Let \(\Lambda \subset \mathbb Z^2\) be finite, with local dimension \(q\), and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Suppose \(A=D\mathbin {\dot\cup }\bigcup _iX_i\) is an ordered two-family partition, with family labels \(f_i\in \{ 0,1\} \). Put \(B=\Lambda \setminus A\) and \(P_i=\bigcup _{j{\lt}i,\, f_j=f_i}X_j\). If \(I_\Omega (X_i:B\cup P_i)\le \varepsilon _i\) for every index, then

This is [ 286 , Lemma 11.1 ] , with no Hamiltonian or geometric hypothesis. Empty pieces, families, and residual sets are allowed. The same bound holds for a unit vector in any finite tensor product of finite-dimensional spaces with site-dependent local dimensions, and the lattice statement above specializes it to sites in \(\mathbb Z^2\) with local dimension \(q\).

Let \(\Omega \) be a normalized pure state on a finite domain, and let \(G,B,C\) be pairwise-disjoint regions. Suppose an ordered two-family partition of \(G\) has remainder \(D\), pieces \(X_i\) and family labels \(f_i\in \{ 0,1\} \). Put \(P_i=\bigcup _{j{\lt}i,\, f_j=f_i}X_j\). Assume \(I_\Omega (X_i:C\cup B\cup P_i)\le \varepsilon _i\) when \(f_i=0\) and \(I_\Omega (X_i:(G\cup B)^c\cup P_i)\le \varepsilon _i\) when \(f_i=1\). Then, as in the entropy estimate preceding the geometric counting argument in [ 287 , Lemma 3.2 ] ,

In the recursion for one cell, an active square of side \(u\) has children of side \(u'=8u^{\gamma }\). For \(K{\gt}0\) and \(\gamma {\lt}\delta \) put \(U(K,\gamma ,\delta )=\max \{ 2,K^{1/(\delta -\gamma )}\} \); then \(Ku^{\gamma }\le u^{\delta }\) for all \(u\ge U(K,\gamma ,\delta )\). In particular, if \(\gamma {\lt}\gamma '\), then \(u_*=U(8,\gamma ,\gamma '){\gt}1\) satisfies \(8u^{\gamma }\le u^{\gamma '}\) for all \(u\ge u_*\). Let \(0{\lt}\gamma '{\lt}1\) and let \(u_*{\gt}1\) be any such threshold. There is a constant \(C{\gt}0\) such that, for every initial side \(s{\gt}0\) and every \(m\) for which the sides of the generations \(0,\dots ,m-1\) are at least \(u_*\),

For every branching bound \(b\ge 0\) there is also \(C{\gt}0\) such that, under the same condition and with \(s\ge 1\),

One may take \(C=\max \{ C_g2^{C_g},C_g\} \), where \(C_g\) is the generation-count constant of Theorem 41.67.1.1 for \(\rho =\gamma '\), \(S_*=u_*\) and branching bound \(b\).

For all \(\gamma ,u_0\in \mathbb R\) one has \(u_0\le u_*(\gamma ,u_0)\) and \(u_*(\gamma ,u_0){\gt}1\), and \(C(\gamma ,u_0)\) and \(K(\gamma ,u_0)\) are positive. If \(\gamma {\lt}1\), then \(\gamma {\lt}\gamma '\) and \(0{\lt}\gamma '{\lt}1\).

Fix \(\gamma {\lt}1\) and \(u_0\), and let \(u_*=u_*(\gamma ,u_0)\), \(C=C(\gamma ,u_0)\) and \(K=K(\gamma ,u_0)\) be the constants of Definition 42.2.7. Tile the plane by an axis-parallel grid of squares of side \(s\ge 1\), choose one cell, and give every other cell a forbidden colour \(0\) or \(1\). The lattice sample of the chosen cell has an ordered two-family partition in which every piece has a scale \(u\in [u_*,s]\) such that the piece is a union of at most \(66^2\) closed lattice rectangles of sup diameter at most \(u\), and every point of the piece is at sup distance greater than \(u^{\gamma }\) from every lattice point of an exterior cell whose colour is the family of the piece and from every earlier piece of the same family. The pieces and the remainder sites number at most \(C(1+\log s)^{K}\) in total.

Fix \(\gamma {\lt}1\) and \(u_0\). There are \(u_*\ge u_0\) with \(u_*{\gt}1\) and constants \(C,K\) with the following property. Tile the plane by an axis-parallel grid of squares of side \(s\ge 1\), choose one cell, and give every other cell a forbidden colour \(0\) or \(1\). The lattice sample of the chosen cell has an ordered two-family partition in which every piece has a scale \(u\in [u_*,s]\) such that the piece is a union of at most \(66^2\) closed lattice rectangles of sup diameter at most \(u\), and every point of the piece is at sup distance greater than \(u^{\gamma }\) from every lattice point of an exterior cell whose colour is the family of the piece and from every earlier piece of the same family. The pieces and the remainder sites number at most \(C(1+\log s)^{K}\) in total.

In the regional dimension estimate used in the proof of [ 286 , Theorem 1.1 ] , let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\in \mathbb N\), and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Every region \(A\subseteq \Lambda \) satisfies \(S_\Omega (A)\le |A|\log q\).

Under the hypotheses of Theorem 42.1.1, suppose also that \(q\ge 1\), \(|D|\le c_D|\partial _\Lambda A|\), and \(\sum _i\varepsilon _i\le c_E|\partial _\Lambda A|\). Then

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. Set \(\delta =\frac12\bigl\| |\Omega \rangle \langle \Omega | -|\Phi \rangle \langle \Phi |\bigr\| _1\). For every region \(A\subseteq \Lambda \),

(When \(q=0\), the existence of unit vectors forces \(\Lambda =A=\varnothing \) and \(|A|\log q=0\).) In the scanner entropy estimate, this inequality provides the regional form of [ 286 , Lemma 2.1 ] .

Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\in \mathbb N\), and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Fix \(X\subseteq \Lambda \) and write \(S_X=S_{\Omega }(X)\). Let \((p_i)\) be the eigenvalues of the reduced state of \(\Omega \) on \(X\), and let \(E\) be a set of spectral indices with mass \(z=\sum _{i\in E}p_i{\gt}0\). Suppose that \(w\in \mathbb R\) and that every \(i\in E\) satisfies \(e^{-S_X-w}\le p_i\le e^{-S_X+w}\). In the corresponding Schmidt decomposition, write \(v_i\) for the complementary Schmidt vectors, and let \(r:E\to \{ 0,\ldots ,|E|-1\} \) be an enumeration of the selected indices. Define

Then

These are the selected-spectrum entropy estimates used in [ 286 , Section 8 ] .

Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\ge 1\) be an integer, and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Suppose that

is a finite ordered two-family partition with labels \(f_i\in \{ 0,1\} \). For each piece \(X_i\), set

Let \(Y_i\subseteq \Lambda \) be birth targets satisfying

and let \(n_i\in \mathbb N\), \(c_D,c_S\in \mathbb R\), and \(c_I\ge 0\) satisfy

for every \(i\), together with

where \(I_\Omega (U:W)=S_\Omega (U)+S_\Omega (W)-S_\Omega (U\cup W)\) is the mutual information of the reduced states of \(\Omega \), and a zero scale carries the weight \(0^{-100}=0\). Then

This is the deduction of the area-law bound from the birth-target mutual-information estimates in the proof of [ 286 , Theorem 1.1, lines 838–852 of Section 11 ] , where \(Y_i=A\cap T_i\).

Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(q\ge 1\) be an integer, and let \(\Omega \in \bigotimes _{v\in \Lambda }\mathbb C^q\) be a unit vector. Let \(A=D\mathbin {\dot\cup }\bigcup _iX_i\) be a finite ordered two-family partition, and set \(P_i=\bigcup _{j{\lt}i,\, f_j=f_i}X_j\). Suppose that \(X_i\subseteq Y_i\subseteq A\), \(Y_i\subseteq N_i\subseteq \Lambda \), and \((\Lambda \setminus A)\cup P_i\subseteq \Lambda \setminus N_i\) for every \(i\). Let \(n_i\in \mathbb N\), \(c_D,c_S\in \mathbb R\), and \(c_I\ge 0\) satisfy

Then

Here again a zero scale carries the weight \(0^{-100}=0\). This is the reduction from exterior-neighborhood mutual information to boundary entropy in [ 286 , Theorem 1.1, lines 815–852 of Section 11 ] .

The fixed parameters satisfy \(0{\lt}\gamma \) and \(1-\varepsilon {\lt}\gamma {\lt}\beta \). For every real constant \(C\), \([L^{1-\varepsilon }+2CL^\gamma ]/L^\beta \to 0\) as the positive integer \(L\) tends to infinity.

For every \(C\in \mathbb R\) and \(\eta {\gt}0\), there is a threshold \(N\ge 2\) depending only on \(C\) and \(\eta \) such that every integer \(L\ge N\) and every nonnegative integer radius \(r\le CL^\gamma \) satisfy \(\lfloor L^{1-\varepsilon }\rfloor +2r{\lt}\eta L^\beta \). Consequently, for every sequence of nonnegative integer radii satisfying \(r(L)\le CL^\gamma \) eventually, \(\lfloor L^{1-\varepsilon }\rfloor +2r(L)=o(L^\beta )\).

Let \(X,Y,Z\) be real random variables on a probability space, with \(Y,Z\ge 0\) almost surely and \(X^2,Y,Z\) integrable. Suppose that \(\mathbb EX^2\le a\), \(\mathbb EY\le b\) and \(\mathbb EZ\le c\), and that \(a/\varepsilon ^2+b/\varepsilon +c/\varepsilon {\lt}1\) for some \(\varepsilon {\gt}0\). Then some realization satisfies \(X,Y,Z\le \varepsilon \).

Let \(c,\alpha {\gt}0\). There is \(B\ge 1\), depending only on \(c\) and \(\alpha \), such that \((1+r)^2e^{-cr^\alpha }\le B\) for every \(r\ge 0\).

Let \(g,c,\alpha {\gt}0\), let \(v,C,C_0\ge 0\) and let \(K\ge 1\) be an integer. Without any upper bound on \(\mathcal I\), the explicit scale control of [ 286 , Proposition 10.2 ] gives \(C'{\gt}0\) and \(N\ge 2\) such that, for every integer \(n\ge N\), all \(\mathcal I,w\ge 0\) and every integer \(0\le s\le C_0n^2\), the time \(t_*\) and radius \(r\) of Definition 42.7.1.1 and \(M_0=\exp ((\mathcal I+3w)/2)\) satisfy

Let \(g,c,\alpha ,v,C,C_0,K\) be as in Theorem 42.7.1.4. There are \(C'{\gt}0\) and \(N\ge 2\) such that, for all \(n\ge N\), all \(\mathcal I,w\ge 0\) and every integer \(0\le s\le C_0n^2\), the following holds with \(t_*,r,M_0\) as there. Let \(X,Y,Z\) be real random variables on one probability space, with \(Y,Z\ge 0\) almost surely, such that

Then one realization satisfies \(X,Y,Z\le n^{-K}\). In the proof of [ 286 , Proposition 10.2 ] , where \(s=|S_0|\) and the three variables are \(\| D(I-P_\Omega )\zeta \| \), \(\| (D-D_{\rm ret})\zeta \| \), and \(M_0\| D_{\rm ret}-L'\| \), taking the three averaged estimates as hypotheses yields this simultaneous bound on a single clock realization.

Let \(\Lambda \) be a finite induced domain, let \(A\subseteq \Lambda \), let \(T\subset \mathbb Z^2\) be finite, and let \(Z\) be the set of endpoints of the cut edges of \(A\). Put \(L=\lfloor n^{1-\varepsilon }\rfloor \) and \(S_0=A\cap T_L\). For every \(C\) there is \(N\) such that for all \(n\ge N\), every integer \(r\le Cn^{(1-\varepsilon )/\alpha }\) and every \(\Lambda \), \(A\), \(T\) with \(\operatorname {dist}_\infty (T,Z){\gt}L+n^\beta \) satisfy \(N_{2r}^\Lambda (S_0)\subseteq A\), establishing the last assertion of Proposition 10.2 of [ 286 ] from its radius bound.

Fix \(q,R,J,\Delta \). Suppose that, under the hypotheses of the amplification bound, for every \(C_0{\gt}0\) and every integer \(M\ge 1\) there are \(C_M\) and \(N\) such that at every \(n\ge \max (2,N)\) some integer \(r\le C_Mn^{(1-\varepsilon )/\alpha }\) satisfies \(I_\Omega (X':J)\le C_Mn^{-M}\) for every \(X'\subseteq X\) and every \(J\subseteq \Lambda \setminus N_{2r}^\Lambda (S_0)\). Then the amplification bound holds, reducing Proposition 10.2 of [ 286 ] to its radius and output bounds.

Let \(D\), \(D_{\rm ret}\) and \(L'\) be square matrices on one finite-dimensional Hilbert space, let \(\zeta \) and \(g\) be vectors, and let \(M_0,\varepsilon \) be real. Suppose that \(Dg=g\), \(\| g\| \le 1\), \(\| \zeta \| \le M_0\),

With \(g\) denoting the ground component of the restoring vector in the proof of Proposition 10.2 of [ 286 ] , the two estimates preceding the polar decomposition hold: \(\| L'\zeta \| \le 1+3\varepsilon \), and if in addition \(M_0\ge 1\) and \(D_{\rm ret}^\dagger w=w\) for a vector with \(\| w\| \le 1\), then \(\| (L')^\dagger w-w\| \le \varepsilon \).

Let \(R'\) be a square matrix on a finite-dimensional Hilbert space \(\mathcal H\), and let \(\phi ,\eta \) be unit vectors in \(\mathcal H\otimes \mathcal K\) for a finite-dimensional space \(\mathcal K\). If \(\| (R'\otimes I)^\dagger \eta -\phi \| \le a\) and \(\| (R'\otimes I)\phi \| \le 1+b\), then

and there is a unitary \(U\) on \(\mathcal H\) with \(\| (U\otimes I)\phi -\eta \| \le a+\sqrt{2a+2b+b^2}\), bounding the direct and unitary errors in the proof of Proposition 10.2 of [ 286 ] without assuming invertibility of \(R'\).

Let \(R_0\) and \(L'\) be square matrices on a finite-dimensional space \(\mathcal H\), and let \(D\), \(D_{\rm ret}\) act on \(\mathcal H\otimes \mathcal K\). Let \(\phi ,\eta \) be unit vectors, put \(\zeta =(R_0\otimes I)\phi \), and suppose that \(\| \zeta \| \le M_0\), that \(D\) fixes a vector \(g\) with \(\| g\| \le 1\), that

and that \(\| ((L'R_0)\otimes I)^\dagger \eta -\phi \| \le a\). Then some unitary \(U\) on \(\mathcal H\) satisfies \(\| (U\otimes I)\phi -\eta \| \le a+\sqrt{2a+6\varepsilon +9\varepsilon ^2}\).

Let \(X,J\subseteq V\) be disjoint, let \(U\) be a unitary on \(\mathcal H_{V\setminus (X\cup J)}\), and let \(\varphi ,\eta \in \mathcal H\) be unit vectors with \(I_\eta (X:J)=0\). For \(\delta =\| (I_{X\cup J}\otimes U)\varphi -\eta \| \), the comparison in the final step of the proof of [ 286 , Proposition 10.2 ] gives

In the setting of Theorem 42.9.4, let \(n\ge 1\) be an integer and suppose that \(\delta \le C_1n^{-K/2}\) and \(\log \dim \mathcal H_X\le C_2n^2\). Then

If \(M,K\) are integers with \(K\ge 4(M+2)\) (as satisfied by the choice \(K\ge 4(M+3)\) in the proof of [ 286 , Proposition 10.2 ] ), then every \(X'\subseteq X\) satisfies \(I_\varphi (X':J)\le 8\sqrt{C_1}\, (1+C_2)\, n^{-M}\), with a constant depending only on \(C_1\) and \(C_2\).

Let \(k_i\) be positive contractions on a finite-dimensional Hilbert space and \(G_i=(I-k_i)^{1/2}\). For a finite word \(w=(i_1,\dots ,i_m)\) the chronological product \(G_{i_m}\cdots G_{i_1}\) is the product over the reversed list of letters. The product over the letters of \(w\) satisfying a predicate \(P\) equals the product over the list of letters of \(w\) filtered by \(P\). If every \(k_i\) annihilates a vector \(\Omega \), then every such product \(D\) satisfies \(D|\Omega \rangle \langle \Omega |=|\Omega \rangle \langle \Omega |\) and \(D^\dagger \Omega =\Omega \).

The map \(\mathcal W\) is unitary. It maps a physical operator written in the eigenbasis of the target to the operator \(L\otimes I\) on the original physical coordinates, and \((A\otimes M)\otimes I_{Z'}\) to the physical operator obtained from \(M\) tensored with \(A\) rotated on \(e'\). The restored vector becomes \(\Omega \) tensored with the ancillary vector \((I\otimes U)\sum _x\sqrt{p_x}|x,x\rangle \), the vector with blanks on both ancillas becomes \(\Omega \) tensored with the blanks, and the copied vector becomes the vector whose value at a configuration \(c\) and an ancillary pair \((s,e)\) is \(|0\rangle (s)\, (U|0\rangle )(c_X)\, \Omega (c')\), with \(c'\) equal to \(c\) off \(X\) and to \(e\) on \(X\). The restoring operator is a sum over the selected indices of physical operators on \(X\cup Y\) tensored with \(p_x^{-1/2}|x\rangle \langle 0|_{s'}\otimes I_{e'}\), and the column operator is a sum of physical operators on \(X\cup Y\) tensored with operators on the ancillas.

Let \(\Omega \) be a unit vector, let \(X,Y\) be disjoint with \(S_0=X\cup Y\), and suppose that the typical projections of \(X\), \(Y\) and \(S_0\) fail with probability at most \(\delta \). In the coordinates \(\Lambda \times (s',e')\) the vectors \(\varphi \) and \(\eta \) are unit vectors, \(\| R_0\| ,\| B_0\| \le M_0=e^{(\mathcal I+3w)/2}\), the column identity \(R_0\varphi =B_0(\Omega \otimes \xi )\) holds for the image \(\xi \) of the blank pair, the physical ground component of \(R_0\varphi \) has norm at most one, and

for every physical operator \(L'\). The column operator acts on \(S_0\) and the ancillas, and the restoring operator on \(S_0\) and \(s'\).

Relabelling the sites along \(\sigma \) moves every regional entropy along \(\sigma \). For unit vectors \(f\) and \(g\), the region \(R\sqcup T\) of \(f\otimes g\) has entropy \(S_f(R)+S_g(T)\). Consequently adjoining sites in a product state does not change the mutual information of two regions of \(A\), and a region of \(B\) has no mutual information with a region of \(A\).

On the enlarged sites, an operator \(P\otimes (b\otimes I_{e'})\) with \(P\) acting on a physical region \(S\) and \(b\) acting on \(s'\) acts on \(S\sqcup s'\), and a physical operator on \(S\) tensored with the identity on both copies acts on \(S\). For unit vectors \(\Omega \) and \(\chi \), the vector \(\Omega \otimes \chi \) has no mutual information between \(e'\) and any physical region. Let \(\varphi \) be the vector whose value at a configuration \(c\) and an ancillary pair \((s,e)\) is \(a(s)\, b(c_X)\, \Omega (c')\), with \(c'\) equal to \(c\) off \(X\) and to \(e\) on \(X\), for unit vectors \(a,b\) and a unit vector \(\Omega \). For every \(X'\subseteq X\) and every physical region \(J\) disjoint from \(X\), the mutual information of \(\varphi \) between the copy of \(X'\) in \(e'\) and \(J\) is \(I_\Omega (X':J)\).

On a finite set of sites with local dimension \(q\), let \(R_0\) and \(L'\) act on a region \(S\). Under the hypotheses of the deterministic steps after the choice of a good realization, there is a unitary acting on \(S\) with \(\| U\varphi -\eta \| \le a+\sqrt{2a+6\varepsilon +9\varepsilon ^2}\). If a unitary acts on a region disjoint from \(X\cup J\), \(I_\eta (X:J)=0\), \(\| U\varphi -\eta \| \le C_1n^{-K/2}\) with \(K\ge 4(M+2)\) and \(\log \dim \mathcal H_X\le C_2n^2\), then every part of \(X\) has mutual information at most \(8\sqrt{C_1}\, (1+C_2)\, n^{-M}\) with \(J\) in \(\varphi \).

Work on the physical sites enlarged by the two copies \(s',e'\) of \(X\). Let the restoring operator \(R_0\) act on \(N\sqcup s'\) and the truncated product \(L'\) act on a physical region \(N\). Under the hypotheses of the deterministic steps after the choice of a good realization, with the physical operators tensored with the identity on both copies, there is a unitary acting on \(N\sqcup s'\) with \(\| U\varphi -\eta \| \le a+\sqrt{2a+6\varepsilon +9\varepsilon ^2}\). If \(X\subseteq N\), \(\varphi \) is the copied vector built from unit blank vectors, \(\varphi \) and \(\eta \) have unit norm, \(\| U\varphi -\eta \| \le C_1n^{-K/2}\) with \(K\ge 4(M+2)\) and \(|X|\log q\le C_2n^2\), then every part of \(X\) has mutual information at most \(8\sqrt{C_1}\, (1+C_2)\, n^{-M}\) in \(\Omega \) with every physical region avoiding \(N\).

Let \(g{\gt}0\), \(C'\ge 0\), \(C_0\ge 0\), \(0{\lt}\alpha \), \(0{\lt}\varepsilon \le 2/5\) and let \(K\) be an integer. There is \(C\) such that at every scale \(n\ge 1\) and for every \(0\le \mathcal I\le C_0n^{1-\varepsilon }\) the radius of Definition 42.7.1.1, chosen with \(w=n^{3/5}\), is at most \(Cn^{(1-\varepsilon )/\alpha }\).

In Proposition 10.2 of [ 286 ] , for every \(q,R,J\) and every gap \(\Delta {\gt}0\) the amplification bound holds.

Let \(q,R\in \mathbb N\), \(J\in \mathbb R\), \(\Delta {\gt}0\), \(C_0{\gt}0\) and \(M\in \mathbb N\). Let \(C_2\) and \(N_2\) be the constant and the threshold produced by the proof of Theorem 42.10.12 at \(C_0\) and \(M\), with the locality constant and rate of the positive constraints at the power \(p=10^6\). Then the amplification bound at the input constant \(C_0\) and the order \(M\) holds with constant \(C_2\) and threshold the larger of \(N_2\) and the radius threshold of \(C_2\).

Suppose that for all \(q\ge 1\), \(R\ge 0\) and \(J,\Delta {\gt}0\) there are template constants \(D_0,C_{\rm tpl}\) for which the sublinear collar bound and the two-family decomposition hold. Then the conclusion of Theorem 39.8.9 holds, proving Theorem 1.1 of [ 286 ] conditionally on Propositions 9.6 and 11.2.

In Proposition 3.1 of [ 287 ] , fix an integer \(N_0\ge 1\) and a constant \(A_0\ge 1\). There are \(\beta {\lt}1\), \(C{\lt}\infty \) and \(u_0{\lt}\infty \) such that the following holds. Let \(u\ge u_0\) and let \(T\) be the full ambient lattice sample of a union of at most \(N_0\) axis-parallel rectangles, with \(\operatorname {diam}_\infty T\le A_0u\). For every physical set \(E\subseteq \Lambda \) with \(\operatorname {dist}_\infty (E,T){\gt}u^\beta \),

The same holds for every subset of \(\Lambda \cap T\) in the first argument, for rectangles with consistently assigned open or closed sides, and after a common translation of the lattice coordinates.

Fix \(q,R,J,\Delta \) and template constants \(D_0,C_{\rm tpl}\). Suppose that the sublinear collar bound holds with constant \(C\) and threshold \(N\), and that the amplification bound holds at \((\max \{ C,1\} ,1)\) with constant \(C_a\) and threshold \(N_a\). Fix \(N_0\) and \(A_0\), and let \(u_0=u_0(C_{\rm tpl},N_0,A_0,N,C_a,N_a)\) be the coarse threshold of Definition 42.11.6. Let \(u\ge u_0\), let \(\Omega \) be a unit ground vector with full-system gap \(\Delta \) of an admissible Hamiltonian of range \(R\) and bound \(J\) on a finite induced domain \(\Lambda \), and let \(T\) be a union of at most \(N_0\) boxes with \(\operatorname {diam}_\infty T\le A_0u\). Then \(I_\Omega (X:E)\le \max \{ C_a,0\} \) for every \(X\subseteq \Lambda \cap T\) and every \(E\subseteq \Lambda \) with \(\operatorname {dist}_\infty (E,T){\gt}u^{\beta _0}\).

Fix \(q,R,J\) and \(\Delta {\gt}0\), and assume the sublinear collar bound for some template constants \(D_0,C_{\rm tpl}\). Fix \(N_0\) and \(A_0\). There are \(\beta {\lt}1\), \(C\) and \(u_0\) with the following property. Let \(u\ge u_0\), let \(\Omega \) be a unit ground vector with full-system gap \(\Delta \) of an admissible Hamiltonian of range \(R\) and bound \(J\) on a finite induced domain \(\Lambda \), and let \(T\) be a union of at most \(N_0\) boxes with \(\operatorname {diam}_\infty T\le A_0u\). Then \(I_\Omega (X:E)\le C\) for every \(X\subseteq \Lambda \cap T\) and every \(E\subseteq \Lambda \) with \(\operatorname {dist}_\infty (E,T){\gt}u^\beta \). The constants are chosen before \(u\), \(\Lambda \), the Hamiltonian, \(T\), \(X\) and \(E\).

Fix \(q\), \(J\in \mathbb R\) and \(\Delta {\gt}0\), and assume the sublinear collar bound at range one. Fix \(N_0\) and \(A_0\). There are \(\beta {\lt}1\), \(C\) and \(u_0\) such that for every \(u\ge u_0\), every open square grid with a Hamiltonian of the model, every unit ground vector \(\Omega \) with full-system gap \(\Delta \), every translation \(c\in \mathbb Z^2\) of the lattice coordinates, every union \(T\) of at most \(N_0\) boxes with \(\operatorname {diam}_\infty T\le A_0u\), every set \(X\) of sites whose translated coordinates lie in \(T\), and every set \(E\) of sites whose translated coordinates are at distance greater than \(u^\beta \) from \(T\), one has \(I_\Omega (X:E)\le C\).

Fix integers \(q\ge 1\), \(R\ge 0\) and \(N_0\ge 0\), real numbers \(J\ge 0\), \(\Delta {\gt}0\) and \(A_0\). There are \(\beta {\lt}1\), \(C\) and \(u_0\) with the following property. Let \(u\ge u_0\), let \(\Lambda \) be a finite induced domain, and let \(\Omega \) be a normalized ground vector with full-system gap \(\Delta \) of an admissible Hamiltonian of range \(R\) and bound \(J\) on \(\Lambda \). Let \(T\) be a union of at most \(N_0\) lattice boxes with \(\operatorname {diam}_\infty T\le A_0u\). Then \(I_\Omega (X:E)\le C\) whenever \(X\subseteq \Lambda \cap T\), \(E\subseteq \Lambda \) and \(\operatorname {dist}_\infty (T,E){\gt}u^\beta \). The constants precede the scale, domain, Hamiltonian, ground vector, rectangle union and physical regions. Empty sampled regions are included.

Fix \(q,R\in \mathbb N\) with \(q\ge 1\), \(J\ge 0\), \(\Delta {\gt}0\) and \(D_0{\gt}2R+10\). The sublinear collar bound at range \(R\), clearance \(D_0\) and template constant \(24\) holds with constant \(C(q,R,J,\Delta )\) and threshold \(N(q,R,J,\Delta )\). Neither depends on \(D_0\).

Fix \(q,R,N_0\in \mathbb N\) with \(q\ge 1\), real numbers \(J\ge 0\), \(\Delta {\gt}0\) and \(A_0\), and put \(u_0=u_0(q,R,J,\Delta ,N_0,A_0)\). Let \(u\ge u_0\), let \(\Lambda \) be a finite induced domain, and let \(\Omega \) be a normalized ground vector with full-system gap \(\Delta \) of an admissible Hamiltonian of range \(R\) and bound \(J\) on \(\Lambda \). Let \(T\) be a union of at most \(N_0\) lattice boxes with \(\operatorname {diam}_\infty T\le A_0u\). Then \(I_\Omega (X:E)\le C_{\rm coarse}(q,R,J,\Delta )\) whenever \(X\subseteq \Lambda \cap T\), \(E\subseteq \Lambda \) and \(\operatorname {dist}_\infty (T,E){\gt}u^{\beta _0}\). The source states [ 287 , Proposition 3.1 ] for site and nearest-neighbour terms only, which is range one; this extends it to range \(R\).

Let \(H=H_A\otimes I+I\otimes H_B\) with \(H_A,H_B\) Hermitian. If a unit vector \(\omega \) satisfies \(H\omega =E_0\omega \) and \(H-E_0I\ge \Delta (I-|\omega \rangle \langle \omega |)\) with \(\Delta {\gt}0\), then \(\omega \) is a product vector, and its reduced state on the first factor has zero entropy.

More precisely, there are real numbers \(a,b\) and normalized factor vectors \(u,v\) such that

The phase is included in the chosen factors, so the product equality is exact. Both factors retain the same lower gap bound \(\Delta \). These factor gap inequalities are the spectral step in the reduction to connected domains in [ 292 , Part I ] .

Let \(k\ge 1\), and for \(i=1,\dots ,k\) let \(H_i\) be a Hermitian operator on a finite-dimensional space \(\mathcal H_i\). Put \(H=\sum _iH_i\) on \(\bigotimes _i\mathcal H_i\), each \(H_i\) acting on its own factor. If a unit vector \(\omega \) satisfies \(H\omega =E_0\omega \) and \(H-E_0I\ge \Delta (I-|\omega \rangle \langle \omega |)\) with \(\Delta {\gt}0\), then there are real numbers \(a_i\) and unit vectors \(\omega _i\) such that

Every factor keeps the gap \(\Delta \). This is the statement in the reduction to connected domains of [ 292 , Part I, Lemma 2.1 ] that each component Hamiltonian inherits the gap inequality with the same gap.

Every support of finite induced-graph diameter lies in one connected component of the domain graph. If \(\partial _\Lambda A=\varnothing \), every such support lies in \(A\) or in \(\Lambda \setminus A\). The set of sites of any union of connected components has no crossing edge.

Under the hypotheses of Theorem 39.8.9, if every admissible support \(X\) satisfies \(X\subseteq A\) or \(X\cap A=\varnothing \), then \(S_\Omega (A)=0\).

Under the hypotheses of Theorem 39.8.9, if \(\partial _\Lambda A=\varnothing \), then \(S_\Omega (A)=0\). This is Lemma 2.2 of [ 286 ] ; only the eigenvector condition and the full-system gap with \(\Delta {\gt}0\) are used.

For range \(R=0\) every regional entropy of the gapped ground vector vanishes. For \(q=1\) or an empty domain every regional entropy of every unit vector vanishes. Every union of connected components, and in particular the empty region and the full domain, has zero regional entropy for the gapped ground vector.

Both \(\partial _{\rm in}A\) and \(Z_A\) are empty when \(A=\varnothing \) or \(A=\Lambda \).

Every site of a finite induced square-lattice domain has degree at most four.

For every cut in a finite induced lattice domain, \(|\partial _\Lambda A|\le 4|\partial _{\rm in}A|\) and \(|\partial _\Lambda A|\le 2|Z_A|\).

Assume that, for fixed \(q,R,J,\Delta \), a nonnegative constant \(C\) gives \(S_\Omega (A)\le C|\partial _\Lambda A|\) uniformly for the finite-domain Hamiltonians and ground vectors of Section 39.8. Then the same constant gives \(S_\Omega (A)\le 4C|\partial _{\rm in}A|\) and \(S_\Omega (A)\le 2C|Z_A|\) uniformly for those instances.

Let \(\Omega \) be a unit vector satisfying (50) and (51) 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 \(w\)

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

Under the hypotheses and notation of Theorem 42.13.2, let \(n\ge 0\) be an integer and \(M\) real, and put \(w=32\sqrt{(1+\vartheta )\mathcal B_A}\, (M\log (n+2)+\mathrm e/2+\log 2)\). The positive eigenvalues \(p_j\) of \(\rho _A\) with \(|-\log p_j-S_\Omega (A)|\le w\) carry total mass at least \(1-(n+2)^{-M}\). This is the concentration statement following Lemma 3.1 of [ 286 ] .

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.

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 [ 286 ] ; the constant depends only on \(q\) and \(R\).

Under the hypotheses and notation of Theorem 42.13.2, let \(q\ge 1\), let \(n\ge 0\) be an integer and \(M\ge 0\), and put

with \(C\) as in Theorem 42.13.6. The positive eigenvalues \(p_j\) of \(\rho _A\) with

carry total mass at least \(1-(n+2)^{-M}\). This is the concentration statement following Lemma 3.1 of [ 286 ] , which is stated there for \(n\ge 1\) and \(M{\gt}0\).

Let \(P,x,Y,F\) partition \(\Lambda \) and let \(\theta \) be a unit vector. Then

where \(S_\theta (x|R)=S_\theta (R\cup x)-S_\theta (R)\).

There are universal constants \(C,c{\gt}0\) such that the following holds. Let \(P,x,Y,F\) partition a finite domain with local dimension \(q\), let \(\theta \) be a unit vector, let \(\sigma \ge 0\) on \(P\cup x\) and \(\tau \ge 0\) on \(x\cup Y\) have trace at most one, put \(\ell =\log (\mathrm e\, q^{|x|})\), and let \(0{\lt}a\ell \le c\). Then

This is  [ 286 , Lemma 5.1 ] with \(U=Y\).

Let \(P_0,P_1,x,U,F\) partition a finite domain with local dimension \(q\), put \(P=P_0\cup P_1\) and \(Y=x\cup U\), let \(\theta \) be a unit vector and let \(0\le h\le 1\) act on \(P_0\cup x\). Put

\(\ell _h=\log (\mathrm e\, q^{|P_0|+|x|})\) and \(\eta _\theta =S_\theta (x|P)+S_\theta (x|U)\). If \(0{\lt}a\ell _h\le 1/8\), then

This is  [ 286 , Lemma 5.3 ] .

Let \(D_0{\gt}R\) be an integer, let \(Q\) be safe with parameter \(D_0\), and let \(Q_0^{+d'}\subseteq Q\). An admissible support with a site of \(A\) in \(Q\) lies in \(A\). If \(d+R\le d'\), no admissible support is split by both \(A\cap Q_0^{+d}\) and \(A\cap Q_0^{+d'}\). If \(R+1\le d'\), at most \(8(R+1)\mu _R(\operatorname {size}(Q_0)+d')\) admissible supports are split by \(A\cap Q_0^{+d'}\), where \(\mu _R=2^{2R(R+1)}\) bounds the number of admissible supports containing a site.

The picture shows the nested rectangles for range \(R=1\), \(d=1\) and \(d'=2\), with two admissible supports that are each split by exactly one contour.

For \(q\ge 1\), \(R\), \(J\ge 0\) and \(\Delta {\gt}0\) there are an integer \(C_{\mathrm{pad}}\ge 1\) and a constant \(C_{\mathrm{buf}}\ge 0\) with the following property, uniformly over every real \(D_0{\gt}2R+10\). Let \(\Omega \) be a unit vector satisfying (50) and (51). For a cut \(A\subseteq \Lambda \), let \(Z\) be the set of endpoints of its crossing edges. Let \(Q\) satisfy \(\operatorname {dist}_\infty (Q,Z){\gt}D_0\operatorname {size}(Q)\), let \(r\ge 1\) be an integer, and let \(Q_0\) be a rectangle of size at most \(r\) with \(Q_0^{+C_{\mathrm{pad}}r}\subseteq Q\). For \(X=A\cap Q_0\) and \(T=(A\cap Q_0^{+C_{\mathrm{pad}}r})\setminus X\),

This is  [ 286 , Lemma 3.2 ] .

Let \(\Lambda \subseteq \mathbb Z^2\) be finite, let \(q\in \mathbb N\) with \(q\ge 1\), and let \(\Omega \) be a unit vector on the domain with local dimension \(q\). Let \(A\subseteq \Lambda \) be arbitrary, let \(Q\) be a nonempty integer rectangle, and put \(s=\operatorname {size}(Q)\). Let \(j\in \mathbb N\) satisfy \(1\le j\le s\), and let \(Y\subseteq Q^{+j}\setminus Q^{+(j-1)}\) be arbitrary. Define \(R=A\cap (Q^{+(j-1)}\setminus Q)\) and \(R_Y=A\cap ((Q^{+(j-1)}\setminus Q)\cup Y)\). Then

This rectangle estimate follows from the partial-row argument of [ 286 , Lemma 9.4, lines 666–667 ] and is used in the rectangle covering argument in the proof of Proposition 9.5, lines 717–729. It includes an empty \(Y\), thin rectangles and missing lattice sites in the domain.

For \(q\ge 1\) the function \(F_{\mathrm{box}}\) is nonnegative, nondecreasing in \(r\), and satisfies \(F_{\mathrm{box}}(r)\le r^2\log q\).

For fixed \(q,R,J,\Delta ,r\) with \(q\ge 1\), write \(F_{\mathrm{box},D_0}(r)\) for the safe-box entropy at safety parameter \(D_0\). If \(D_0,D_1\in \mathbb N\) and \(D_0\le D_1\), then \(F_{\mathrm{box},D_1}(r)\le F_{\mathrm{box},D_0}(r)\).

Partition each axis of a rectangle \(Q\) of size at most \(Mr\) into chunks of \(r\) sites and one final shorter chunk. The products of chunks are at most \(M^2\) pairwise disjoint rectangles of size at most \(r\) covering \(Q\). If a point of one child lies in the \(Cr\)-padding of another, their chunk positions differ by at most \(C\) along each axis. A child at least \(C\) positions from the first chunk and \(C+2\) from the last along both axes has its \(Cr\)-padding inside \(Q\).

Let \(\Omega \) be a unit vector and write \(S=S_\Omega \). Let \(X_i\) be pairwise disjoint regions carrying colors in \(\{ 0,\dots ,N-1\} \), and suppose each \(i\) in a set \(G\) has a buffer \(T_i\) contained in the union of the regions of the other colors, with \(S(X_i\mid T_i)\le S(X_i)/2+\beta \). Then

Fix an integer \(D_0{\gt}2R+10\). With \(C_{\mathrm{pad}},C_{\mathrm{buf}}\) from Theorem 42.15.3, \(P=2C_{\mathrm{pad}}+1\) and \(p=1/(2P^2)\), for every \(M\) and \(r\ge 1\),

For fixed \(q\ge 1\), \(R\), \(J\ge 0\) and \(\Delta {\gt}0\), there are constants \(C\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), such that for every integer \(D_0{\gt}2R+10\) and every integer \(r\ge 1\), \(F_{\mathrm{box},D_0}(r)\le Cr^{1+e_0}\). This is the integer-clearance specialization of [ 286 , Proposition 3.3 ] ; the real-clearance statement follows from Theorem 44.24.14.

Let \(\Lambda \subset \mathbb Z^2\) be finite, let \(A\subseteq \Lambda \), and let \(Z\) be the endpoints of its crossing edges. Let \(D_0,j\in \mathbb N\), and let \(Q,Q'\) be integer rectangles, each a product of two finite nonempty integer intervals. Suppose that every point of \(Q\) has sup-norm distance greater than \(D_0\operatorname {size}(Q)\) from every point of \(Z\), and that

Then every point of \(Q'\) has sup-norm distance greater than \(D_0\operatorname {size}(Q')\) from every point of \(Z\). Here the size counts sites along the longer axis and \(Q^{+j}=Q+([-j,j]^2\cap \mathbb Z^2)\). This is the clearance step in [ 286 , proof of Proposition 9.5, lines 723–729 ] .

Fix a local dimension \(q\ge 1\), an interaction range \(R\), a term norm bound \(J\ge 0\), and a gap \(\Delta {\gt}0\). There exist constants \(C\ge 0\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), such that for every integer \(D_0{\gt}2R+10\), every finite induced square-lattice domain \(\Lambda \), every Hamiltonian \(H\) on \(\Lambda \) as in Definition 39.8.2 with a unit ground vector \(\Omega \) at energy \(E_0\) and full-system gap \(\Delta \), every cut \(A\subseteq \Lambda \), and every safe integer rectangle \(Q\),

Here \(Q\) is a product of two finite nonempty integer intervals, \(\operatorname {size}(Q)\) is the maximum of its side lengths in sites, and safety means that every site of \(Q\) has sup-norm distance greater than \(D_0\operatorname {size}(Q)\) from every endpoint of a crossing edge of \(A\). This is the pointwise integer-clearance specialization of [ 286 , Proposition 3.3 ] . The full real-clearance conclusion follows from Theorem 44.24.14.

For the initial recurrence parameters,

This is the logarithmic form of the exponent \(e_0\), following the exponent calculation in [ 286 , proof of Proposition 3.3, lines 651–663 ] .

For the initial recurrence parameters,

Let \(0{\lt}e_0\le 1\), \(0{\lt}\varepsilon \le 1\) and \(0{\lt}\ell {\lt}1\). Then

For the initial exponent \(0{\lt}e_0{\lt}1\), use \(\ell \) and \(e_*\) from Definition 39.12.1.1. Then the round count \(m\) of Theorem 42.19.1, with \(\varepsilon =e_*\), satisfies \(m\le B\).

With the initial exponent \(e_0\) and the parameters of Definition 39.12.1.1,

Let \(q\ge 1\), \(R\in \mathbb N\), \(J\ge 0\) and \(\Delta {\gt}0\). Fix transport constants \(c_t,C_{\rm ent},e_{\rm ent},C_{\rm en},e_{\rm en}\) with \(c_t{\gt}0\) satisfying the finite-family transport bound. Set \(C_0\) to the explicit initial safe-box constant and let \(G\) be the physical one-round improvement with truncation constant \(C_{\rm tr}(R,J,\Delta ,2)\), as in Theorem 44.24.13. Define

There is a single integer \(m\le B\), independent of the safety parameter and scale, such that for every \(D_0{\gt}2R+10\) and \(r\ge 1\),

The coefficient is evaluated at the actual round count \(m\); no comparison of \(C_m\) with \(C_B\) is asserted.

Adjacent sites \(x,y\) satisfy \(|x_a-y_a|\le 1\) for \(a=1,2\) and \(|\rho (x)-\rho (y)|\le 1\). In particular, if \(x\in K(c,r)\), then \(y\in K(c,r+1)\). For any cut \(A\), an unordered pair belongs to \(\partial _\Lambda A\) precisely when its sites are adjacent and exactly one belongs to \(A\). Consequently, for adjacent \(x,y\),

If \(r\ge 0\), then \(|K(c,r)|\le (2r+1)^2\). If also \(r\le 4\), then \(|K(c,r)|\le 81\).

For every \(r\ge 0\), the number of unordered induced edges crossing the sampled square satisfies \(|\partial _\Lambda K(c,r)|\le 8r+4\).

The family contains \(m\ge 1\) patches and \(m{\gt}u/2\). Every selected radius satisfies \(u\le r_j\le 2u\) and every selected patch satisfies \(|\partial _\Lambda K_j|\le 16u+4\). For indices \(i{\lt}j\), one has \(r_i+2\le r_j\). For \(i\le j\), one has \(K_i\subseteq K_j\); clipping can make these inclusions equalities.

Suppose \(e\in \partial _\Lambda K(c,r)\). If \(s+1\le r\), then the two sites of \(e\) are disjoint from \(K(c,s)\). If \(r+1\le t\), then both sites belong to \(K(c,t)\).

Let \(e=\{ x,y\} \) be an induced nearest-neighbour edge. If \(e\in \partial _\Lambda K_j\), then, for every \(i{\lt}j\), \(\{ x,y\} \cap K_i=\varnothing \), and, for every \(j{\lt}k\), \(\{ x,y\} \subseteq K_k\). Consequently, an edge crosses at most one patch in the family, and the crossing-edge sets of distinct patches are disjoint.

Let \(u\ge 4\), \(\kappa \in \mathbb R\), and assign each selected patch the same weight \(t_j=\kappa /m\). Then

One has \(R\le s_R\le R+1\), and \(s_R\ge 2\) when \(R\ge 1\). At \(R=1\) the count, radii and patches are those of Definition 42.20.6. The family has \(m_R\ge 1\) patches and \(m_R{\gt}u/s_R\). If \(u\ge 0\), every radius satisfies \(u\le r_j\le 2u\). For \(i{\lt}j\) one has \(r_i+s_R\le r_j\), and for \(i\le j\) one has \(K_i\subseteq K_j\).

The endpoints \(x,y\) of a walk of length \(\ell \) in the induced graph satisfy \(|\rho (x)-\rho (y)|\le \ell \). Hence any two sites of a support of graph diameter at most \(R\) satisfy \(|\rho (x)-\rho (y)|\le R\). If the values of \(\rho \) on a set \(X\) differ pairwise by at most \(R\), then at most one patch \(K_j\) of Definition 42.20.1.1 both meets \(X\) and misses a site of \(X\).

If the values of \(\rho \) on \(X\) differ pairwise by at most \(R\), and \(X\) has a site \(y\) outside \(K(c,r)\), then every site of \(X\) in \(K(c,r)\) lies in \(\mathcal N(c,r,R)\). If \(r\ge 0\), then \(|\mathcal N(c,r,R)|\le R(8r+4)\), and for the patches of Definition 42.20.1.1 with \(u\ge 0\), \(|\mathcal N(c,r_j,R)|\le R(16u+4)\).

Let \(R\ge 1\), \(u\ge 0\) and \(\kappa \in \mathbb R\), and assign each patch of Definition 42.20.1.1 the weight \(t_j=\kappa /m_R\). Then

Assume the square-grid entropy area law for a family of states with constant \(C_A\), and put \(C_S=\max \{ 1,16C_A\} \). For every \(L\), every state \(\Omega \) of the family, every centre \(c\), every \(u\ge 0\), and every \(0\le j{\lt}m\), the nested patch \(K_j\) in the integer square of side \(L\) satisfies \(S_{U\Omega }(K_j)\le C_S(1+u)\). This is the entropy bound in the proof of Proposition 4.1 with its constant explicit.

Assume the square-grid entropy area law. Let \(q\ge 2\) be an integer and \(J,\Delta {\gt}0\). There is a constant \(C_S\ge 1\) with the following property. For every \(L\), every original square Hamiltonian \(H\) with interaction bound \(J\), every unit vector \(\Omega \) and real \(E_0\) with \(H\Omega =E_0\Omega \) and \(H-E_0I\ge \Delta (I-|\Omega \rangle \langle \Omega |)\), every centre \(c\), every \(u\ge 0\), and every \(0\le j{\lt}m\), the nested patch \(K_j\) in the integer square of side \(L\) satisfies the entropy bound in the proof of Proposition 4.1:

The constant is chosen before \(L\), \(H\), \(\Omega \), \(c\) and \(u\). The area law is a hypothesis here; its derivation from the spectral gap is the separate rectangular specialization of the finite-domain area law, which is not yet established.

Let \(Q\) be a nonempty integer rectangle and let \(j\in \mathbb N\). Then

Let \(Q\) be a nonempty integer rectangle and put \(s=\operatorname {size}(Q)\). Let \(j,L,K\in \mathbb N\) satisfy \(j\le L\le s\) and \(L{\lt}2^{K+1}\). Put \(S_j=Q^{+j}\setminus Q\) and \(N_K=\# \{ (r,z)\in \mathcal P_K(S_j):r=K\} \). Then

Let \(Q\) be a nonempty integer rectangle and put \(s=\operatorname {size}(Q)\). Let \(j,K,k\in \mathbb N\) satisfy \(j\le s\), \(k{\lt}K\) and \(2^{k+1}\le s\). Put \(S_j=Q^{+j}\setminus Q\) and \(N_k=\# \{ (r,z)\in \mathcal P_K(S_j):r=k\} \). Then

Let \(S\subseteq \mathbb Z^2\) be finite, let \(K\in \mathbb N\), and let \(f:\mathbb N\to \mathbb R\) be arbitrary. Then

In the covering argument of [ 286 , Lemma 9.4, lines 656–664 ] , this finite summation by scale holds without sign conditions on \(f\), including when \(S=\varnothing \) or \(K=0\).

Let \(S\subseteq \mathbb Z^2\) be finite and let \(K,L\in \mathbb N\) satisfy \(2^K\le L\). Let \(e,A,B\in \mathbb R\) with \(e{\gt}0\) and \(B\ge 0\). Write \(N_k=\# \{ (r,z)\in \mathcal P_K(S):r=k\} \). Suppose \(2^K N_K\le A\) and \(2^k N_k\le B\) for every \(0\le k{\lt}K\). Then

In the covering argument of [ 286 , Lemma 9.4, lines 656–664 ] , this finite moment estimate holds including when \(S=\varnothing \) or \(K=0\).

Let \(Q\) be a nonempty integer rectangle and put \(s=\operatorname {size}(Q)\). Let \(j,L,K\in \mathbb {N}\) satisfy \(j\le L\le s\) and \(2^K\le L{\lt}2^{K+1}\), and let \(e{\gt}0\). Set \(S_j=Q^{+j}\setminus Q\), with both rectangles viewed as sets of lattice sites. Then the capped maximal-contained dyadic partition satisfies the geometric weighted covering estimate of [ 286 , proof of Proposition 9.5, lines 717–729 ] :

For every \(k\in \mathbb {N}\) and \(z\in \mathbb {Z}^2\), the lattice points of \(R_{k,z}\) are exactly \(C_{k,z}\), and \(\operatorname {size}(R_{k,z})=2^k\). For a finite physical domain \(\Lambda \) and a region \(A\subseteq \Lambda \), its physical region is \(A\cap C_{k,z}\).

Fix a finite lattice domain \(\Lambda \), a region \(A\subseteq \Lambda \), a finite lattice set \(S\), and \(K\in \mathbb {N}\). The physical regions \(A\cap R_{k,z}\) for \((k,z)\in \mathcal P_K(S)\) are pairwise disjoint, and their union is \(A\cap S\). For a unit physical vector \(\Omega \) on \(\Lambda \) with local dimension \(q\in \mathbb {N}\), one has

This is the physical summation step of [ 286 , Lemma 9.4, lines 651–664 ] .

Fix a finite lattice domain \(\Lambda \), a physical region \(A\), and \(D_0\in \mathbb {N}\). Let \(Q\) be a safe integer rectangle and put \(s=\operatorname {size}(Q)\). Suppose \(j,L,K\in \mathbb {N}\) satisfy \(j\le L\), \(2^K\le L\), and

Then each selected square of \(\mathcal P_K(Q^{+j}\setminus Q)\), interpreted as its integer rectangle, is safe. This is a sufficient clearance condition for the covering argument in [ 286 , proof of Proposition 9.5, lines 723–729 ] .

Fix a finite lattice domain \(\Lambda \) with local dimension \(q\), a unit physical vector \(\Omega \), and a region \(A\). Let \(S\) be a finite set of lattice sites, let \(K\in \mathbb {N}\), and let \(e,C\in \mathbb {R}\). Suppose that every square \(R_{k,z}\) selected by \(\mathcal P_K(S)\) satisfies

Then

This is the conditional summation step in [ 286 , proof of Lemma 9.4, lines 651–664 ] .

Fix a finite lattice domain \(\Lambda \) with local dimension \(q\), a unit physical vector \(\Omega \), a region \(A\), and \(D_0\in \mathbb {N}\). Let \(S\) be a finite set of lattice sites, let \(K\in \mathbb {N}\), and let \(e,C\in \mathbb {R}\). Suppose that each square selected by \(\mathcal P_K(S)\) is safe, and that every safe integer rectangle \(R\) satisfies

Then

This is the conditional summation step in [ 286 , proof of Lemma 9.4, lines 651–664 ] .

Fix a finite lattice domain \(\Lambda \) with local dimension \(q\), a unit physical vector \(\Omega \), a region \(A\), and \(D_0\in \mathbb {N}\). Let \(e{\gt}0\) and \(C\ge 0\), and assume that every safe integer rectangle \(R\) satisfies

Let \(Q\) be safe and put \(s=\operatorname {size}(Q)\). For \(j,L,K\in \mathbb {N}\) with \(j\le L\le s\), \(2^K\le L{\lt}2^{K+1}\), and \(D_0L+L\le D_0s\), one has

This is the conditional shell entropy estimate used in [ 286 , proof of Proposition 9.5, lines 717–729 ] .

Fix a local dimension \(q\in \mathbb N\) with \(q\ge 1\), an interaction range \(R\in \mathbb N\), a term norm bound \(J\ge 0\), and a gap \(\Delta {\gt}0\). There exist constants \(C\ge 0\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), with the following property simultaneously for every integer safety parameter \(D_0{\gt}2R+10\). Let \(\Lambda \) be any finite induced square-lattice domain, let \(H\) be an admissible Hamiltonian on \(\Lambda \) with range \(R\) and term norm bound \(J\), and let \(\Omega \) be its normalized ground vector at energy \(E_0\) with full-system projector gap \(\Delta \). Let \(A\subseteq \Lambda \) be any cut, let \(Q\) be any safe integer rectangle, and put \(s=\operatorname {size}(Q)\). For all \(j,L\in \mathbb N\) satisfying \(j\le L\le s\) and \(D_0L+L\le D_0s\), one has

As the initial-exponent consequence of [ 286 , Proposition 3.3, lines 590–604 ] for the rectangle-shell entropy input in the proof of Proposition 9.5, lines 717–729, the constant \(C\) and exponent \(e_0\) are independent of the safety parameter, domain, Hamiltonian, ground energy, ground vector, cut, rectangle, and shell scales.

Let \(C_2,\eta ,e\ge 0\) be real numbers, and let \(s\in \mathbb N\) with \(s\ge 1\). Put \(L_s=\lfloor \lceil C_2s\rceil ^\eta \rfloor \). Then

Fix a local dimension \(q\ge 1\), an interaction range \(R\in \mathbb N\), a term norm bound \(J\ge 0\), and a gap \(\Delta {\gt}0\). There exist constants \(C\ge 0\) and \(e_0\in (0,1)\), depending only on \(q,R,J,\Delta \), with the following property. For each real \(C_2{\gt}0\) there is an integer threshold \(N\ge 1\) such that the following holds.

Let \(D_0{\gt}2R+10\) be an integer, let \(\Lambda \) be any finite induced square-lattice domain, and let \(H\) be an admissible Hamiltonian on \(\Lambda \) with range \(R\) and term norm bound \(J\). Let \(\Omega \) be its normalized ground vector at energy \(E_0\) with full-system projector gap \(\Delta \). Let \(A\subseteq \Lambda \) be any cut and \(Q\) any safe integer rectangle of size \(s\ge N\). Put \(\ell =(1-e_0)/200000\), \(\eta =1-\ell \), and \(L_s=\lfloor \lceil C_2s\rceil ^\eta \rfloor \). For every integer \(0\le j\le L_s\), one has

The exponent \(1+\eta e_0\) is strictly smaller than \(1+e_0\). This estimates the shell; the improvement of the safe-box exponent still requires the remaining physical comparison argument.

Let \(b,e,e_0,\eta ,C_2\in \mathbb R\) satisfy \(0{\lt}b\le e\le e_0\), \(\eta \ge 0\), and \(C_2\ge 0\). Then

Fix \(C_2{\gt}0\) and \(e_0\in (0,1)\), and set \(\ell =(1-e_0)/200000\) and \(\eta =1-\ell \). There is an integer \(N\ge 1\), depending only on \(C_2\) and \(e_0\), with the following property. Let \(b{\gt}0\), let \(\Lambda \) be any finite induced square-lattice domain, let \(q\in \mathbb N\), and let \(\Omega \) be a unit vector on its configurations. Let \(D_0\ge 1\) be an integer, let \(A\subseteq \Lambda \) be any cut, and let \(Q\) be any safe integer rectangle with \(s=\operatorname {size}(Q)\ge N\). Suppose that \(b\le e\le e_0\) and that \(C\in \mathbb R\) satisfies, for every safe integer rectangle \(P\),

Put \(L_s=\lfloor \lceil C_2s\rceil ^\eta \rfloor \). For every integer \(0\le j\le L_s\), one has

Let \(S,C,\tau ,E,B_u,B_l\in \mathbb R\) satisfy \(S\ge 0\), \(C\ge 0\), \(\tau {\gt}0\) and

Let \(F,d,r_u,r_l,r_e\) be real sequences. Suppose that \(r_u(k),r_l(k),r_e(k)\) tend to zero as \(k\to \infty \), and that for every sufficiently large \(k\),

Then

The entropy \(S\) and all fixed coefficients are independent of \(k\). The two bounds \(B_u\) and \(B_l\) include any substitutions of the typical entropy and auxiliary dimension made before these scalar comparisons.

Fix \(e_0\in (0,1)\) and \(C,C_E\in \mathbb R\), and let \(\nu =(1-e_0)/2000\). There is an integer \(N\ge 1\), depending only on these fixed coefficients, with the following property. For every integer \(n\ge N\), put

Then

This is the numerical threshold used after the fixed-scale replica limit. It does not require a replica convergence rate uniform in \(n\).

Fix a unit state \(\Omega _t\) on a finite subset of the square lattice, a region \(X\), and a spectral set \(E\) for its reduced density on \(X\). Write \(S=S_{\Omega _t}(X)\). Suppose that the selected mass \(z\) is positive and that every selected eigenvalue \(p_i\) satisfies

Let \(d_*=|E|\), let \(\psi \) be the normalized selected Schmidt vector, with the selected Schmidt indices enumerated in its auxiliary register, and let \(S'\) be the entropy of that register. Put \(\eta =w-\log z\). Fix \(C\ge 0\), \(0{\lt}\tau {\lt}1/2\), \(a{\gt}0\), \(E_{\rm def}\ge 0\) and real bounds \(B_{\rm sh},B_{\rm exc}\) such that

Let \(F,d,r_u,r_l,r_e\) be real sequences. Suppose that \(r_u(k),r_l(k),r_e(k)\) tend to zero and that, for all sufficiently large \(k\),

Before the shell and mismatch terms are estimated in [ 286 , Proposition 9.5 ] , these rough comparisons imply

retaining the same physical state and selected spectrum while leaving the physical derivation of the comparisons separate.