Schmidt truncation, with the retained \(X\) factor relabelled by an auxiliary factor \(R\) of dimension \(d\). Supply labels \(\lambda _k\) such that, for all sufficiently large \(k\),

Use the same vectors \(\mathrm{pre}_k=\pi _R^{\lambda _k} (\widetilde\Omega \otimes \mathrm{Bell}_{CR})^{\otimes k}\) in every actual fill and charge, with auxiliary dimensions \(d,d\) and band exponent \(a/2\). There exists \(c\ge 0\) such that, for all sufficiently large \(k\), every round \(s+1\le nm\), every real interpolation parameter \(p\), and either actual root \(\mathcal M_{s,k}(p)\), one has

The root at real \(p\) is evaluated at its projection onto \([0,1]\). Both \(c\) and the replica threshold are chosen before \(s\) and \(p\).

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 \(2nm\beta _0\).

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.

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 \(c_0,C_{\mathrm{ent}},e_{\mathrm{ent}},C_{\mathrm{en}},e_{\mathrm{en}}\), 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 28.3.69 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 28.3.78, 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.

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.

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 28.3.69 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 (92). 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.

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 \(A\).

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 \(t\),

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

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 ] .

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.

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 \(P\cup Y\subseteq P'\).

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 ] .

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.

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

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.

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.

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

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.

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

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 \(1/(2M)\ge [2(C_1+1)]^{-1}/(nD)\).

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.

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 28.3.75, 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 \(q\ge 1\), 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 \(k+1\le nm\).

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 28.3.76, 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 (120) 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 \(|u|\le c_0/\sqrt{\mathcal B}\).

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

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 \(|A\cap T_L|\le (C+1)n^2\).

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.

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 \(p\in [0,1]\).

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.

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.

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.

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.

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 28.3.69, 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 (76), with \(0\le e\le \epsilon _n\). For some real \(\theta \), the same vector satisfies

For this same vector, both marginal inequalities of Theorem 28.3.75 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.

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.

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.

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.

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 \(V\) 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 28.3.70,

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 28.3.72. 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.

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 \(Y\).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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 \(Q_j\subseteq S_0\).

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 ] .

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 \(1-\ell -\mu \).

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) ] .

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.

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.

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 \(c_0\).

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 ] .

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.

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 \(\theta \).

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.

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 (120) 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 \(\widetilde\Omega \).

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.

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.

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

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 \(L,m\).

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.

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 28.3.76 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 28.3.76. 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 \(\mathrm e^{-s_X-w}\le p_i\le \mathrm e^{-s_X+w}\).

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 \(k+1\le nm\).

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 \(v\le 9nrs\).

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 \(1\le \ell \le \ell _q(\lceil C_r\log ^2 n\rceil )\),

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.

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.

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 \(j\le L\),

For every region \(S\subseteq I\),

This equivalence remains valid for matrices with an arbitrary spectator index set \(J\).

Suppose \(B_{ab}\in \mathcal A_S\) for every \(a,b\in J\). For every \(y\in I\),

where \(\mathbf1_S(y)\) is one on \(S\) and zero outside \(S\). This is the initial support estimate in [ 247 , Section 10 ] , for arbitrary finite spectators and arbitrary complex \(B\).

Suppose \(B_{ab}\in \mathcal A_S\) for every \(a,b\in J\). If \(y\notin S\), then \(d_y(B)=0\).

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 \(\frac12\| |\widetilde\Omega \rangle \langle \widetilde\Omega |-P_\Omega \| _1 \le (2\epsilon _n/g)^{1/2}\).

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 \(E=e\).

Let \(V\) be a finite graph embedded injectively in \(\mathbb {Z}^2\), with edge displacement at most one. Let \(\mathcal F\) be a finite family of distinct finite supports such that any two sites in a support admit a walk in the full graph of length at most the nonnegative integer \(R\). Put \(v_R=1+2R(R+1)\) and \(\mu _R=2^{v_R-1}\). Each original support has at most \(v_R\) sites, and at most \(\mu _R\) original supports contain any specified site. For \(J\ge 0\) and real weights \(w_X\le J\),

If also \(w_X\ge 0\) for each \(X\in \mathcal F\), then

The continuation bound holds for each original support \(X\) and permits repetitions. Both bounds apply to \(w_X=\lVert h_X\rVert \) for elements \(h_X\) of any seminormed additive group with \(\lVert h_X\rVert \le J\). These are the interaction budgets used in [ 247 , Sections 2 and 4 ] .

The diamond \(D_R(a)\) has cardinality \(v_R=1+2R(R+1)\). Let \(\iota :V\to \mathbb {Z}^2\) be injective. A finite set \(S\subseteq V\) with \(\ell (\iota (a),\iota (x))\le R\) for each \(x\in S\) has at most \(v_R\) elements. If every graph edge has displacement at most one, the same bound holds when every \(x\in S\) is reached from \(a\) by a graph walk of length at most \(R\), or when \(d_G(a,x)\le R\). This is the local counting bound of [ 247 , Section 2 ] .

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 ] .

Put \(v_R=1+2R(R+1)\) and \(\mu _R=2^{v_R-1}\). At most \(\mu _R\) admissible supports contain a given site, and for every \(A\subseteq \Lambda \) at most \(|\partial _\Lambda A|\, v_R\mu _R\) admissible supports meet both \(A\) and its complement.

Writing \(\Omega \) in the coordinates of \(\mathcal H_A\otimes \mathcal H_{A^c}\),

Let \(H\) be a complex matrix indexed by a finite set, let \(E_0\) be real, and let \(\Omega \) be a unit vector satisfying (343) and (344) with \(\Delta {\gt}0\). Then the eigenspace of \(H\) at \(E_0\) is \(\operatorname {span}_{\mathbb C}\{ \Omega \} \). This is an elementary consequence of the global gap, giving the uniqueness assertion in [ 247 , Proposition 4.5 ] .

Let \(\Omega \) be a unit vector satisfying (343) and (344) with \(\Delta {\gt}0\) for a Hamiltonian whose terms have norm at most \(J\ge 0\), and let \(A\subseteq \Lambda \). Let \(K\) take the value \(-\log p_j\) with probability \(p_j\) for the eigenvalues \(p_j\) of \(\rho _A\), and put \(\vartheta =J/\Delta \). Then \(\log \mathbb E\, \mathrm e^{uK}\le uS_\Omega (A)+512\mathrm e\vartheta \mathcal B_Au^2\) for \(|u|\le 1/(32\sqrt{(1+\vartheta )\mathcal B_A})\), and for every real \(w\)

For any finite ambient set \(U\) disjoint from \(Z\), inclusion into the lattice sends \(\partial _\Lambda (A\cap U)\) injectively into \(\partial U\). In particular, \(|\partial _\Lambda (A\cap U)|\le |\partial U|\).

Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\in \mathbb N\), and let \(\Omega ,\Phi \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be unit vectors. Put \(\delta =\frac12\bigl\| |\Omega \rangle \langle \Omega | -|\Phi \rangle \langle \Phi |\bigr\| _1\). Every region \(A\subseteq \Lambda \) satisfies

If \(q=0\), the unit-vector hypotheses force \(\Lambda \) and \(A\) to be empty; in this case \(|A|\log q\) is read as zero. This is the regional form of [ 247 , Lemma 2.1 ] used in its scanner entropy estimate.

Let \(q\ge 1\) and put \(C=v_R\mu _R\log ^2(\mathrm e\, q^{v_R})\). Then \(C\ge 1\), and for every \(A\subseteq \Lambda \),

This is the bound on the tail parameter stated after Lemma 3.1 of [ 247 ] ; the constant depends only on \(q\) and \(R\).

If \(0\le k\le 1\), \(k\) is supported on \(C(a)\), and \(d_G(a,y)=\infty \), then \(d_y(\mathcal E_k(B))\le d_y(B)\).

Suppose \(0\le k\le 1\), \(k\) is supported on \(C(a)\), and \(n\ge R_a\). If \(\| k-E_{N_l(a)}(k)\| \le \varepsilon _l\) for every \(l\le n\), then

The coefficient has no factor depending on \(\dim \mathcal K\). This is the finite-radius consequence of exact component support in [ 247 ] , before choosing a decaying error bound.

Every physical operator \(A\) supported on \(C(a)\) satisfies \(E_{N_n(a)}(A)=A\) and \(\| A-E_{N_n(a)}(A)\| =0\) for \(n\ge R_a\). In particular, if \(k\) is a positive contraction supported on \(C(a)\), then \(\mathcal E_{k,n}(B)=\mathcal E_k(B)\) and \(\Delta _{n+1}(B)=0\) for all \(n\ge R_a\) and all \(B\). The same radius works for every observable.

In the setting of Theorem 33.9.10, with \(q\geq 1\), assume that at most \(K_g(d+1)^k\) sites lie at distance exactly \(d\) from any site. Let \(N_l(a)=\{ x:d_G(a,x)\leq l\} \). Then

with \(v=2v_Rb_0e^{2R}\) and \(C=2e^RK_g\, 2^kk!\, e^{1/2}/(1-e^{-1/2})\).

The balls \(N_l(a)\) increase with \(l\). Every \(n\ge R_a\) satisfies \(N_n(a)=C(a)\). If \(d_G(a,y)=\infty \), then \(y\notin N_n(a)\) for every finite \(n\).

The set in (12) contains zero and is bounded above by \(2\| B\| \). In particular, its supremum is finite and \(0\le d_y(B)\le 2\| B\| \). Every unitary \(U\in \mathcal A_{\{ y\} }\) satisfies \(\| [\widehat U,B]\| =\| [B,\widehat U]\| \le d_y(B)\). If these commutator norms are all at most a real number \(r\), then \(d_y(B)\le r\).

If \([\widehat U,B]=0\) for every \(U\in \mathcal A_{\{ y\} }\), then \(d_y(B)=0\). In particular, \(d_y(0)=0\), and every operator \(S\) on the spectator satisfies \(d_y(\mathbb 1_{\mathcal H}\otimes S)=0\).

For every \(c\in \mathbb {C}\),

Consequently \(d_y(-B)=d_y(B)\) and \(d_y(B-C)\le d_y(B)+d_y(C)\). For any finite set \(F\) and any family \((B_k)_{k\in F}\), \(d_y(\sum _{k\in F}B_k)\le \sum _{k\in F}d_y(B_k)\).

For every \(U\in \mathcal A_{I\setminus K}\), \([E_K^{\mathcal K}(B),\widehat U]=0\).

Suppose \(q{\gt}0\), and let \(\varepsilon :I\to \mathbb {R}\) satisfy \(\| [B,\widehat U]\| \le \varepsilon _y\) for each \(y\notin K\) and every unitary \(U\in \mathcal A_{\{ y\} }\). Then