In [ 286 , Proposition 8.1 ] , the statement of the norm comparisons holds with \(a_0=1/10\), \(C=4\), \(C'=4+25C_0\) and \(c=c_0/10\), where the label sequence is chosen before the status family and the weight, and the two lower comparisons hold for every vector of norm at most one in \(\mathcal S_k\) carrying the label \(\lambda _k\) on both auxiliary systems, in particular for \(\mathrm{pre}_k\).

The scanner statement in Definition 44.2.1 holds for \(J\ge 0\), \(\Delta {\gt}0\) and \(C_T\ge 0\), establishing [ 286 , Proposition 9.2 ] with all four norm comparisons under the coefficient convention in  [ 75 ] .

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 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 43.1.3.

Fix a finite index set \(J\), a finite nested list \(\mathcal R\) of physical regions disjoint from \(X\), and \(K\geq 8\). Following the label-sequence application of the rough upper comparison in [ 286 , Section 8, lines 130–169 and 332–381 ] , there is a constant \(c\geq 0\) for which the following holds eventually in \(k\), with \(c\) and the eventual threshold chosen before \(Z,U,Y,T,B_{\mathrm{sh}}\) and \(B_{\mathrm{exc}}\). 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

Fix finite sets \(\Lambda ,T\subseteq \mathbb Z^2\) with \(T\ne \varnothing \), a positive local dimension \(q\), and a unit vector \(\Omega _t\) on \(\Lambda \). Let the collar scanner have \(K\) ordered bands, initial near region \(A\subseteq \Lambda \), offset scale \(m{\gt}0\), and charge capacity \(M{\gt}0\). Its depth is the ambient distance \(d_T(x)=\min _{t\in T}\| x-t\| _\infty \), and its target is \(X=A\cap T\). Fix an integer \(L\) with \(8Km\leq L\) and use the horizon \(nm\). Fix a positive-mass spectral set \(E\) of the reduced density on \(X\), with \(d=|E|\) and mass \(z{\gt}0\). Let \(\lambda _k\) be one label sequence whose selected Schmidt vector satisfies the eventual mass bound \(\| (I\otimes P_{\lambda _k})\psi ^{\otimes k}\| ^2 \geq [2(k+1)^{d^2}]^{-1}\). Let \(a{\gt}0\), \(6a\leq r\), and \(6a\leq 1\). Suppose that, for every completed or pre-charge status through the scan horizon, each of its five physical marginals satisfies

Here \(Z_Q\) is the surprisal of the reduced density on \(Q\). All statuses retain their completed or pre-charge physical partitions. For each terminal \(j\) and band \(g\), put

where \(U_{jg}\) is the positive-depth near region and \(Y_{jg}\) is the middle region. At every fill or charge interpolation, let \(T_p\) be its original weighted mean tree, indexed by these terminals, and let \(N_{k,p}\) be the squared filtered norm of the original prevector, with metric parameter \(a/2\). There is \(c\geq 0\) such that, eventually in \(k\), simultaneously at all scan times and all real interpolation parameters,

The constant and replica threshold are chosen before the scan time, interpolation parameter, and mean tree. The interpolation parameter is projected onto \([0,1]\), as in the collar scanner.

In Theorem 44.4.1, suppose additionally that \(\Omega _t\) is an exact eigenvector of a given physical Hamiltonian with eigenvalue \(e\). One can choose a single label sequence whose original prevector has the exact mean-replica energy \(e\), full copy symmetry, and the prescribed label on both original auxiliary registers. Its selected mass satisfies the stated polynomial lower bound, and \(\log \dim \lambda _k-kS(\psi _C)=o(k)\). The same sequence and physical prevector satisfy (21) at every fill and charge root.

Fix \(q\geq 1\), an interaction range \(R\in \mathbb N\), a term norm bound \(J\geq 0\), a gap \(\Delta {\gt}0\), and a target-size coefficient \(C_T\geq 0\). There are positive truncation, target-bound and common-status-bound coefficients \(C_{\mathrm{tr}},C_{\mathrm{target}},C_B\), chosen before the exponent family and physical instance. 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 \(\kappa \leq e\), and a real \(\omega \) satisfying the explicit margin

There is \(N\geq 2\), depending on this exponent family and margin, such that the following holds for \(n\geq N\). Let \(\Lambda \subseteq \mathbb Z^2\) be finite, let \(T\subseteq \mathbb Z^2\) be finite and nonempty, and let \(H=\sum _i h_i\) have interaction range \(R\) and term norm bound \(J\) on \(\Lambda \). Suppose \(\| \Omega \| =1\), \(H\Omega =E_0\Omega \), and \(H-E_0I\succeq \Delta (I-|\Omega \rangle \langle \Omega |)\). Use a collar scanner with the induced lattice graph, anchors in their labelled interaction supports, \(M{\gt}0\), depth \(d_T(x)=\min _{t\in T}\| x-t\| _\infty \), and target \(X=A\cap T\). The scales are

Assume \(|T|\leq C_Tn^2\), each ambient dilation layer \(T_d\setminus T_{d-1}\) has at most \(n\) sites for \(1\leq d\leq L\), and every endpoint of a cut edge of \(A\) has ambient distance greater than \(2L+10r_0\) from every point of \(T\). One truncated unit ground vector \(\Omega _t\) satisfies the energy, phase, trace-distance, all-cut tail and target-window estimates of Theorem 43.3.77. In particular \(m,K\geq 1\). The same vector has the common status bound and moment constants

For \(0{\lt}W\leq n^\omega \), put \(a=W/K\). Then \(6a\leq c_0/\sqrt B\) and \(6a\leq 1\). Every label sequence satisfying the selected mass condition for this same target spectral set satisfies (21), with \(K_{\mathrm{mom}}=C_0B\). The coefficients are uniform in the physical instance; the replica-copy threshold and remainder constant may depend on the fixed finite-domain instance and its label sequence. With (23) explicit, this gives the target-instance consequence rather than the complete comparison assertion of Proposition 9.6 in [ 286 ] .

Fix a finite lattice domain \(\Lambda \), interaction range \(R\), term norm bound \(J\), positive physical and auxiliary dimensions \(q,d\), and \(a{\gt}0\). There are \(c,C_f\geq 0\), chosen before the scanner, finite horizon, physical states, moment parameters, copy number, labels and interpolation parameter, with the following property. Let the scanner be indexed by the ordered admissible interaction supports at range \(R\), with offset scale \(m{\gt}0\) and charge capacity \(M{\gt}0\). Let \(\widetilde H\) be the sum of the truncated energy terms, formed from a fixed reference state, and suppose that a unit vector \(\widetilde\Omega \) satisfies

For every terminal and band, assume the original marginal moment estimate

for \(U=Q_{jg},Y_{jg},(Q_{jg}\cup Y_{jg})^c\), where \(K_{\rm mom}\geq 0\), \(10a\leq r_{\rm mom}\) and \(10a\leq 1\). Here \(Z_U\) is the surprisal under the spectral probability distribution, with zero weights contributing zero. Entropies and moments refer to the same original state \(\widetilde\Omega \). Fix \(k{\gt}0\) and a nonzero prevector of norm at most one, fixed by simultaneous copy symmetry and both auxiliary label projectors \(\ell _C,\ell _R\), and assume

Define

At every fill or charge interpolation before the horizon, let \(N_k\) be its squared filtered norm, \(\mu _k\) its mean energy, and \(w_j\) its terminal weights under the inclusion into the common terminal family. Its filtered vector has norm one, \(N_k{\gt}0\), and \(\delta _k=(\mu _k-e_0)/\gamma \geq 0\). For every \(0{\lt}\tau \leq \tfrac 12\), put

Then

The same prevector, labels, gap and moment data serve all these interpolations, including a scanner with zero bands.

For every label sequence \(\ell _k\) and every \(k\), \(\varepsilon _k\geq 0\) and

Fix \(a{\gt}0\) and \(c,C_f\in \mathbb R\). If a label sequence \(\ell _k\) satisfies \(\log \dim \ell _k-kS'=o(k)\), then \(R_k^-\to 0\) as \(k\to \infty \) through positive integers; in the physical application, this asymptotic holds for the Schmidt–Bell label sequence.

Under the scanner and gap conditions of Theorem 44.4.1.1, use the horizon \(nm\), fix \(K_{\rm mom}\geq 0\), \(a{\gt}0\), \(10a\leq r_{\rm mom}\) and \(10a\leq 1\), and take the augmented replica prevector of a label sequence \(\ell _k\) whose original-coordinate vectors are nonzero and have norm at most one for every \(k{\gt}0\), with \(\log \dim \ell _k-kS'=o(k)\) (as holds for the physical Schmidt–Bell prevector). Assume the marginal moment estimate of that theorem for all five regions of Definition 43.3.1.2 at every completed status through \(nm\) and every pre-charge status before \(nm\); these five bounds imply all the near, middle and far marginal bounds. Then there exist \(c,C_f\geq 0\), fixed before \(k\), scan time and interpolation parameter, such that \(R_k^-\to 0\) and every fill or charge obeys

Here \(N_k{\gt}0\), \(\delta _k\geq 0\), \(L_g\), the unit filtered vector and the terminal weights are those of the same interpolation in the previous theorem.

Fix a local dimension \(q\geq 1\), an interaction radius \(R\in \mathbb N\), an interaction bound \(J\geq 0\), a gap \(\Delta {\gt}0\) and a coefficient \(C_T\geq 0\) for the target-size bound. There are positive constants \(C_{\rm tr},C_{\rm target},C_B\) with the following property. For fixed scanner exponents and a real weight exponent \(\omega \) satisfying

there is an integer threshold \(N\geq 2\) before every physical instance. Let \(\Omega \) be an original unit ground state, of energy \(E_0\) and gap \(\Delta {\gt}0\), for the finite-domain local Hamiltonian. The scanner graph is the induced domain graph, its depth is the target’s ambient depth, and each anchor belongs to its original admissible support. Suppose \(n\geq N\), the target is nonempty with size at most \(C_Tn^2\), its ambient layers have size at most \(n\), and the target clearance is greater than \(2L+10r_0\). Retain the explicit scales

Then \(K\geq 1\), \(m{\gt}0\), \(8Km\leq L\) and \(B\geq 1\).

Put \(g=\Delta /\nu \), where \(\nu \) is the positive physical normalization, and \(\varepsilon =\min \{ n^{-1000},g/4\} \). One unit truncated ground state \(\widetilde\Omega \) has energy \(e\) and gap \(g/2\), with

This same state has a target typical set of width \(w=n^{3/5}\) and mass \(z\) satisfying

The target cut weight is at most \(C_{\rm target}n(\log n)^{12}\), and every eigenvalue in this set lies between \(\exp (-S(\widetilde\Omega _X)-w)\) and \(\exp (-S(\widetilde\Omega _X)+w)\). Put

For every completed or pre-charge history within the physical horizon and every one of its five marginal regions \(Q\), the same state satisfies

Finally, for every \(0{\lt}W\leq n^\omega \), with \(a=W/K\),

The physical state and target set are chosen before \(W\) and before any auxiliary label sequence.

For fixed \(d,c,K,C\), the remainder in (48) tends to zero as \(k\to \infty \).

Suppose the ambient target is nonempty, the scanner graph is the induced domain graph, its depth is the target’s ambient depth, \(n,m,M{\gt}0\), \(r_0\leq D\), \(D\geq 1\), \(4D\leq m\) and \(8Km\leq L\). Assume every ambient target layer up to \(L\) has size at most \(n\) and every whole-row positive-depth prefix of the truncated state has entropy at most \(B_{\rm sh}\). There are nonnegative constants \(c,C_i\), before the replica count, such that eventually the final squared filtered norm is positive and

Here \(N_k^{\rm initial}\) and \(N_k^{\rm final}\) are the squared filtered norms at the fill endpoints \((0,0)\) and \((H,0)\).

Fix \(q\geq 1\), a positive normalized gap \(g\) and a volume coefficient \(C_{\rm vol}\). For all sufficiently large \(n\) (above a threshold depending only on \(q\), \(g\) and \(C_{\rm vol}\)), suppose the original and truncated vectors are unit, the compact collar has at most \(C_{\rm vol}n^2\) sites, and the trace distance between the two states is at most \(\sqrt{2\min \{ n^{-1000},g/4\} /g}\). If every original-state positive-depth prefix up to \(L\) has entropy at most \(C_e(nL^e+n)\), then the endpoint bound (49) holds with

All geometric and same-sequence premises of the preceding theorem are retained.

Let \(k{\gt}0\), \(A_k=\log N_k^{\rm initial}-\log N_k^{\rm final}\), and \(\beta _k\leq C_\beta \log (k+2)\). Assume the endpoint inequality (49). For any real rate \(T\),

In the scanner entropy bound, \(T=C_{\rm ent}aK\, a^{1/4}\ell ^{e_{\rm ent}}\).

Let \(k,n,m,D,K\) be positive integers and \(a,C_1,b\) positive real numbers, set \(H=nm\) and \(\rho =ka/(2(C_1+1)nD)\). If a selected point satisfies \(Q\leq B/(\rho H b)\) and \(B/k\leq R\), then at that point

Here \(b=p_1-p_0\) for the selected parameter interval.

Let \(a,W,\tau {\gt}0\). Suppose the same scalar rate \(R\) satisfies

Then

Fix a unit physical truncated ground vector with energy \(e\) and positive gap \(\gamma \), the original reference state defining its truncated terms, and a target set \(E\) of positive spectral mass \(z\). Let \(d=|E|{\gt}0\), \(K{\gt}0\), \(W=aK\) and \(a{\gt}0\). Suppose the collar scanner has the target depth, positive window and charge-slot counts, and \(8Km\le L\). Every completed and pre-charge status marginal of this same vector satisfies the common moment estimate with parameters \(K_{\rm mom}\ge 0,r\), and \(10a\le r,1\).

The given physical label sequence has nonzero initial vectors of norm at most one, the polynomial selected-state projection mass, and \(\log \dim \lambda _k-kS'=o(k)\). Then nonnegative constants for the upper and lower replica remainders can be chosen before all charge stages and parameters. Eventually in \(k\), at every charge stage and every real interpolation parameter, the filtered norm is positive, \(\delta =(\mu -e)/\gamma \ge 0\), and for every \(0{\lt}\tau \le 1/2\),

where both \(R_+(k)\) and \(R_-(k)\) tend to zero at fixed physical data and weight. The same exact terminal tree appears on both sides.

Under the selector’s physical graph, anchor, layer, multiplicity, clearance, separation and round-step smallness hypotheses, one common error sequence \(r_k\ge 0\), \(r_k\to 0\), works before all symmetric nonzero initial vectors and all intervals \([p_0,p_1]\) with \(0{\lt}p_0{\lt}p_1{\lt}1\). If the same initial vector is an eigenvector of the augmented truncated copy mean with energy \(e\), the selected point also satisfies

Its selected charge-defect bound is unchanged.

Suppose the selector–energy conclusion, the accepted coefficient conclusion and the uniform sharp comparison hold for the same physical data. Define

Then the same selected point obeys \(\delta _k\le (e+\operatorname {cap}_k+r_k)/(g/2)\). This upper bound can replace \(\delta \) in (56).

Let \(V\) be finite, let \(q,d,k\in \mathbb N\), let \(\Omega \) be a complex vector on the physical configurations, and let \(\lambda \) be a label of the symmetric group on \(k\) copies. Let \(v\) be the corresponding physical Schmidt–Bell prevector in the global coordinates, and let \(\mathrm{pre}=Uv\) be its expression in the augmented site coordinates, where \(U\) is the coordinate isometry. For a physical matrix \(H\), write \(H_{\rm glob}\) and \(H_{\rm aug}\) for its replica mean, with identities on the two auxiliary registers, in these two coordinate systems. If \(v\ne 0\) and \(H_{\rm glob}v=e v\) for a real \(e\), then

Here \(\mathcal H_{\rm sym}\) is the subspace invariant under simultaneous permutation of the copies. This coordinate statement is used in [ 286 , Section 8, lines 130–153 and 421–438 ] .

Let \(k,n,m,D,K\) be positive integers, let \(d\in \mathbb N\), and let \(a,C_1,w{\gt}0\). Put \(W=aK\) and \(\rho =ka/(2(C_1+1)nD)\). For real \(A,B,T,b,c,C_i,C_b,\mathcal Q\), suppose

Then

This is the scalar normalization of the selected charge in [ 286 , Section 9, lines 457–490 ] .

Fix an interaction range \(R\in \mathbb N\), \(J\geq 0\), \(\Delta {\gt}0\) and \(C_T\geq 0\). There is \(C_{\rm tr}{\gt}0\) such that, for every integer \(q\geq 1\), every \(C_1{\gt}0\), every integer anchor-multiplicity bound \(m_*\geq 0\) with \(3m_*\leq C_1\), and every real exponent \(P\), there are \(C_{\rm target},c_t{\gt}0\) and real \(C_{\rm ent},e_{\rm ent}\) with the following property. For every scanner exponent tuple, there are \(C\geq 1\), \(C_l\geq 0\) and \(N\geq 2\) before all the physical data and entropy inputs below. Thus \(C_{\rm tr}\) is independent of \(q\), \(C_1\), \(m_*\) and \(P\). This is the stronger uniform choice of Theorem 43.3.78, derived from the uniform marginal-tail theorem. The printed truncation proposition permits dependence on \(q\); the scanner uses its radius without asserting dimension uniformity [ 286 , Proposition 4.5 and Section 9, lines 49–59 ] .

Let \(\Lambda ,T\subset \mathbb Z^2\) be finite, with \(T\ne \varnothing \), and let \(H\) be a range-\(R\) Hamiltonian of interaction bound \(J\), with unit ground vector \(\Omega \) and projector gap \(\Delta \). Use the induced lattice graph, anchors belonging to their interaction supports, and depth \(d(x)=\operatorname {dist}_\infty (x,T)\). For the scan on \(A\subseteq \Lambda \), assume \(n\geq N\) and the scales

The exponent conditions are \(0{\lt}\ell {\lt}1\), \(0{\lt}\kappa {\lt}\mu {\lt}1-\ell \), \(\nu {\gt}0\) and \(0{\lt}e_{\rm in}{\lt}1\) with \(\kappa \leq e_{\rm in}\). The charge constant is \(C_1\). Put \(M=\lceil C_1nD\rceil \). Suppose \(|T|\leq C_Tn^2\), every ambient dilation layer through depth \(L\) has at most \(n\) sites, and at most \(m_*\) interaction supports have any given anchor. Every point of \(T\) is at distance greater than \(2L+10r_0\) from every endpoint of a cut edge between \(A\) and its complement. Also suppose \(2r_0\leq D\), \(1\leq D\) and \(4D\leq m\). For arbitrary real \(C_e,\eta \), assume \(S_\Omega (\{ x\in A:0{\lt}d(x)\leq j\} )\leq C_e(nL^\eta +n)\) for every integer \(0\leq j\leq L\).

Put \(X=A\cap T\) and \(S_0=A\cap T_L\), where \(T_L\) is the ambient dilation of \(T\). Let \(g{\gt}0\) be the normalized amplified gap in Theorem 43.3.77, and set

Then \(m,M{\gt}0\), \(K\geq 8\), \(8Km\leq L\), \(|S_0|\leq (C_T+1)n^2\) and \(X\subseteq S_0\). There is a unit ground vector \(\widetilde\Omega \) of the truncated Hamiltonian \(\widetilde H\), with energy \(e_t\) and projector gap \(g/2\), such that

Write \(p_i\) for the target-marginal eigenvalues and \(S_X=S_{\widetilde\Omega }(X)\). Their surprisal tail outside width \(w\) has mass at most \(\delta _t\). The typical set \(E=\{ i:p_i{\gt}0,\ |{-\log p_i}-S_X|\leq w\} \) has cardinality \(d{\gt}0\) and mass \(z\) with \(0{\lt}1-\delta _t\leq z\leq 1\); every \(i\in E\) obeys \(\exp (-S_X-w)\leq p_i\leq \exp (-S_X+w)\).

Let \(\psi \) be the normalized Schmidt vector restricted to \(E\), and let \(S'\) be its auxiliary entropy. One sequence \(\lambda _k\) and its physical Schmidt–Bell prevectors \(\mathrm{pre}_k\) satisfy \(\mathrm{pre}_k\ne 0\) and \(\| \mathrm{pre}_k\| \leq 1\) for all \(k{\gt}0\), and \(\log \dim \lambda _k-kS'=o(k)\). For all sufficiently large \(k\),

The state \(\widetilde\Omega \), the set \(E\), the compressed vector \(\psi \) and the sequences \((\lambda _k)\) and \((\mathrm{pre}_k)\) are fixed before \(W\). For any \(W{\gt}0\), put \(a=W/K\) and \(\ell _{\dim }=1+(2r_0+1)^2\log q\), and suppose \(a\ell _{\dim }\leq c_t\). Define

There are nonnegative \(C_b,c_U,c_R,C_i\), a sequence \(0\leq b_k\leq C_b\log (k+2)\), and \(r_k\geq 0\) tending to zero. Write

The remainders \(R_k^{\rm Bell}\) and \(R_k^{\rm end}\) tend to zero. For every fixed \(0{\lt}p_0{\lt}p_1{\lt}1\) and all sufficiently large \(k\), there is a charge round \(s_k{\lt}nm\) and \(p_k\in [p_0,p_1]\) with the following properties. Let \(\mathcal Q_k\) be the entropy defect at this charge, and let \(\mathcal M_k\) be its root metric at \(p_k\). Define

Here \(N_k\) is the positive square root of \(N_k^2\), whose positivity is part of the conclusion, and \(\overline{\widetilde H}_k\) is the mean truncated physical Hamiltonian on the replicas, extended by the identity on both auxiliary registers. Put

At this single point,

The remainder sequences and their coefficients are chosen before the interpolation interval. They may depend on the fixed physical instance and weight; the selected round and parameter may depend on \(k\).

Let \(B,a,c_0\in \mathbb R\). If \(B\geq 1\) and \(a\sqrt B\leq \min \{ c_0,1\} /10\), then \(10a\leq c_0/\sqrt B\) and \(10a\leq 1\). This is the elementary smallness implication used in [ 286 , Proposition 9.2, sharp-comparison clause ] .

Fix \(R\in \mathbb N\), \(J\geq 0\), \(\Delta {\gt}0\) and \(C_T\geq 0\). There is \(C_{\rm tr}{\gt}0\) such that, for every integer \(q\geq 1\), every \(C_1{\gt}0\), every anchor-multiplicity bound \(m_*\in \mathbb N\) with \(3m_*\leq C_1\), and every real exponent \(P\), there are \(C_{\rm target}{\gt}0\) and \(0{\lt}c\leq 1/2\), chosen before the scanner exponents and entropy coefficient. The truncation constant is the one chosen independently of \(q\) in Theorem 44.8.3. For every admissible exponent tuple \((\ell ,\mu ,\kappa ,\nu ,e_{\rm in})\), there are \(C\geq 1\) and \(C_l\geq 0\). For every \(C_e\geq 0\), there are \(C_\delta {\gt}0\), \(C_{l,\delta }\geq 0\) and \(N\geq 2\), all chosen before the finite physical instance and \(W\).

Use the original gapped Hamiltonian, nonempty target, ambient distance function, anchor multiplicity, layer-cardinality bound, cut clearance, and rounded scales of Theorem 44.8.3. Suppose \(n\geq N\), and put \(X=A\cap T\). For every integer \(0\leq j\leq L\), assume

Then \(2r_0\leq D\), \(1\leq D\), and \(4D\leq m\). There is one truncated vector \(\widetilde\Omega \), typical set \(E\), positive selected mass \(z\), and sequence of labels and physical Schmidt–Bell prevectors satisfying all the state, projected-mass and dimension conclusions of Theorem 44.8.3, before \(W\) is chosen.

For any \(W\geq 1\), set \(a=W/K\), \(\varepsilon =n^{-\nu }\), and assume \(ar_0^2\leq c\). With the same \(B_{\rm rough}\) and \(T_{\rm ent}\) as in (59), with \(\eta =e_{\rm in}\), one has

There is a nonnegative sequence \(\rho _k\to 0\) such that, for all sufficiently large \(k\), the same selected charge \(s_k{\lt}nm\) and parameter \(p_k\in [\varepsilon /2,\varepsilon ]\) satisfy

At this same point all four conclusions of (60) hold, with \(p_0=\varepsilon /2\) and \(p_1=\varepsilon \). In particular, the energy cap retains its unexpanded density and truncation terms, and the filtered prevector has positive norm. Every remainder tends to zero with the finite instance and \(W\) fixed. The exponent conditions do not require \(\delta _n\to 0\). The case \(C_e=0\) is included. This is the density consequence used in [ 286 , Proposition 9.2, lines 475–490 ] .

Fix \(R\in \mathbb N\), \(J\geq 0\), \(\Delta {\gt}0\) and \(C_T\geq 0\). There is \(C_{\rm tr}{\gt}0\), chosen before the integer \(q\geq 1\), the charge constant \(C_1{\gt}0\) and the anchor-multiplicity bound \(m_*\). With this same truncation constant, under the hypotheses of Theorem 44.8.5 and with the error exponent fixed at \(P=100\), there are common coefficients \(C_\delta {\gt}0\) and \(C_{l,\delta }\geq 0\) for the density and energy bounds. They and the size threshold are chosen after the scanner exponents and \(C_e\geq 0\), but before the finite physical system and the total weight. The constants \(C\geq 1\) and \(C_l\geq 0\) in the original energy bound are retained separately.

The same truncated state \(\widetilde\Omega \), typical set, selected mass, label sequence and physical prevector sequence satisfy all conclusions of that theorem before \(W\) is chosen. For \(W\geq 1\), let \(a=W/K\) and assume \(ar_0^2\leq c\). Put \(\varepsilon =n^{-\nu }\) and

If \(e\) is the truncated ground energy and \(g/2\) its guaranteed gap, then

There are nonnegative sequences \(\rho _k,\rho _{E,k}\to 0\) such that, for every sufficiently large \(k\), one charge round and one common parameter \(p_k\in [\varepsilon /2,\varepsilon ]\) satisfy

Here the defect is the normalized mean energy minus \(e\), divided by \(g/2\). At the same point, the original density bound, original energy bound, positive filtered norm and rough upper norm estimate remain valid. Every replica remainder is uniform in the selected round and parameter with the finite instance and \(W\) fixed. The coefficient \(C_\delta \) may be positive when \(C_e=0\). This is the density-to-energy consequence used in [ 286 , Proposition 9.2, lines 368–381 and 492–533 ] .

Fix \(q\geq 1\), an interaction range \(R\in \mathbb N\), and \(C_r{\gt}0\). There exists \(C_B{\gt}0\), depending only on these parameters, with the following property. Assume the geometric hypotheses of Theorem 43.3.1.3, with \(K,m,M{\gt}0\) and \(r_0=\lceil C_r(\log n)^2\rceil \). Put \(\mathcal B=C_BnD(\log n)^{12}\); then \(\mathcal B\geq 1\). Let \(H\) be a local Hamiltonian of range \(R\), let \(\Omega \) be a unit vector, and let \(\Delta {\gt}0\). For the associated truncated Hamiltonian, prescribe a ground vector \(\widetilde\Omega \) of energy \(e_t\) and projector gap \(g/2\), where \(g\) is the normalized gap used in the truncation. Set

At \(X=A\cap T\), prescribe a spectral set \(E\) of cardinality \(d{\gt}0\) and mass \(z{\gt}0\). Let \(\psi \) be its normalized Schmidt truncation, with the retained factor relabelled by \(\mathbb C^d\), and let \(S'\) be the entropy of that factor. Prescribe labels \(\lambda _k\) and \(\mathrm{pre}_k=\pi _R^{\lambda _k} (\widetilde\Omega \otimes \mathrm{Bell}_{CR})^{\otimes k}\). Assume \(\mathrm{pre}_k\ne 0\) and \(\| \mathrm{pre}_k\| \leq 1\) for \(k{\gt}0\), the eventual projected-mass bound \(\| (I\otimes \pi _R^{\lambda _k})\psi ^{\otimes k}\| ^2 \geq [2(k+1)^{d^2}]^{-1}\), and \(\log \dim \lambda _k-kS'=o(k)\).

Fix \(W\geq 1\), put \(a=W/K\), and assume \(ar_0^2\leq 1/2\). At each charge round \(s{\lt}nm\) and real parameter \(p\), let \(\mathcal M_k\) be its root metric with band exponent \(a/2\), using \(p\) projected onto \([0,1]\). Write

The mean physical Hamiltonian is extended by the identity on the auxiliary factors. There are \(c_R,C_{f,R}\geq 0\), fixed before \(k,s,p\), such that the remainder \(R_k^{(a)}\) in (141) tends to zero and, for every \(k{\gt}0\), one has \(N_k^2{\gt}0\), \(\| v_k\| =1\) and \(D_k\geq 0\). For every \(0{\lt}\tau \leq 1/2\),

Here \(h\) is binary entropy.

Suppose in addition that \(a\sqrt{\mathcal B}\leq \min \{ c_0,1\} /10\). For each terminal \(j\) of the fixed family, including terminals of weight zero, and each band \(b\), let \(Q_{jb}\) be its physical receiving region, \(Y_{jb}\) its middle region, and \(U_{jb}\) its positive-depth near region. Define

Let \(w_j\) be the weights of the charge tree at \((s,p)\). There are \(c_U,c_L,C_{f,L}\geq 0\), fixed before \(k,s,p,\tau \), such that \(R_k^+=[\log 4+(d^2+c_UK)\log (k+2)]/k\to 0\) and the remainder \(R_k^-\) of Definition 44.4.1.2, with coefficients \(c_L,C_{f,L}\), tends to zero. For all sufficiently large \(k\), simultaneously for every \(s{\lt}nm\), every real \(p\) and every \(0{\lt}\tau \leq 1/2\), set

Then

The prescribed state, spectral set, and labels are unchanged in all three estimates. This is the uniform comparison step used in [ 286 , Proposition 9.2, lines 386–395 ] .

Suppose that for every integer \(R\ge 0\) there are real constants \(D_0{\gt}2R+10\) and \(C_{\rm tpl}\ge 24\) for which the two-family decomposition of Definition 39.8.44 holds. Then, for every pair of integers \(q\ge 1\), \(R\ge 0\) and every \(J,\Delta {\gt}0\), there is a constant \(C\ge 0\) with the following property. For every finite induced domain \(\Lambda \), every admissible Hamiltonian of range \(R\) and interaction bound \(J\) with a unit ground vector \(\Omega \) and global spectral gap at least \(\Delta \), and every \(A\subseteq \Lambda \),

The constant is independent of the domain, Hamiltonian, ground vector, and cut. This is the reduction in the proof of [ 286 , Theorem 1.1, Section 11 ] , with the geometric decomposition as its hypothesis; Theorem 39.8.47 supplies that hypothesis.

Fix \(q\ge 1\), \(R\ge 0\), \(J\), \(\Delta {\gt}0\) and template constants \(D_0,C_{\rm tpl}\). Suppose that the assertions of the sublinear collar bound hold with the constant \(C_1\) for all \(n\ge N_1\); that the assertions of the amplification bound for the input constant \(\max \{ C_1,1\} \) and the order \(M=100\) hold with the constant \(C_2\) for all \(n\ge \max \{ 2,N_2\} \); and that \(C_3\) is a constant of the two-family decomposition at the lower scale \(n_*(N_1,N_2,C_2)\). Then \(C(q,C_2,C_3)\ge 0\), \(n_*(N_1,N_2,C_2)\ge 2\), and for every finite induced domain \(\Lambda \), every admissible Hamiltonian with a gapped ground vector \(\Omega \) and every \(A\subseteq \Lambda \),

Fix \(q\ge 1\), \(R\ge 0\), \(J\ge 0\) and \(\Delta {\gt}0\). Assume that the constants \(c_0,C_{\rm ent},e_{\rm ent},C_{\rm en},e_{\rm en}\), with \(c_0{\gt}0\), satisfy the finite-family transport bound of Definition 43.3.1.14. Then the constant \(C=C_{\rm AL}(q,R,J,\Delta ;c_0,C_{\rm ent},e_{\rm ent},C_{\rm en},e_{\rm en})\) of Definition 44.10.3 is nonnegative, and for every finite induced domain \(\Lambda \), every admissible Hamiltonian with a gapped ground vector \(\Omega \) and every \(A\subseteq \Lambda \),

The finite-family transport bound of Definition 43.3.1.14 holds for

Fix \(q\ge 1\), \(R\ge 0\), \(J\ge 0\) and \(\Delta {\gt}0\). Then \(C=C_{\rm AL}(q,R,J,\Delta )\) is nonnegative, and for every finite induced domain \(\Lambda \), every admissible Hamiltonian with a gapped ground vector \(\Omega \) and every \(A\subseteq \Lambda \),

Let \(V\ge 0\) and \(C_0\ge 0\), let \(G\) be a real function with \(G(C)\ge 0\) for \(C\ge 0\), let \(N\) take natural values, and put

1. If \(F\) is nondecreasing on \([0,\infty )\), \(x\le F(x)\) for \(x\ge 0\), and \(G(C)+V\, N(C)^2\le F(C)\) for \(C\ge 0\), then \(C_j\le F^{M}(C_0)\) for all \(j\le M\).

2. If \(A\ge 0\), \(K\ge 1\) is an integer and \(f(y)=A(1+y)^K\), then \(1+f^{j}(x)\le \bigl((A+1)(1+x)\bigr)^{(2K)^j}\) for every \(x\ge 0\) and every \(j\).

3. If \(A\ge 1\), \(K\ge 1\) and \(G(C)+V\, N(C)^2\le A(1+C)^K\) for \(C\ge 0\), then \(C_j\le \bigl((A+1)(1+C_0)\bigr)^{(2K)^M}\) for all \(j\le M\).

In the notation of Definition 44.10.9, \(A\ge 1\) and

Fix \(q\ge 1\), \(R\ge 0\), \(J\), \(\Delta \), \(0{\lt}e_0{\lt}1\), truncation and transport constants and an initial constant \(C_0\ge 0\), and let \(m\) be the number of bootstrap rounds from \(e_0\) to \(e_*=2\cdot 10^{-6}\). For every integer \(M\ge m\), the safe-box constant \(C_{\rm box}\) of Theorem 44.24.13 obtained from \(C_0\) satisfies

In particular, for \(J\ge 0\) and \(\Delta {\gt}0\) this holds for the safe-box constant of Definition 44.10.3, with \(e_0\) and \(C_0\) the exponent and the constant of the explicit initial box estimate and with the truncation constant at the factor \(2\).

For all \(q\), \(R\), \(J\) and \(\Delta \), \(C_{\rm AL}(q,R,J,\Delta )=C_{\rm AL}^{C_{\rm box}(q,R,J,\Delta )}(q,R,J,\Delta )\).

Fix \(q\ge 1\), \(R\ge 0\), \(J\ge 0\) and \(\Delta {\gt}0\).

1. Under the hypotheses of Theorem 44.10.4, let \(B\) be at least the safe-box constant \(C_{\rm box}\) of Definition 44.10.3. Then \(C=C_{\rm AL}(q,R,J,\Delta ;B;c_0,C_{\rm ent},e_{\rm ent},C_{\rm en},e_{\rm en})\) is nonnegative, and for every finite induced domain \(\Lambda \), every admissible Hamiltonian with a gapped ground vector \(\Omega \) and every \(A\subseteq \Lambda \), \(S_\Omega (A)\le C\, |\partial _\Lambda A|\).

2. For every \(B\ge C_{\rm box}(q,R,J,\Delta )\), the same holds with \(C=C_{\rm AL}^B(q,R,J,\Delta )\).

Fix \(q\ge 1\), \(R\ge 0\), \(J\ge 0\) and \(\Delta {\gt}0\). Then \(C_{\rm box}(q,R,J,\Delta )\le B_*(q,R,J,\Delta )\). Consequently \(C=C_{\rm AL}^*(q,R,J,\Delta )\) is nonnegative, and for every finite induced domain \(\Lambda \), every admissible Hamiltonian with a gapped ground vector \(\Omega \) and every \(A\subseteq \Lambda \), \(S_\Omega (A)\le C\, |\partial _\Lambda A|\).

Every ground state at range \(R\) is a unit vector, and every ground state of the model at \(J\) and \(\Delta \) is a ground state at range one.

Fix natural numbers \(q\) and \(R\) and reals \(J\), \(\Delta \) and \(C\ge 0\). Suppose that \(S_\Omega (A)\le C\, |\partial _\Lambda A|\) for every finite induced domain \(\Lambda \), every Hamiltonian on \(\Lambda \) whose terms are supported on sets of diameter at most \(R\) and have norm at most \(J\), every gapped ground vector \(\Omega \) of it with gap at least \(\Delta \) and every \(A\subseteq \Lambda \). Then the square-grid entropy area law of Definition 39.8.31 holds for the ground states at range \(R\) of Definition 44.10.17 with the constant \(C\).

Fix \(q\ge 1\), \(R\ge 0\), \(J\ge 0\) and \(\Delta {\gt}0\). Then the square-grid entropy area law of Definition 39.8.31 holds for the ground states at range \(R\) of Definition 44.10.17 with the constant \(C_{\rm AL}(q,R,J,\Delta )\) of Definition 44.10.6, and also with the constant \(C_{\rm AL}^*(q,R,J,\Delta )\) of Definition 44.10.15.

Fix \(q\ge 1\), \(J\ge 0\) and \(\Delta {\gt}0\). Then the square-grid entropy area law of Definition 39.8.31 holds for the ground states of the model at \(J\) and \(\Delta \) with the constant \(C_{\rm AL}(q,1,J,\Delta )\) of Definition 44.10.6.

If the polynomial PEPS approximation at range \(R\) of Definition 44.10.22 holds, then for every integer \(q\ge 2\) and all reals \(J,\Delta {\gt}0\) there are \(C,c{\gt}0\) such that, for every \(L\ge 2\), every unit gapped ground vector, with gap \(\Delta \), of a Hamiltonian in the model of Definition 39.8.10 with term bound \(J\) has a PEPS approximation with these constants.

Fix integers \(q\ge 2\) and \(R\ge 1\) and reals \(J,\Delta {\gt}0\), and let \(C_{{\rm PEPS},R}\) and \(c_{{\rm PEPS},R}\) be the constants of Definition 44.10.24. For every \(L\ge 2\), every ground state \(\Omega \) at range \(R\) on the grid of side \(L\) has a nonzero PEPS \(\Phi \) on the \(L\times L\) grid with \(D_e\le C_{{\rm PEPS},R}L^{c_{{\rm PEPS},R}}\) for every edge \(e\) and

In particular the polynomial PEPS approximation at range \(R\) of Definition 44.10.22 holds.

Fix the local dimension \(q\ge 1\), range, interaction bound \(J\ge 0\), physical gap \(\Delta {\gt}0\), target-size coefficient \(C_T\ge 0\), anchor multiplicity \(\mathfrak m\), charge coefficient \(C_1{\gt}0\) with \(3\mathfrak m\le C_1\), and a real decay parameter \(P\). There are positive \(C_{\rm tr},C_{\rm target},C_B,c_t\), real propagation coefficients and exponents, coefficient constants \(C\ge 1\), \(C_\ell \ge 0\), and a threshold \(N\ge 2\) preceding every finite physical instance, as follows.

Let \(\Omega \) be an original gapped ground vector on the finite domain. The scanner uses the induced domain graph, its depth is the ambient depth of a nonempty target \(T\), and its anchors lie in their original supports with multiplicity at most \(\mathfrak m\). Assume \(n\ge N\), \(|T|\le C_Tn^2\), and retain the explicit scales

The target’s ambient layers through \(L\) have size at most \(n\), and its clearance exceeds \(2L+10r_0\). Assume explicitly that \(2r_0\le D\), \(1\le D\) and \(4D\le m\). For real \(C_e,\eta \), suppose every original-state positive-depth prefix through \(L\) has entropy at most \(C_e(nL^\eta +n)\).

Put

One unit truncated ground vector \(\Omega _t\), with energy \(0\le e\le \varepsilon \) and gap \(g/2\), has the stated quantitative trace distance from \(\Omega \). Its target typical set \(E\) has mass \(0{\lt}1-n^{-100}\le z\le 1\) and the complete eigenvalue window \([\exp (-S_X-w),\exp (-S_X+w)]\). Its target cut weight is at most \(C_{\rm target}n(\log n)^{12}\). In particular \(d=|E|{\gt}0\).

One label sequence \(\lambda _k\) for this same physical vector is chosen before \(W\). Its initial vectors are nonzero and have norm at most one for positive replica counts, and \(\log \dim \lambda _k-kS'=o(k)\) for the entropy \(S'\) of the compressed typical auxiliary state. For every \(0{\lt}W\le n^{1/20000}\), set \(a=W/K\) and let \(\ell _{\dim }\) be the physical propagation logarithmic dimension bound. Assume explicitly \(a\ell _{\dim }\le c_t\).

Replica-remainder constants and nonnegative sequences can then be chosen before every \([p_0,p_1]\) with \(0{\lt}p_0{\lt}p_1{\lt}1\) and every selected charge point. Eventually in \(k\), there are a charge stage \(s_k\) and \(p_k\in [p_0,p_1]\) such that

At this same point,

with the density cap from (58). For every \(0{\lt}\tau \le 1/2\), the exact terminal-weight comparison (56) holds with \(K_{\rm mom}=C_0B\) and \(\delta \) replaced by the right side of (89). All replica remainder limits fix the physical system and weight first.

Fix scanner exponents satisfying \(0{\lt}\ell {\lt}1\), \(0{\lt}\kappa {\lt}\mu {\lt}1-\ell \), \(\nu {\gt}0\), \(0{\lt}e{\lt}1\) and \(\kappa \leq e\). Fix \(C\geq 0\) and \(C_\ell ,\omega ,r\in \mathbb R\), and suppose

There is an integer \(N\geq 2\), depending only on these fixed quantities, with the following property. For every integer \(n\geq N\), retain the scales

Then \(K\geq 1\), and every \(1\leq W\leq n^\omega \), with \(a=W/K\), satisfies

For the source’s final parameters \(e=2/1000000\), \(\mu =1/4\), \(\nu =1/1000\), \(\ell =1/100000\) and \(\omega =1/20000\), the two gaps at \(r=\nu \) are

Fix \(C\geq 0\) and \(C_\ell \in \mathbb R\). At the final scanner parameters there is an integer threshold \(N\geq 2\) such that, for every \(n\geq N\) and every \(1\leq W\leq n^{1/20000}\), the rounded scales in (91) satisfy \(K\geq 1\) and

Fix a local dimension \(q\geq 1\), a normalized gap \(g{\gt}0\) and a volume coefficient \(C_{\rm vol}\). For all sufficiently large \(n\), uniformly over the finite lattice, two unit vectors \(\Omega \) and \(\widetilde\Omega \), and a region \(X\) of size at most \(C_{\rm vol}n^2\), the trace-distance bound

implies

Let \(n\geq 2\), \(q\geq 1\), \(L\leq n\), \(D\geq 1\), \(e,C_e\geq 0\), \(W\geq 1\), and \(D\leq 2n^e\). Suppose the positive selected mass and target estimates satisfy

Then the rough expression obeys

Let \(n,m\) be positive integers, \(W,\varepsilon ,C_1{\gt}0\), \(a,C_r,C_t\geq 0\), and let \(e,C_l,B,G,R\) be real. If \(B\leq C_rWn^{1+e}\) and \(G\leq C_t(\log n)^{C_l}\), then

No sign restriction on \(B,G,R\) is required.

Adopt the exponents and rounded scales of Lemma 43.9.1, and suppose its estimates hold with constant \(C_s\geq 1\). Let \(W,C_1{\gt}0\), put \(a=W/K\), and suppose \(C_r,C_t\geq 0\) and

Set

Then, for every real \(R\),

Let \(d,\delta ,\rho ,A,t\geq 0\), let \(g{\gt}0\), and suppose \(d\leq \delta +\rho \) and \(0\leq p\leq \epsilon \). If

then

No sign assumption is made on \(e,r\) or \(E\).

Fix \(A,g\in \mathbb R\), with division by zero assigned the value zero for this scalar extension. If \(\rho _k\to 0\) and \(r_k\to 0\) along any fixed filter on their index set, then

The coefficients are fixed before this limit is taken. In the physical application \(g{\gt}0\), so this is the ordinary quotient by the spectral gap.

Fix \(q\geq 1\), \(C_e,C_t\geq 0\), \(C_1{\gt}0\) and a real logarithmic exponent \(C_l\). There is a size threshold \(N\geq 2\) before all physical scalar inputs, remainders and selected points. For \(n\geq N\) use the final rounded scales

Assume the displayed target-mass, entropy, selected-dimension and shell bounds, with \(e\) equal to the source box-error exponent, and \(G\leq C_t(\log n)^{C_l}\). Let \(R_k\to 0\) for this fixed instance. Suppose that eventually in \(k\) there is a point \((x,p)\), with \(p\in [p_0,p_1]\), for which a quantity \(d_k(x,p)\) satisfies

together with any attached proposition \(\mathcal P_k(x,p)\). Then eventually the same selected point satisfies \(d_k(x,p)\leq \varepsilon \) and \(\mathcal P_k(x,p)\).

Fix the local dimension, interaction radius, interaction and positive spectral-gap bounds, a nonnegative target-volume coefficient, a positive scanner coefficient \(C_1\), and the anchor multiplicity controlled by \(C_1/3\). Fix also the energy-tail exponent \(P\). There are positive source constants \(C_{\rm tr},C_{\rm target},C_B,c_t\), real round-energy coefficients and exponents, and cap constants \(C\geq 1\), \(C_l\geq 0\), chosen before every entropy coefficient \(C_e\geq 0\). For each such \(C_e\) there is a size threshold before the finite physical data.

Let \(\Omega \) be the original unit gapped ground state of the given finite-domain local Hamiltonian. The scanner uses the domain graph, anchors in their original interaction supports and ambient depth from the nonempty target set \(T\). Impose the source quadratic target size, whole-row layer counts, anchor multiplicity, collar clearance and rounded scales. Retain the geometric premises \(2r_0\leq D\) and \(4D\leq m\). Write \(X=A\cap T\), \(L=\lfloor n^{1-\ell }\rfloor \), and let \(e_{\mathrm{box}}\) be the final source box-error exponent. Assume the original entropy bounds

Then there is one truncated ground state \(\widetilde\Omega \), one target spectral window \(E\) of mass \(z\), and one physical Schmidt–Bell label sequence before every weight \(1\leq W\leq n^\omega \). They retain the source gap, energy, trace, cut-tail and window bounds.

Set \(a=W/K\) and retain the round-step smallness premise \(a\, \mathrm{transportLogDimBound}(q,r_0)\leq c_t\). For each such fixed weight, eventually in replica number \(k\) there is a charge point \((s,p)\) with

Here \(Q_k\) is the charge entropy defect of the same prevector. At this unchanged point all the previous energy and sharp-comparison conclusions hold. In particular, writing \(e_0\) for the truncated ground energy and \(g/2\) for its physical gap,

The full sharp inequality keeps the same mapped terminal weights, label entropy, physical gap, loss \(37K_{\rm mom}a\), mass term and both replica remainders. Its auxiliary parameter remains universally quantified over \(0{\lt}\tau \leq 1/2\) at this same point.

At the final scanner exponents,

Fix \(C\geq 0\), \(C_\ell \in \mathbb R\) and \(P\geq \nu \). For every integer \(n\geq 1\), every \(1\leq W\leq n^\omega \), every \(0\leq x\leq n^{-8\nu }\) and every \(p\in [n^{-\nu },2n^{-\nu }]\), the expression in (114) satisfies

Fix \(C\geq 0\), \(C_\ell \in \mathbb R\), \(g{\gt}0\) and \(P\geq \nu \). There exists an integer \(N\geq 2\) such that, for every \(n\geq N\), every \(1\leq W\leq n^\omega \), every \(0\leq x\leq n^{-8\nu }\), every \(p\in [n^{-\nu },2n^{-\nu }]\) and every \(e\leq n^{-1000}\),

The threshold is chosen before \(n,W,x,p,e\).

Under the hypotheses of Theorem 44.15.4, the same threshold works for every additional real remainder \(r\):

At the final source exponents, the exact slacks for density rate \(8\nu =1/125\), with interval exponent still equal to \(\nu \), are

For every fixed \(C\geq 0\) and real \(C_l\), there is a threshold before \(n\) and \(W\) such that the floor and natural-division scales obey \(K\geq 1\) and, uniformly over \(1\leq W\leq n^\omega \),

Under the same explicit target-mass, target-entropy, selected-dimension, shell-entropy and logarithmic-cap hypotheses as the modular selected density theorem, retain its initial selected bound and interval denominator \(\varepsilon =n^{-\nu }\). With all fixed coefficients before the size threshold, eventually in replica number the unchanged selected point satisfies density at most \(\eta =n^{-8\nu }\) and every proposition already attached to that point.

Use the same original gapped-ground-state, graph, anchor, depth, whole-row geometry, target-volume, clearance and scale premises as the concrete source-density theorem. Fix \(C_1{\gt}0\) and its anchor multiplicity before all thresholds. Fix \(P\geq \nu \). The positive source constants, real round-energy coefficients and exponents, and cap constants \(C\geq 1\), \(C_l\geq 0\) precede the nonnegative entropy coefficient \(C_e\), which precedes the enlarged size threshold. The original target and prefix bounds are

Then one truncated state, one target window and one physical label sequence precede every weight \(1\leq W\leq n^\omega \). Retain \(2r_0\leq D\), \(4D\leq m\) and \(a\, \mathrm{transportLogDimBound}(q,r_0)\leq c_t\) for \(a=W/K\). For each fixed admissible instance and weight, eventually there is a charge point \((s,p)\) for the same prevector with

The original cap-based energy inequality is also retained. In the full sharp comparison, the factor multiplying \(\max \{ 0,2S'+g_{\max }\} /\tau \) is replaced by this derived energy upper bound, for every \(0{\lt}\tau \leq 1/2\) at the same point. All other entropy, mass, moment and replica-remainder terms remain unchanged, including the loss \(37K_{\mathrm{mom}}a\).

Let the scanner depth be the ambient depth of the nonempty target template \(T\), and let \(8Km\leq L\). At every completed or pre-charge terminal and every original band, put

These four regions are pairwise disjoint and cover the physical sites. The near region is \(X\cup U\), the positive near-middle region is \(U\cup Y\), and the complement of the same truncation set is contained in \(V\).

For the same physical vector, let

Every terminal satisfies

If \(K{\gt}0\), any given terminal mean tree, with its unchanged weights \(w_j\), gives

where \(S_0\) is the same truncation set.

Under the preceding collar hypotheses, with \(m,M,K{\gt}0\), let \(\Omega _t\) be unit, and retain its positive-mass spectral window \(E\), of width \(w\), with mass \(z{\gt}0\). Let \(S'\) be the selected auxiliary entropy, and define \(G_{\rm pre},G_{\rm post}\) from the mapped charge tree at the same charge stage and parameter. Then

For any unit original vector \(\Omega \), the same right-hand side bounds \(2I_\Omega (X:S_0^c)\) after adding

Assume the induced domain graph, ambient target depth, \(n{\gt}0\), \(r_0\leq D\), \(1\leq D\), \(4D\leq m\), \(8Km\leq L\), and the layer-cardinality bound \(n\) through depth \(L\). Suppose the same unit vector obeys \(S(\text{positive prefix }i)\leq B_{\rm sh}\) for every \(0\leq i\leq L\). Put

Every terminal through horizon \(nm\) satisfies

For \(K{\gt}0\), every terminal-tree average satisfies \(0\leq G_{\rm pre}\leq B\), and the maximum over all terminal/band pairs, with zero inserted, satisfies \(0\leq g_{\max }\leq \max (0,B)\).

Fix \(q\geq 1\), a positive gap parameter \(g\), and a compact volume coefficient \(C_{\rm vol}\). At sufficiently large \(n\), uniformly before choosing the scanner, vectors, source entropy coefficient \(C_e\), or exponent \(\eta \), the preceding bounds hold with

whenever the original unit vector obeys \(S_\Omega (\text{positive prefix }i)\leq C_e(nL^\eta +n)\) for all \(0\leq i\leq L\), the compact prefix has volume at most \(C_{\rm vol}n^2\), and the unit vector \(\Omega _t\) obeys

For scalar data define

Assume \(W,\tau {\gt}0\), \(\delta ,S,c,B_{\rm pre}\geq 0\), and

If the sharp comparison is

and the information bridge gives \(2I\leq 2S'+G_{\rm pre}-G_{\rm post}+2c+C\), then \(2I\leq F+r\).

Keep all the physical and scalar data fixed. If \(r_k,u_k,l_k\to 0\) and eventually

then \(2I\leq F(S,c,B_{\rm pre},\log z,W,\tau ,\delta ,A,C)\).

Under the complete hypotheses of the stronger source-selected sharp energy theorem, enlarge its size threshold to include the uniform positive-prefix continuity threshold. Retain its same truncated vector \(\Omega _t\), typical spectral set \(E\), mass \(z\), selected dimension \(d\) and physical label sequence. Put

For every \(1\leq W\leq n^\omega \) satisfying the original round-step smallness with \(a=W/K\), and every \(0{\lt}\tau \leq 1/2\), define

Then the original-state collar information satisfies

The fixed physical and round-step constants precede \(C_e\); \(C_e\) precedes the size threshold and the finite physical system. The state, spectral set and labels precede \(W\).

Retain the scales

There are fixed \(C_a\geq 1\) and \(N\geq 2\) such that, for all integers \(n\geq N\), one has \(K\geq 1\), \(a{\gt}0\), \(0{\lt}\tau \leq 1/2\), and

For \(n\geq 2\) and \(1-n^{-100}\leq z\leq 1\), \(z{\gt}0\), the mass estimate below holds. The binary estimate additionally assumes \(a{\gt}0\) and \(a^{-1}\leq n^{0.74994}\), as established above.

The binary estimate uses \(h(t)\leq t(1-\log t)\) for \(0{\lt}t{\lt}1\).

Fix nonnegative \(C_e,C_0,C_B,C_a,C_{\rm tr},C_{\dim }\) and \(\log q\). In addition to (127) and the selected-mass bounds, assume

The complete error is at most

The decimal powers in this display denote exact terminating rationals, not numerical approximations.

For the fixed coefficients in (131), one threshold works for every remaining scalar datum at the rounded scales. With

it gives

The exact maximal power is \(62497/62500=0.999952\). Its margin to \(1-1/100000\) is \(19/500000{\gt}0\).

For fixed \(q\geq 1\), \(g{\gt}0\) and \(C_{\rm vol}\geq 0\), the source estimate

is covered by \(C_{\rm tr}=\sqrt{2/g}\) in the preceding theorem. If \(|X|\leq C_{\rm vol}n^2\), its first-region dimension bound is covered by \(C_{\dim }=C_{\rm vol}\log q\). In addition, for fixed \(C_r\geq 0\) and \(c_t{\gt}0\), a uniform threshold gives

Fix \(q\geq 1\), an exponent \(e{\gt}0\) and a safe-box coefficient \(C_{\rm box}\geq 0\). There is \(C_e\geq 0\), chosen before every finite domain, unit vector, cut, safety parameter, template and collar scale, with the following property. Let \(T\) be a template with \(C_{\rm tpl}\geq 24\), separated from the cut at safety parameter \(D_0\geq 1\), and suppose

for every safe integer rectangle \(Q\). Then

The same shell bound holds after replacing a complete prefix by all preceding depth rows and an arbitrary subset of the final row.

Fix \(q\geq 1\), \(D_0\geq 1\) and \(C_{\rm box}\geq 0\). Suppose the uniform safe-box function at fixed \(q,R,J,\Delta ,D_0\) satisfies \(F_{\rm box}(r)\leq C_{\rm box}r^{1+e_*}\) for all \(r\geq 1\), where \(e_*=2\cdot 10^{-6}\). One coefficient \(C_e\geq 0\), chosen before the finite physical system, gives the two displayed entropy inputs for every original gapped ground vector and every separated template target of a scanner whose depth is the ambient target distance. In particular one may use \(L=\lfloor n^{1-10^{-5}}\rfloor \) whenever \(1\leq L\leq s_0\).

For a finite physical region \(X\) with local dimension \(q\geq 1\), if \(|X|\leq C_{\rm vol}n^2\), then

Fix \(q\geq 1\), an interaction radius \(R\), a norm bound \(J\geq 0\), a spectral gap \(\Delta {\gt}0\), a nonnegative quadratic target-volume coefficient \(C_T\), a positive scanner coefficient \(C_1\), an anchor-multiplicity bound \(m_{\rm anc}\) with \(3m_{\rm anc}\leq C_1\), and a tail power \(P\geq \nu \). There is a positive radius coefficient \(C_{\rm tr}\) such that, for every fixed nonnegative source entropy coefficient \(C_e\), there are constants \(C{\gt}0\) and \(N\geq 2\), chosen before the finite physical system, with the following property.

Let the original vector \(\Omega \) be a gapped ground state and let the given physical scanner have a nonempty ambient target and obey all the source graph, support-anchor, template-volume, ambient-depth, layer-cardinality, anchor-multiplicity and boundary-clearance hypotheses. Retain the scales

where the final division is natural-number division, together with \(n\geq N\), \(2r_0\leq D\), \(1\leq D\) and \(4D\leq m\). Put \(X=A\cap T\) and let \(S_0\) be the truncation set at depth \(L\). Assume the genuine original-state source entropy inputs

Then

In particular, the exact maximum exponent is \(0.999952\) and the displayed paper exponent is \(0.99999\).

Fix \(C_r\geq 0\) and the final exponents \(\mu =1/4\), \(\kappa =1/1000000\) and \(\ell =1/100000\). There is one \(N\geq 2\) such that every integer \(n\geq N\), with

satisfies

The division defining \(K\) is natural-number division.

The conditional final-scale physical information theorem holds without separately assuming \(2r_0\leq D\), \(1\leq D\) or \(4D\leq m\). Retain its original gapped ground state, floor and ceiling scales, scanner graph and support anchors, template-volume bound, ambient depth and layer bounds, anchor multiplicity, cut clearance, and source target/prefix entropy bounds. Then one constant \(C{\gt}0\) and one uniform threshold give

The radius coefficient is selected before the entropy coefficient; the final constant and size threshold precede every physical instance.

For every nonnegative integer \(L\), the truncation set of the constructed scanner is exactly

This identity requires no separation, scale or entropy estimate.

Fix the physical parameters \(q\geq 1\), \(R\), \(J\geq 0\), \(\Delta {\gt}0\), an integer safe-box clearance \(D_0\geq 1\), and \(C_{\rm tpl}\geq 24\). Suppose one fixed \(C_{\rm box}\geq 0\) satisfies the genuine uniform original-state safe-box estimate

Then there are \(C{\gt}0\) and \(N\geq 2\), preceding every physical instance, such that the following holds. Let \(\Omega \) be an original gapped ground state on any finite induced domain \(\Lambda \), let \(A\) be any cut, and let \(T\) be a separated template with parameters \((n,s_0)\), \(n\geq N\) and \(L=\lfloor n^{1-1/100000}\rfloor \leq s_0\). For the regions \(X=A\cap T\) and \(S_0=A\cap T_L\),

No scanner, support-anchor choice, row count, multiplicity estimate, target/prefix entropy bound or auxiliary replica datum is assumed in this theorem.

Fix a finite physical domain, positive local and auxiliary dimensions \(q,d\), an interaction range and bound, a band count \(K\in \mathbb N\), and \(0{\lt}a\leq 1/2\). Put \(W=aK\). There are \(c,C_f\geq 0\) before the scanner, horizon, physical vectors, copy number, auxiliary label and interpolation parameter with the following property. Consider any collar scanner with \(K\) bands and \(m,M{\gt}0\). Let \(\widetilde H\) be the sum of its truncated energy terms, formed from the reference vector. Suppose \(\| \widetilde\Omega \| =1\), \(\gamma {\gt}0\), and

Fix \(k{\gt}0\) and a nonzero prevector of norm at most one, invariant under simultaneous permutation of the copies and fixed by the same label \(\lambda \) on both auxiliary factors. For real \(S',\varepsilon \), assume \(2kS'-\varepsilon \leq 2\log \dim \lambda \) and set

At every fill or charge round before the chosen finite horizon and every real interpolation parameter, write \(\mathcal N_k\) for the squared filtered norm and \(\mu _k\) for the mean energy. The normalized filtered vector has norm one, \(\mathcal N_k{\gt}0\), and \(\delta _k=(\mu _k-e_0)/\gamma \geq 0\). For every \(0{\lt}\tau \leq 1/2\),

Here \(h\) is the binary entropy; this is the augmented-coordinate form of the rough lower comparison in [ 286 , Section 8, lines 97–169 and 585–651 ] .

Fix \(K\in \mathbb N\) and real \(a,c,C_f,S'\). For a label sequence \(\lambda _k\), put \(\varepsilon _k=2|\log \dim \lambda _k-kS'|\) and, for \(k{\gt}0\), define

If \(\log \dim \lambda _k-kS'=o(k)\), then \(R_k^{(a)}\to 0\). All coefficients and the label sequence are fixed in this limit.

Under the physical and gap hypotheses of Theorem 44.23.1.1, take the physical Schmidt–Bell prevector associated with a label sequence \(\lambda _k\) whose original-coordinate vectors are nonzero and have norm at most one for every \(k{\gt}0\), and assume \(\log \dim \lambda _k-kS'=o(k)\). The constants \(c,C_f\geq 0\) may be chosen before the scanner, reference and truncated vectors, label sequence and \(S'\), with the finite domain, dimensions, \(K\) and \(a\) already fixed.

Every fill and charge round before the full horizon \(nm\), at every real interpolation parameter and every \(k{\gt}0\), obeys (139) with remainder \(R_k^{(a)}\) from (141), with positive squared filtered norm, a unit normalized filtered vector, and nonnegative energy defect \(\delta _k\). For this label sequence, \(R_k^{(a)}\to 0\) uniformly over all rounds and interpolation parameters, so a single replica threshold suffices for any positive error tolerance.

Fix a finite physical site space, positive local and auxiliary dimensions \(q,d\), and a band count \(K\geq 8\). Put \(a=4/K\). There are \(c,C_f\geq 0\), chosen before the scanner, horizon, physical vectors, copy number, label and interpolation parameter, with the following property for every scanner with \(K\) bands and positive parameters \(m,M\). Let \(\widetilde H\) be the sum of its truncated energy terms, formed from the reference vector. Suppose the unit physical vector \(\widetilde\Omega \) satisfies

Fix \(k{\gt}0\) and a nonzero prevector of norm at most one, fixed by simultaneous copy symmetry and the same auxiliary label \(\ell \) on both auxiliary factors. For real \(S'\) and \(\varepsilon \), assume \(2kS'-\varepsilon \leq 2\log \dim \ell \) and put

At any fill or charge interpolation before the horizon, let \(\mathcal N_k\) be its squared filtered norm and \(\mu _k\) its mean energy. Its filtered vector has norm one, \(\mathcal N_k{\gt}0\), and \(\delta _k=(\mu _k-e_0)/\gamma \geq 0\). For every \(0{\lt}\tau \leq 1/2\),

No marginal-moment estimate or bound \(\delta _k\leq \tau \) is assumed.

For every \(k\), \(\varepsilon _k\geq 0\) and \(2kS'-\varepsilon _k\leq 2\log \dim \ell _k\).

If \(\log \dim \ell _k-kS'=o(k)\), then \(R_k\to 0\) for each fixed \(K,c,C_f\). For every \(\epsilon _{\rm rep}{\gt}0\), there is a threshold \(k_0\) such that \(R_k{\lt}\epsilon _{\rm rep}\) whenever \(k\geq k_0\). This threshold is independent of the round and its interpolation parameter.

Under the scanner and gap hypotheses of Theorem 44.23.1.4, take the augmented physical replica prevector associated with a label sequence \(\ell _k\) whose original-coordinate vectors are nonzero and have norm at most one for every \(k{\gt}0\), and assume \(\log \dim \ell _k-kS'=o(k)\). The constants \(c,C_f\) can be fixed before the scanner and physical vector, with the finite dimensions and band count already fixed. Every fill and charge before the full horizon obeys (144) with this sequence and \(R_k\to 0\), where the normalized filtered vector and nonnegative energy defect are those in Theorem 44.23.1.4.

Fix any scanner exponent tuple, an integer \(q\geq 1\), a coefficient \(C_r\geq 0\) and a threshold \(c_t{\gt}0\). There is \(N\geq 2\), chosen before the system size or any finite physical data, such that every \(n\geq N\) satisfies \(K_n\geq 8\), \(a_n{\gt}0\) and

No sharp-comparison exponent margin is assumed.

Fix any scanner exponent tuple and \(C_r\geq 0\). There is \(N\geq 2\), before all finite physical data, such that every \(n\geq N\) satisfies

Fix scanner exponents and \(C_r\geq 0\). There is a threshold \(N\geq 1\) before the finite domain, cut, rectangle, interaction range, charge coefficient and safety parameter, with the following property. Let \(s\geq N\) and \(D_0\geq 1\), and suppose the rectangle has strict clearance greater than \(D_0s\) from every endpoint of an edge crossing \(A\). For the scanner above and \(L=\lfloor n^{1-\ell }\rfloor \),

The last inequality holds for every \(t\in T\) and every cut endpoint \(z\). Every chosen anchor lies in its original support, and at most \(2^{(1+2R(R+1))-1}\) labels have a given anchor. The target is exactly \(A\cap Q\), the positive-depth prefix at every nonnegative integer \(j\) is exactly \(A\cap (Q^{+j}\setminus Q)\), and the truncation set at every \(L'\) is exactly \(A\cap Q^{+L'}\).

Fix a local dimension \(q\geq 1\), a scanner exponent tuple, a positive squared-logarithmic radius coefficient, a positive charge constant, a nonnegative target-volume coefficient, an anchor-multiplicity bound, a polynomial tail exponent, and the two round-energy coefficients. There are \(C\geq 1\), \(C_l\geq 0\) and \(N\) before every finite physical system, auxiliary dimension, replica vector, round and interpolation. Under the rounded scales and the domain, row, anchor and clearance hypotheses, for \(W\geq 0\) and a nonzero symmetric replica vector, the energy-error term is bounded by

The bound uses the absolute value of the given round-energy coefficient. The existing all-time and old-prefix history estimates give the physical energy bound at these generic exponents.

For positive physical and auxiliary dimensions, \(\Delta {\gt}0\) and total weight \(W\geq 0\), at any selected charge point the round-energy conclusion and the preceding coefficient cap imply \(E_{\rm def}\leq (e+\mathrm{cap}+r_k)/(g/2)\). The truncated eigenvalue \(e\), physical prevector, selected point and mean energy are unchanged. Nonnegativity of the energy-error integral justifies replacing its signed coefficient by its absolute value.

Fix \(q\geq 1\), range \(R\), \(J\geq 0\), \(\Delta {\gt}0\), the nonnegative target-volume coefficient and positive charge constant with its anchor-multiplicity bound, and a real tail exponent. The truncation and round-energy constants are chosen before a scanner exponent tuple; the physical energy coefficients and size threshold are then chosen before every finite instance and current entropy input. Assume a nonempty ambient target and the graph, depth, rounded scales, rows, anchors and weak scanner clearance, and the original-state positive-prefix bound \(C_e(nL^\eta +n)\). The numerical radius and separation conditions are explicit here. For total weight four, \(K\geq 8\), and the round-step smallness condition, one truncated vector, typical spectral set and Schmidt–Bell label sequence yield a selected point in every fixed interval \(0{\lt}p_0{\lt}p_1{\lt}1\) for all large replica counts. That same point satisfies the endpoint density bound, the normalized-energy cap, and both rough norm comparisons. The upper, lower, endpoint and energy remainders vanish for the fixed physical instance. No improved entropy or low-energy point is claimed here.

Fix the scanner tuple, \(q\geq 1\), \(C_e,C_t\geq 0\), \(C_1{\gt}0\) and a logarithmic exponent. One \(C_\delta {\gt}0\) and threshold precede every current \(e\geq \kappa \) and finite scalar instance. The current target, shell, selected-dimension and mass bounds, together with the rough endpoint numerator and logarithmic round-energy cap, give eventually

The replica remainder is absorbed only after the instance is fixed. Every proposition attached to the original selected point survives.

Fix \(0{\lt}e_0{\lt}1\), nonnegative cap and density coefficients, real logarithmic exponents and a positive physical gap \(\gamma \). One threshold works for all \(e\leq e_0\). At the bootstrap scales, if \(0\leq \rho \leq \delta \), \(p\in [\epsilon ,2\epsilon ]\), \(p\leq 1\), and the truncated eigenvalue is at most \(n^{-1000}\), then the normalized selected cap is at most \((1+\gamma ^{-1})n^{-\nu /2}\). The physical application uses \(\gamma =g/2\) and retains \(r_k/\gamma \).

For \(e_*{\lt}e_0{\lt}1\), \(q\geq 1\) and \(C_e\geq 0\), one positive coefficient and size threshold work for every \(e_*{\lt}e\leq e_0\). They bound the exact typical-mass, whole-prefix, mismatch, binary-entropy and original-state continuity error by \(C n^{1+(1-\ell )e}\). The hypotheses include \(z{\gt}0\), \(z\geq 1-n^{-100}\) and \(0\leq \tau \leq 1\); the displayed sum keeps \(7(w-\log z)-\log z\) and the additional \(n^{-247}\) continuity error.

Fix \(e_*{\lt}e_0{\lt}1\), \(C_e\geq 0\) and the physical and geometric constants of the rough source theorem. There are positive \(C_r,C_{\rm out}\) and \(N\geq 2\) before every current \(e_*{\lt}e\leq e_0\) and finite system. Use the fixed bootstrap exponents, the rounded scales and the graph, target, depth, anchors, rows and clearance. If the original target entropy is at most \(C_e n^{1+e}\) and every original positive-depth prefix entropy is at most \(C_e(nL^e+n)\), then the same original target satisfies

Fix \(b{\gt}0\) and \(C_{\rm box}\geq 0\). One nonnegative \(C_e\) precedes every integer safety parameter \(D_0\geq 0\), finite state, weakly safe rectangle, row scale \(n\geq s\), collar \(1\leq L\leq s\), and current exponent \(e\geq b\). If \((D_0+1)L\leq D_0s\) and the same unit vector obeys the pointwise safe-box bound at this one exponent, then its target and all positive-depth prefixes obey the source bounds above. One may take \(C_e=C_{\rm box}(25+64/(2^b-1))\). This uses the weak-rectangle dyadic shell cover, including thin rectangles; it does not invoke template separation.

For \(q\geq 1\), a positive-scale uniform box bound at exponent \(1+e{\gt}0\) gives the corresponding entropy estimate for every original physical safe rectangle, including a possible zero-size case by its cardinal bound.

Let \(q\geq 1\), \(C,p\geq 0\) and \(N\in \mathbb N\). If every safe original physical rectangle of size \(s\geq N\) has entropy at most \(Cs^p\), then for every \(r\geq 1\) the box supremum is at most \((C+N^2\log q+1)r^p\).

Fix \(q,R\in \mathbb N\) with \(q\geq 1\) and \(J,\Delta \in \mathbb R\). Write \(F_{\mathrm{box},D}\) for the safe-box entropy at safety parameter \(D\), with the same fixed physical parameters \(q,R,J,\Delta \). Let \(D\in \mathbb N\) satisfy \(D\geq 3\), and let \(C,p\geq 0\). Suppose that \(F_{\mathrm{box},D+1}(r)\leq Cr^p\) for every integer \(r\geq 1\). Then, for every integer \(r\geq 1\),

Fix \(q\geq 1\), \(R\), \(J\geq 0\), \(\Delta {\gt}0\), \(e_*{\lt}e_0{\lt}1\) and \(C_{\rm box}\geq 0\). There is \(C{\gt}0\) before every \(e_*{\lt}e\leq e_0\) and every integer \(D_0\geq 1\) such that

For \(q\geq 1\), \(R\in \mathbb N\), \(J\geq 0\) and \(\Delta {\gt}0\), there is \(C{\gt}0\) before every integer \(D_0{\gt}2R+10\) and every positive integer \(r\) such that

Fix integers \(q\geq 1\) and \(R\geq 0\), a term norm bound \(J\geq 0\), and a gap \(\Delta {\gt}0\). There is \(C{\gt}0\), depending only on \(q,R,J,\Delta \), such that the following holds simultaneously for every real \(D_0{\gt}2R+10\).

Let \(\Lambda \subseteq \mathbb Z^2\) be any finite domain, let \(H\) be an interaction Hamiltonian of range \(R\) with term norms at most \(J\), and let \(\Omega \) be a normalized eigenvector of energy \(E_0\) satisfying the full-system projector-gap inequality with gap \(\Delta \). Let \(A\subseteq \Lambda \), and let \(Z\) be the endpoints of the edges of the induced lattice graph crossing \(A\). If \(Q\) is a product of two finite nonempty integer intervals and every \(t\in Q\) and \(z\in Z\) satisfy \(D_0\operatorname {size}(Q){\lt}\| t-z\| _\infty \), then

Here size counts sites along the longer axis, and \(A\cap Q\) contains only physical sites of \(\Lambda \). Empty domains, empty cuts, empty physical intersections, disconnected domains, and domains with holes are permitted. If \(Z\) is empty, the safety condition is vacuous, agreeing with distance to the empty set being infinite.

Fix \(q,R,D_0\in \mathbb N\) and \(J,\Delta ,C_{\rm tpl}\in \mathbb R\) with

There are \(C{\gt}0\) and \(N\geq 2\), chosen before every finite instance, such that the following holds. For all natural \(n,s_0\) with \(n\geq N\), every finite induced domain \(\Lambda \), every local Hamiltonian with parameters \((q,R,J)\), every original gapped ground vector \(\Omega \) with gap \(\Delta \), every cut \(A\), and every template \(T\) with parameters \((C_{\rm tpl},n,s_0)\), assume the source conditions

Here the cut set is the ambient set of endpoints of edges crossing \(A\) in the induced domain. Set \(X=A\cap T\) and \(S_0=A\cap T_L\). Then

The same \(C,N\) apply to all the displayed finite data. The theorem does not assert that they are independent of the fixed \(D_0\) or \(C_{\rm tpl}\). No nonempty physical-core hypothesis is required.

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 \(B\) 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 \([\widehat U,\mathcal E_K(B)] =\mathcal E_K([\widehat U,B])\).

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

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 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 \(\Delta _l(B)=0\).

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

Fix \(l\ge 0\) and real numbers \(\varepsilon _{\rm next}, \varepsilon _{\rm prev}\) satisfying (177) and (178). Then

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

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 \(n\).

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.

As the finite-radius consequence of component support in [ 286 ] before choosing a decaying error bound, 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

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

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 \(a,b\in J\). If \(y\notin S\), then \(d_y(B)=0\).

As the initial support estimate of [ 286 , Section 10 ] for arbitrary finite spectators and arbitrary complex \(B\), 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 \(S\).

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

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 \(\eta _0=2\).

As the finite-radius one-event estimate underlying [ 286 , Section 10 ] , 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

where every regional sum is evaluated on the original operator \(B\).

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

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

Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\), and take \(A,b\) from (196). Fix \(l\in \mathbb {N}\). Suppose that, for every integer \(j\) with \(0\le j\le l\),

Then

Let \(C\ge 0\), \(c{\gt}0\), and \(0{\lt}\alpha \le 1\), and take \(A,b\) from (196). 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.

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 (196). 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 \((w_l)_{l\ge 0}\) 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.

Following the finite event-label sum of [ 286 , Section 10 ] , 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 (196). 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 \(B\), and the estimate holds for arbitrary finite spectators and arbitrary complex \(B\).

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 \(y\in I\),

As the finite-kernel weighted row estimate of [ 286 , Section 10 ] , 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 (212) holds for all sites and radii and (213) 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 \(y\).

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 (210) 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 (212) for every site and radius, and the anchor-fiber bound (213) for every site. Set \(A=2+8\sqrt C\exp (c/2)\), and use (210) 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.

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.

Combining [ 286 , Proposition 4.3 ] with the finite channel estimate of [ 286 , Section 10 ] , fix integers \(p\ge 1\) and \(R\ge 0\), and real constants \(J\ge 0\) and \(\Delta {\gt}0\), and 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.

Let \(q\in \mathbb {N}\) with \(q{\gt}0\) be any local dimension and \(\Lambda \subseteq \mathbb {Z}^2\) any finite domain, 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 \(H-E_0\mathbb 1\ge \Delta (\mathbb 1-|\Omega \rangle \langle \Omega |)\).

With the spectral-filter time kernel \(f=f_{p,\Delta /2}\) and the positive constraints of Theorem 43.2.2,

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

For any predicate \(P\) on \(\mathcal F_{\Lambda ,R}\) and \(F_P=\{ X\in \mathcal F_{\Lambda ,R}:P(X)\} \), every retained \(k_X\) uses the original full Hamiltonian \(H\) and ground vector \(\Omega \) in (227). Let \(\mathsf A^{(N,P)}\) denote the kernel (210) on \(G_\Lambda \), with label set \(F_P\), the restricted anchors, \(D=2(2+8\sqrt C\exp (c/2))\), and \(b=c/2\); its incidence counts are

For 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 \(G_\Lambda \), every \(y\in \Lambda \) satisfies

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, and the gap assumption concerns the original full Hamiltonian \(H\).

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.