Proof. On-site and nearest-neighbor interactions have graph range at most one. Each inner boundary vertex is incident with between one and four crossing edges, so \(|\partial_{\rm in}A|\le|\partial_\Lambda A| \le4|\partial_{\rm in}A|\). Counting both endpoints gives \(|Z_A|\le2|\partial_\Lambda A|\le4|Z_A|\). These inequalities include empty boundaries. ◻

Proof. We include the mixture argument for the bound in (Winter 2016, Lemma 2). If \(\tau=\sum_j p_j\tau_j\), adding a classical register that records \(j\) shows \[0\le S(D|E)_\tau-\sum_jp_jS(D|E)_{\tau_j}\le H(p).\] For the lower bound, condition on that register and use strong subadditivity. For the upper bound, use the ordinary mixing upper bound on \(S(DE)\) and concavity of \(S(E)\). The ordinary bound follows by recording a duplicate of the classical register: its conditional entropy given \(DE\) is nonnegative after this further conditioning.

If \(\delta>0\), write the Jordan decomposition \(\rho-\sigma=\delta(\tau_+-\tau_-)\), where \(\tau_\pm\) are states. Then \[\frac{\rho+\delta\tau_-}{1+\delta} =\frac{\sigma+\delta\tau_+}{1+\delta}.\] Apply the mixture bounds to both expressions and use \(|S(D|E)_{\tau_+}-S(D|E)_{\tau_-}|\le2\log d_D\). This proves the first inequality. The second follows from the elementary bound \(h(u)\le C\sqrt u\). The case \(\delta=0\) is immediate. Finally, \(I(D:E)=S(D)-S(D|E)\), so the two displayed entropy bounds apply. ◻

Proof. Such a set \(A\) is a union of graph components. Every support of finite graph diameter lies in one component, so \(H\) is the tensor sum of component Hamiltonians. Its ground eigenspace is the tensor product of their ground eigenspaces. If the former is one-dimensional, so is each latter space. The ground vector factors over the components and therefore over the cut. ◻

Proof. Put \(M(u)=\sum_jp_j^{1-u}\) and \(L(u)=\log M(u)\). Let \(P\) be the support projection of \(\rho\). On \(P\mathcal H_B\) define \(F=\rho^{-u/2}\), and extend \(F\) by the identity on the kernel of \(\rho\). The normalized tilted vector and its marginal are \[\Theta=\frac{F\Psi}{\sqrt{M(u)}},\qquad \sigma=\mathop{\mathrm{Tr}}_{B^c}\lvert \Theta\rangle\langle \Theta\rvert =\frac{\rho^{1-u}}{M(u)},\] where the last power is zero on the kernel. Since \(F\) is invertible on the whole space, \[ F\mathsf H F^{-1}\Theta=E_0\Theta. \tag{8}\] An interaction supported in \(B^c\) is unchanged by this conjugation. An interaction supported in \(B\) has unchanged expectation, because \[\mathop{\mathrm{Tr}}\sigma Fh_iF^{-1}=\mathop{\mathrm{Tr}}\sigma h_i\] by \([F,\sigma]=0\) and cyclicity. Thus only crossing interactions contribute to the excitation energy of \(\Theta\).

Fix such an interaction. For every complex \(z\), use the support convention \[\sigma^z=P\exp\bigl(z\log(\sigma|_{P\mathcal H_B})\bigr)P, \qquad f_i(z)=\langle\Theta,\sigma^z h_i\sigma^{-z}\Theta\rangle.\] This is an entire scalar function. Although \(\sigma^0=P\), the vector \(\Theta\) lies in its support, so \(f_i(0)=\langle\Theta,h_i\Theta\rangle\). At \[z_u=-\frac{u}{2(1-u)}\] the two scalar normalization factors cancel, giving \[ f_i(z_u)=\langle\Theta,Fh_iF^{-1}\Theta\rangle. \tag{9}\] Indeed, on the support, \(\sigma^{z_u}=M(u)^{-z_u}F\) and \(\sigma^{-z_u}=M(u)^{z_u}F^{-1}\). The endpoint support projections in the expectation remove any matrix elements entering the kernel.

On the imaginary axis the powers are unitary on the support, and hence \(|f_i(\mathrm i y)|\le c_0\). Decompose \(h_i\) across the cut by matrix units on its portion in \(B\): \[h_i=\sum_{\alpha=1}^{N_i}c_\alpha\otimes d_\alpha, \qquad N_i\le d_i^2, \qquad \lVert c_\alpha\rVert\lVert d_\alpha\rVert\le c_0.\] Here each factor acts as the identity outside the appropriate portion of \(D_i\). If \(s_j\) are the positive Schmidt probabilities of \(\Theta\), a product contributes terms with scalar coefficient \(s_j^{1/2+z}s_k^{1/2-z}\). For \(|\Re z|\le1/4\), \[\bigl|s_j^{1/2+z}s_k^{1/2-z}\bigr|\le s_j+s_k.\] Moreover, in the two Schmidt bases, Cauchy–Schwarz bounds every row and column sum of \(|(c_\alpha)_{jk}(d_\alpha)_{jk}|\) by \(\lVert c_\alpha\rVert\lVert d_\alpha\rVert\). Compression to the Schmidt supports does not increase either norm. Summing first over a row or a column therefore gives \[|f_i(z)|\le2c_0d_i^2\qquad(|\Re z|\le1/4).\] These bounds are uniform in \(\Im z\). Apply the three-lines theorem on each of the two strips between real parts \(0\) and \(\pm1/4\). Writing \(\ell_i=\log(\mathrm e d_i)\), we obtain \[ |f_i(z)|\le\mathrm e c_0 \qquad\left(|\Re z|\le\delta_i:=\frac1{8\ell_i}\right), \tag{10}\] because \(\log(2d_i^2)\le2\ell_i\). The case \(c_0=0\) is immediate; otherwise this application may be made to \(f_i/c_0\).

Hermiticity gives \(f_i(-\overline z)=\overline{f_i(z)}\). The even Taylor coefficients at zero are therefore real, and the odd coefficients are purely imaginary. Cauchy’s coefficient estimate on the disk of radius \(\delta_i\), followed by summation of the even terms, yields, for real \(|x|\le\delta_i/2\), \[ \bigl|\Re(f_i(x)-f_i(0))\bigr| \le2\mathrm e c_0x^2\delta_i^{-2} =128\mathrm e c_0\ell_i^2x^2. \tag{11}\] In particular, it is the real linear term that vanishes; we do not assert that \(f_i'(0)=0\).

For \(u\) in the interval of the lemma, \(|u|\le1/32\) and \(|z_u|\le|u|\). Since \(\ell_i\le\sqrt{\mathcal B}\), the estimate (11) applies to every crossing interaction. Using (8) and (9), we conclude that \[ 0\le\langle\Theta,(\mathsf H-E_0)\Theta\rangle \le128\mathrm e c_0\mathcal B u^2. \tag{12}\] The full-system gap inequality \(\mathsf H-E_0\ge g_0(\mathop{\mathrm{id}}-\lvert \Psi\rangle\langle \Psi\rvert)\) now gives \[\frac{M(u/2)^2}{M(u)} =|\langle\Psi,\Theta\rangle|^2 \ge1-128\mathrm e\,\vartheta\mathcal B u^2.\] The last subtracted quantity is at most \(\mathrm e\vartheta/[8(1+\vartheta)]<1/2\). Taking logarithms and using \(-\log(1-v)\le2v\) for \(0\le v\le1/2\) gives \[L(u)\le2L(u/2)+256\mathrm e\,\vartheta\mathcal B u^2.\] Iterate this inequality. Every successive halving remains in the permitted interval, for either sign of \(u\), and \(2^jL(u/2^j)\to uS\) because \(L(0)=0\) and \(L'(0)=S\). The sum of the remaining geometric series is at most two, proving (6). Finally apply the exponential Markov inequality with \(u=\pm[32\sqrt{(1+\vartheta)\mathcal B}]^{-1}\). The quadratic term in (6) is at most \(\mathrm e/2\). Adding the two one-sided estimates proves (7). ◻

Proof. If \(X\) is empty, the conclusion is immediate. Suppose henceforth that \(X\ne\varnothing\). We will compare two estimates for the largest norm obtained by applying positive filters on nested regions around \(X\). The filters need not commute. Stationarity and the gap give a comparison involving the entropies of those regions. For the opposite comparison, we pin a Schmidt vector of \(X\) and then average over its index. This removes \(X\) from each region and contributes a single entropy cost \(S(X)\). The difference of the two comparisons will therefore control a weighted sum of conditional entropies of \(X\).

Choose an integer \(D>2R+2\). Temporarily leave the integer \(C_{\mathrm{pad}}>D+1\) variable, and put \[m=\left\lfloor\frac{(C_{\mathrm{pad}}-1)r}{D}\right\rfloor, \qquad d_j=r+jD,\qquad X_j=A\cap Q_0^{+d_j}\quad(1\le j\le m),\qquad X_0=X.\] All these rectangles lie in the safe parent \(Q\). Any interaction meeting an \(X_j\) lies wholly in \(A\): otherwise a graph path of length at most \(R\) from a support site in \(X_j\) to a support site of the opposite color would contain a point of \(Z\) within ambient distance \(R\) of \(Q\). This contradicts safety. The same observation rules out any nearest-neighbor color-crossing edge incident with \(X_j\). Consequently the edge boundary of \(X_j\) has at most \(4(r+2d_j)\) edges, all crossing the ambient rectangle’s perimeter. Missing sites in \(\Lambda\) only reduce this count.

Every designated interaction support splits at most one of the cuts \(X_j\mid X_j^c\). More precisely, a support split by \(X_j\) is disjoint from every \(X_i\) with \(i<j\) and contained in every \(X_i\) with \(i>j\). Indeed, successive rectangles have an ambient margin \(D>R\), whereas the ambient diameter of the support is at most its graph diameter \(R\). A term that does not split a contour is either contained in it or disjoint from it. The support budget and (13) therefore give, with constants independent of \(C_{\mathrm{pad}}\), \[ \#\{\text{interactions split by }X_j\}\le C(r+d_j), \qquad \mathcal B_j\le C(r+d_j), \tag{15}\] where \(\mathcal B_j\) is the parameter in Lemma 5 for the marginal on \(X_j\).

Assign a positive weight \(a_j\) to each contour. The energy estimate below charges contour \(j\) by a constant times \((r+d_j)a_j^2\). We choose weights of total mass two: in the eventual conditional-entropy comparison this will give the factor two needed for (14). At this fixed mass, Cauchy–Schwarz gives \[4=\left(\sum_j a_j\right)^2 \le\left(\sum_j(r+d_j)a_j^2\right) \left(\sum_j\frac1{r+d_j}\right).\] Equality holds for weights proportional to \(1/(r+d_j)\). Accordingly, put \[ Z_r=\sum_{j=1}^m\frac1{r+d_j},\qquad a_j=\frac{2}{Z_r(r+d_j)},\qquad \mathcal E=\sum_{j=1}^m(r+d_j)a_j^2=\frac4{Z_r}. \tag{16}\] Their sum is two, and their quadratic cost attains this lower bound. Integral comparison gives the uniform lower bound \[ Z_r\ge\frac1D\log\frac{C_{\mathrm{pad}}+1}{2+D}. \tag{17}\] For example, compare the sum with the integral of \((2r+Dx)^{-1}\) over \(1\le x\le m+1\), use \(2r+D(m+1)>(C_{\mathrm{pad}}+1)r\), and use \(r\ge1\) in the denominator. Thus we may make \(\mathcal E\) as small as needed, uniformly in \(r\). We take \(C_{\mathrm{pad}}\) large enough that \(a_j\le1/2\) for all \(j\). We shall increase it once more after the energy estimate below; all constants up to that choice are independent of \(C_{\mathrm{pad}}\).

Proof. Choose an integer \(M\) below, and consider a safe parent rectangle of size at most \(Mr\). If its size is at most \(r\), its entropy is already bounded by \(F_{\mathrm{box}}(r)\). Otherwise partition each axis into chunks of \(r\) sites and, if needed, one final shorter chunk. Their Cartesian products give at most \(M^2\) children \(Q_i\), each safe and of size at most \(r\). Put \(X_i=A\cap Q_i\).

Call a child interior if its \(C_{\mathrm{pad}}r\) padding lies in the parent. There are at most \[B_M=4(C_{\mathrm{pad}}+2)M\] noninterior children: in either coordinate only the first or last \(C_{\mathrm{pad}}+2\) chunk positions can violate this containment. This estimate remains valid when the parent is thin and all of its children are noninterior.

For an interior child its padded rectangle meets at most \((2C_{\mathrm{pad}}+5)^2-1\) other children. Order all children by a uniform random permutation. The probability that this child comes after every one of those other children is at least \[p_0=(2C_{\mathrm{pad}}+5)^{-2}>0.\] On that event, its preceding physical sites contain the entire buffer in Lemma 6. The corresponding term of the entropy chain rule is therefore at most \(S(X_i)/2+C_{\mathrm{buf}}r\). In all other cases it is at most \(S(X_i)\). The latter bound also holds for noninterior children. Taking the expected entropy chain rule consequently gives \[\begin{align*} F_{\mathrm{box}}(Mr) &\le\left[(1-p_0/2)M^2+(p_0/2)B_M\right] F_{\mathrm{box}}(r)+C M^2r. \tag{27}\end{align*}\] For completeness, the probability of the buffer event may vary among children. Its lower bound \(p_0\) controls the negative term \(-p_0S(X_i)/2\), since \(S(X_i)\ge0\); its positive error can always be bounded by \(C_{\mathrm{buf}}r\). Thus the displayed common bound holds without assuming equal event probabilities. Parents of size at most \(r\) satisfy the same inequality once its coefficient is at least one.

Choose \(M\ge16(C_{\mathrm{pad}}+2)\) and \(M\ge2\). Then the coefficient in (27) is at most \[\lambda=(1-p_0/4)M^2, \qquad M<\lambda<M^2.\] Iterate the recursion at \(r=1,M,M^2,\ldots\). The inhomogeneous terms are bounded by a geometric series with ratio \(M/\lambda<1\), yielding \(F_{\mathrm{box}}(M^j)\le C\lambda^j\) for all \(j\ge0\). Set \[1+e_0=\log_M\lambda =2+\frac{\log(1-p_0/4)}{\log M}\in(1,2).\] For a general \(r\ge1\), comparison with the next power of \(M\) and monotonicity of \(F_{\mathrm{box}}\) prove (26) after a fixed change of constant. ◻

Proof. Fix \(B_y\) and define \[F(X,t)=\sup_{\substack{\mathop{\mathrm{supp}}A\subseteq X\\\|A\|\le1}} \|[\tau_t(A),B_y]\|.\] For \(t\ge0\), fix one such \(A\), put \(f(t)=[\tau_t(A),B_y]\), and let \(Q_X(t)=\sum_{j:X_j\cap X\ne\varnothing}\tau_t(h_j)\). Since the terms disjoint from \(X\) commute with \(A\), differentiating and using the Jacobi identity gives the exact equation \[f'(t)=i[Q_X(t),f(t)] -i\sum_{j:X_j\cap X\ne\varnothing} [\tau_t(A),[\tau_t(h_j),B_y]].\] The propagator of the homogeneous equation is unitary conjugation. Variation of constants, followed by the supremum over \(A\), consequently gives \[ F(X,t)\le 2\|B_y\|\mathbf1_{\{y\in X\}} +2\sum_{j:X_j\cap X\ne\varnothing}\|h_j\| \int_0^t F(X_j,u)\,du. \tag{32}\] The same argument with \(-H\) handles negative times.

Starting from \(X=X_i\), iterate (32). At each step the sum of the available interaction norms is at most \(\kappa:=v_Rb_0\), by (29). A chain \(X_i,X_{j_1},\ldots,X_{j_m}\) with consecutive intersections and with \(y\in X_{j_m}\) has \(d_\Lambda(a_i,y)\le(m+1)R\). The remainder after \(M\) iterations is at most \(2\|B_y\|(2\kappa t)^M/M!\), since \(F(X,u)\le2\|B_y\|\); it tends to zero. For every \(\mu>0\), the resulting chain sum satisfies \[\begin{align*} F(X_i,t) &\le 2\|B_y\|\sum_{m\ge0} \frac{(2\kappa t)^m}{m!} \mathbf1_{\{d_\Lambda(a_i,y)\le(m+1)R\}}\\ &\le 2\|B_y\|e^{\mu R-\mu d_\Lambda(a_i,y)} \exp(2\kappa e^{\mu R}t). \end{align*}\] This proves (30). If the distance is infinite, no finite chain contributes and the vanishing remainder proves exact zero.

For completeness, write the expectation in (28) as a product of on-site twirls \(T_y\). These are commuting contractions, and telescoping that product shows, for any \(B\), \[\|B-E_{i,l}(B)\| \le\sum_{y\notin N_l(a_i)}\|B-T_y(B)\| \le\sum_{y\notin N_l(a_i)}\sup_{U_y}\|[B,U_y]\|,\] where \(U_y\) ranges over on-site unitaries. Sites in another component contribute zero. The number of sites at graph distance \(d\) is at most \(C(d+1)^2\), because the graph ball lies in an ambient lattice ball. Applying (30) and summing \(\sum_{d>l}(d+1)^2e^{-cd}\le C'e^{-c'l}\) proves the exponential bound in (31). Contractivity gives its other bound. ◻

Proof. We give the derivative estimate responsible for the exponent. For real \(u\in(-1,1)\) set \(w=1-u^2\). Choose \(\eta_p>0\) sufficiently small. On the complex disk \(|z-u|\le\eta_pw\) one has \[1-z^2=w(1+\zeta),\qquad |\zeta|\le3\eta_p, \qquad \operatorname{Re}(1+\zeta)^{-p}\ge c_p>0.\] The last inequality follows by continuity at \(\zeta=0\), with \(p\) fixed. The function \(e\exp[-(1-z^2)^{-p}]\) is analytic on the disk, and Cauchy’s estimate gives \[\left|\frac{d^m}{du^m}\bigl(e\exp[-(1-u^2)^{-p}]\bigr)\right| \le C m!\eta_p^{-m}w^{-m}e^{-c_pw^{-p}}.\] For each fixed \(m\) this tends to zero at both endpoints. Extension by zero is therefore smooth. To make the estimate uniform in \(u\), substitute \(s=w^{-p}\) and maximize \(s^{m/p}e^{-c_ps}\). Its maximum is bounded by \(C^{m+1}(m!)^{1/p}\), using \(m!\ge(m/e)^m\); the case \(m=0\) is absorbed in \(C\). Scaling from \(u\) to \(\omega\) yields \[ \|\chi^{(m)}\|_\infty\le C_1^{m+1}(m!)^{1+1/p}. \tag{34}\]

Integration by parts \(m\) times has no boundary terms. Consequently, for \(t\ne0\), \[|f(t)|\le C\frac{C_1^m(m!)^{1+1/p}}{|t|^m} \le C\left(\frac{C_1m^{1+1/p}}{|t|}\right)^m.\] For sufficiently large \(|t|\), choose \(m=\lfloor\eta |t|^{p/(p+1)}\rfloor\) with a fixed, sufficiently small \(\eta>0\). The expression in parentheses is at most \(1/2\), proving the decay in (33); bounded \(t\) is covered by increasing \(C\). In particular \(f\in L^1(\mathbb R)\). Fourier inversion gives the first identity in (33). It can also be obtained directly by inserting \(e^{-\varepsilon t^2}\), interchanging the two integrals, and letting the resulting Gaussian approximate identity converge to the continuous function \(\chi\). Reality and evenness follow from those properties of \(\chi\). ◻

Proof. Use Lemma 9 with \(\delta=\Delta/2\), and define \[ M_i=\int_{\mathbb R}f(t)\tau_t(h_i)\,dt -\langle\Omega,h_i\Omega\rangle I. \tag{37}\] The integral converges in norm and \(M_i\) is Hermitian. In an eigenbasis of \(H\), its uncentered ground-to-energy-\(E\) matrix element equals the corresponding matrix element of \(h_i\) multiplied by \(\chi(E-E_0)\). All excited energies satisfy \(E-E_0\ge\Delta\), where \(\chi\) vanishes. The ground component is unchanged because \(\chi(0)=1\), and the scalar in (37) removes it. Hence \(M_i\Omega=0\). Also, since \(\tau_t(H)=H\) and \(\int f=1\), \[ \sum_i M_i=H-E_0I,\qquad \|M_i\|\le J(\|f\|_1+1). \tag{38}\] Choose \(c_*:=\max\{1,J(\|f\|_1+1)\}\) and set \(k_i=|M_i|/c_*\). For a Hermitian operator, \(|M_i|\ge M_i\) and \(\ker|M_i|=\ker M_i\). These observations and (38) prove (35).

It remains to check locality after taking the absolute value. The scalar in (37) is fixed by \(E_{i,l}\). For \(|t|\le\varepsilon l\), choose \(\varepsilon>0\) small enough that (31) is at most \(Ce^{-c'l}\) for \(A=h_i\). For the remaining times use its bound \(2J\). Thus \[\begin{align*} \|M_i-E_{i,l}(M_i)\| &\le Ce^{-c'l}\|f\|_1 +2J\int_{|t|>\varepsilon l}|f(t)|\,dt \le C'e^{-c''l^\alpha}. \tag{39}\end{align*}\] The last tail estimate follows by substituting \(u=|t|^\alpha\) and absorbing the resulting fixed power of \(u\) into the exponential. The case \(l=0\) follows from the uniform norm bound.

We use the dimension-independent inequality \[ \|U^{1/2}-V^{1/2}\|\le\|U-V\|^{1/2} \qquad(U,V\ge0). \tag{40}\] Here is an order proof. The integral formula \[T^{1/2}=\frac1\pi\int_0^\infty T(T+sI)^{-1}s^{-1/2}\,ds\qquad(T\ge0)\] follows from scalar integration and the spectral theorem. It shows that the square root is operator monotone: inversion reverses the order of positive invertible operators, and \(T(T+sI)^{-1}=I-s(T+sI)^{-1}\). With \(\epsilon=\|U-V\|\), monotonicity and scalar functional calculus give \(U^{1/2}\le(V+\epsilon I)^{1/2}\le V^{1/2}+\sqrt\epsilon I\). Interchanging \(U,V\) proves (40).

Apply this with \(U=M_i^2\) and \(V=(E_{i,l}M_i)^2\). Both factors in the latter square are Hermitian, and their norms are at most \(c_*\). Therefore \[\big\||M_i|-|E_{i,l}M_i|\big\| \le\bigl(2c_*\|M_i-E_{i,l}M_i\|\bigr)^{1/2} \le Ce^{-cl^\alpha}.\] The operator \(|E_{i,l}M_i|\) is supported in \(N_l(a_i)\). Since \(E_{i,l}\) fixes it and is contractive, \[\||M_i|-E_{i,l}(|M_i|)\| \le2\big\||M_i|-|E_{i,l}M_i|\big\|.\] After division by \(c_*\) this proves (36); its order bounds follow from positivity and unitality of \(E_{i,l}\).

Finally, every original support lies in one component. The Hamiltonian is a tensor sum over components, so \(\tau_t(h_i)\) is supported in the component of \(a_i\). The integral, scalar centering, and functional calculus in the above construction all stay in that component’s operator algebra. This proves the exact component assertion. ◻

Proof. Apply (40) to \(k_i,k_{i,l}\) and to \(I-k_i,I-k_{i,l}\), using (36). The order and ground-vector assertions follow from functional calculus. Each displayed channel is a sum of two maps of the form \(B\mapsto A^*BA\); this proves complete positivity, while the sum of the squared Kraus operators is \(I\). More explicitly, the map \(v\mapsto G_iv\oplus K_iv\) is an isometry and \(\mathcal E_i(B)\) is the compression of \(B\oplus B\) by that isometry. This proves contractivity, and the same proof works for the local maps and after adjoining an identity tensor factor. Finally, \[\|ABA-A'BA'\| \le(\|A\|+\|A'\|)\|A-A'\|\|B\|,\] also on every auxiliary extension. Applying this to both summands proves (42), where the completely bounded norm is the supremum of these extended operator norms. An operator commuting with the ball algebra commutes with both local square roots, so its image is itself times \(G_{i,l}^2+K_{i,l}^2=I\). ◻

Proof. We first prove the summation estimate for any fixed \(\alpha\in(0,1)\) for which (36) holds, and any integer \(r_0\ge1\). At each finite distance \(d\) from \(S_0\) there are at most \(C\mu_R|S_0|(d+1)^2\) labels: every anchor at that distance belongs to \(N_d(x)\) for some \(x\in S_0\), and anchor multiplicity is bounded by \(\mu_R\). Labels at infinite distance have no error by definition. The triangle inequality therefore gives \[\begin{align*} \|\widetilde H-H_F\| &\le C|S_0|\sum_{d\ge0}(d+1)^2 \exp\bigl(-c\max\{r_0,\lfloor d/2\rfloor\}^{\alpha}\bigr) \le C'|S_0|e^{-c'r_0^\alpha}. \tag{48}\end{align*}\] To justify the final inequality, the terms \(d\le2r_0+2\) have sum at most \(C(r_0+1)^3e^{-cr_0^\alpha}\). For \(d>2r_0+2\) use \(\lfloor d/2\rfloor\ge d/3\); comparison with the integral of \((x+1)^2e^{-c(x/3)^\alpha}\), followed by \(u=x^\alpha\), bounds the remaining tail by a fixed power of \(r_0+1\) times \(e^{-c''r_0^\alpha}\). All such powers are absorbed by decreasing the positive exponential constant. Thus (48) has no factor involving \(|\Lambda|\).

Now take \(\alpha=1/2\), \(|S_0|\le C_0n^2\), and \(r_0=\lceil C_1(\log n)^2\rceil\). The right-hand side of (48) is at most \(C'C_0n^{2-c'\sqrt{C_1}}\). Taking \(C_1\) sufficiently large proves (44), including its \(g/4\) bound for all \(n\ge2\).

Positivity gives \(\widetilde E_0\ge0\), and testing \(\widetilde H\) on \(\Omega\) gives \(\widetilde E_0\le\epsilon_n\). The min–max principle and (44) put the second eigenvalue of \(\widetilde H\) at least \(g-\epsilon_n\). The difference between its first two eigenvalues is consequently at least \(g-2\epsilon_n\ge g/2\), which also proves uniqueness. Moreover, \[g\bigl(1-|\langle\Omega,\widetilde\Omega\rangle|^2\bigr) \le\langle\widetilde\Omega,H_F\widetilde\Omega\rangle \le\widetilde E_0+\epsilon_n\le2\epsilon_n.\] The trace distance of two pure states equals the square root of one minus their squared overlap, giving the second estimate in (45). Choose the overlap real and nonnegative, and use \(1-s\le1-s^2\) for \(0\le s\le1\) to obtain the first estimate.

Finally, if a designated support meets \(X\subseteq S_0\), then its anchor has finite distance from \(S_0\). If \(r_i>r_0\), formula (43) gives \(r_i=\lfloor d_i^0/2\rfloor<d_i^0\), so that its ball cannot meet \(S_0\). Hence every label in \(\mathcal C_X\) has \(r_i=r_0\). If its anchor lies in \(X\), take a shortest path from the anchor to a point of the ball outside \(X\); otherwise take one to a point inside \(X\). This path stays in the ball and contains an edge of \(\partial_\Lambda X\). Thus the anchor is within \(r_0\) of an endpoint of a crossing edge. There are at most \(2b_X\) endpoints, at most \(C(r_0+1)^2\) sites in each such ball, and at most \(\mu_R\) labels per anchor. This proves the first bound in (46). The ball volume bound also gives \(\log d_i=|\widetilde X_i|\log q\le C(r_0+1)^2\). Summing its squared version proves (47). Since \(r_0+1\le C(\log n)^2\) for \(n\ge2\), its stated consequence follows. ◻

Proof. Purity and strong subadditivity give the displayed identities and \(\eta\ge0\). It suffices initially to take \(\sigma,\tau\) faithful. Write \(d_U=S(x|U)_\theta\) and \(d_P=S(x|P)_\theta\). Consider \[G(z)=\sigma^{az}\widehat\rho_P^{-az} \tau^{a(1-z)}\widehat\rho_U^{-a(1-z)}\theta, \qquad 0\le\Re z\le1.\] The midpoint is the vector in (52). All imaginary powers here are unitary. In particular \(G\) is bounded on this strip, with bounds allowed at this intermediate step to depend on its fixed invertible factors.

On the left boundary, discard the leftmost unitary factors and move the remaining imaginary powers past the commuting real powers. With \(\rho=\rho_{xU}\), \(Q=I_x\otimes\widehat\rho_U\), and \(\tau_u=Q^{-iau}\tau Q^{iau}\), this gives \[\|G(iu)\|=M(u):=\|\tau_u^aQ^{-a}\theta\|.\] Hölder’s inequality and (51) imply \[M(u)^2=\mathop{\mathrm{Tr}}\tau_u^{2a}K\le Z, \qquad \log Z\le-2ad_U+C a^2\ell^2.\] We retain the slack in this bound: a large slack already improves the left boundary, while a small slack will control the distance from the right-boundary input to \(\theta\). Choose a universal \(C_0\) larger than half the constant in this last bound and set \[ \lambda(u)=-ad_U+C_0a^2\ell^2-\log M(u)\ge0, \qquad \xi=K^r/\mathop{\mathrm{Tr}}K^r. \tag{53}\] The normalized Hölder overlap then satisfies \[ \mathop{\mathrm{Tr}}\tau_u^{2a}\xi^{1-2a}=M(u)^2/Z\ge e^{-2\lambda(u)}. \tag{54}\]

We need to relate this overlap to \(\rho\), independently of the smallest eigenvalues of either marginal. Taking \(\rho\) in place of \(\tau_u\) for this comparison gives \[\|\rho^aQ^{-a}\theta\| \ge\langle\theta,\rho^aQ^{-a}\theta\rangle =\mathop{\mathrm{Tr}}\rho^{1+a}Q^{-a} =\mathbb Ee^{aL}\ge e^{-ad_U}.\] The trace is real and nonnegative. Thus \(1-\mathop{\mathrm{Tr}}\rho^{2a}\xi^{1-2a}\le C a^2\ell^2\). The function \(b\mapsto1-\mathop{\mathrm{Tr}}\rho^{[b]}\xi^{[1-b]}\) is nonnegative and concave on \([0,1]\): expand in their two eigenbases for concavity, and use Hölder for nonnegativity. For \(2a\le1/2\), concavity and its nonnegative value at zero give \[1-\mathop{\mathrm{Tr}}\sqrt\rho\sqrt\xi \le \frac{1-\mathop{\mathrm{Tr}}\rho^{2a}\xi^{1-2a}}{4a}.\] Consequently \[ H:=\|\sqrt\rho-\sqrt\xi\|_2\le C\sqrt a\,\ell. \tag{55}\]

We next show how the slack in (53) controls the state arriving at the opposite boundary. Define \[\theta_u=\tau^{-iau}Q^{iau}\theta =Q^{iau}\tau_u^{-iau}\theta.\] This is a unit vector with the same \(P\) marginal as \(\theta\). We claim \[ \|\theta_u-\theta\| \le C(1+|u|)\big(\sqrt{\lambda(u)}+\sqrt a\,\ell\big). \tag{56}\] Here and in the verification all kernel phases are extended by one. First, \[\|(Q^{-iau}-\widehat\rho^{-iau})\theta\|^2 \le a^2u^2\mathbb EL^2\le C a^2u^2\ell^2.\] For the remaining comparison of \(\tau_u\) and \(\rho\) insert the phase of \(\xi\). For any densities \(A,B\) and real \(b\), \[ \| (\widehat A^{ib}-\widehat B^{ib})\sqrt B\|_2 \le C(1+|b|)\|\sqrt A-\sqrt B\|_2. \tag{57}\] To check this, expand in their two spectral bases. For positive eigenvalues with ratio \(v\), use \(|1-v^{ib}|\le C(1+|b|)|1-\sqrt v|\). This follows from the mean-value theorem when \(1/4\le v\le4\), and the bound two outside that interval. At a zero eigenvalue of \(A\) the same assertion follows from the bound two and \(|1-\sqrt0|=1\); zero eigenvalues of \(B\) contribute nothing to the left side.

Replacing \(\sqrt\rho\) by \(\sqrt\xi\) in the \(\tau_u,\xi\) phase difference costs at most \(2H\). On \(\sqrt\xi\), give a pair of eigenvalues \(p\) of \(\xi\) and \(q\) of \(\tau_u\) the probability \(p|\langle p,q\rangle|^2\) and put \(v=\log(q/p)\). Faithfulness of \(\tau_u\) makes this finite, and \[\mathbb Ee^v\le1,\qquad \mathbb Ee^{2av}\ge e^{-2\lambda(u)}\] by (54). On \(v\le0\), \[|1-e^{-iauv}|^2\le C(1+u^2)(1-e^{2av}),\] and \[\mathbb E[(1-e^{2av})1_{v\le0}] \le 1-e^{-2\lambda(u)} +\mathbb E[(e^{2av}-1)1_{v>0}] \le 2\lambda(u)+2a.\] The last step uses \(e^{2av}-1\le2a(e^v-1)\) for \(v>0\). On \(v>0\) the squared phase difference is at most \(C a^2u^2 e^v\). These estimates, followed by (57) for \(\xi,\rho\) and (55), prove (56).

The phase comparison now lets us recover the other conditional entropy. On the right boundary, the remaining \(P\) phases merely conjugate \(\sigma\). Apply (51) and Hölder on \(xP\) to the input \(\theta_u\), whose \(P\) marginal is unchanged. It follows that \[\log\|G(1+iu)\| \le-aS(x|P)_{\theta_u}+C a^2\ell^2.\] The continuity bound for conditional entropy in Lemma 3, using only \(\dim x\), and (56) now give \[\begin{align*} \log\|G(1+iu)\| \le{}&-ad_P+C a^2\ell^2\\ &+C a\ell(1+|u|)^{1/2} \big(\lambda(u)^{1/4}+a^{1/4}\ell^{1/2}\big). \end{align*}\] On the left boundary the bound is exactly \(-ad_U+C_0a^2\ell^2-\lambda(u)\).

For clarity, the logarithmic interpolation used here assigns density \[\frac{du}{2\cosh(\pi u)}\] to each boundary line when evaluated at the midpoint. Each line has total mass \(1/2\). This follows by mapping the strip to the upper half-plane by \(z\mapsto e^{\pi iz}\) and using harmonic measure at \(i\). Applied to the logarithm of a scalar analytic function it follows from the submean inequality, first on bounded domains and then by a limit. Pairing \(G\) with a unit vector in its midpoint direction gives the vector-norm assertion. All boundary norms above are bounded above and away from zero for the fixed faithful factors, so this application and the limit are legitimate.

Pair the two boundary estimates at the same \(u\). Young’s inequality gives \[-\lambda+C a\ell(1+|u|)^{1/2}\lambda^{1/4} \le C\big(a\ell(1+|u|)^{1/2}\big)^{4/3}.\] After integration, this term, the quadratic errors, and the remaining \(a^{5/4}\ell^{3/2}\) term are all at most \(C a^{5/4}\ell^2\). The entropy contribution is \(-a(d_P+d_U)/2\), proving (52) for faithful \(\sigma,\tau\). Approximation by faithful densities removes that restriction by continuity of positive powers. Finally normalizing subdensities only increases the left side; the zero subdensity case is immediate. ◻

Proof. The proof writes \(\eta\) as the integral of a nonnegative resolvent defect. Two moment bounds, involving only \(d\), control the ends of the integral; the entropy controls the part between them. Initially let \(\rho_f=\rho_{xUF}\) be faithful, and put \[\sigma_f=I_x/d\otimes\rho_{UF},\qquad \rho_s=\rho_{xU},\qquad \sigma_s=I_x/d\otimes\rho_U,\qquad d=\dim x.\] On their Hilbert–Schmidt matrix spaces define positive definite operators \[A=L_{\sigma_f}R_{\rho_f^{-1}},\qquad B=L_{\sigma_s}R_{\rho_s^{-1}},\] where \(L\) and \(R\) denote left and right multiplication. Define the isometry \[V(X\sqrt{\rho_s})=(X\otimes I_F)\sqrt{\rho_f}.\] It is an isometry by taking the partial trace of \(\rho_f\). Taking matrix elements in the same way with \(\sigma_f\) shows \(V^*AV=B\): both sides have the form \(\mathop{\mathrm{Tr}}X^*\sigma_s Y\). Put \(b_0=\sqrt{\rho_s}\) and \(a_0=Vb_0=\sqrt{\rho_f}\). For \(v>0\), let \[\delta_v=(A+vI)^{-1}a_0-V(B+vI)^{-1}b_0.\] Expansion of its weighted squared norm, using \(V^*(A+vI)V=B+vI\), gives the resolvent compression identity (Carlen and Vershynina 2020, Lemma 2.1): \[ \|(A+vI)^{1/2}\delta_v\|_2^2 =\langle a_0,(A+vI)^{-1}a_0\rangle -\langle b_0,(B+vI)^{-1}b_0\rangle =:h(v)\ge0. \tag{59}\] The logarithmic resolvent formula gives \[ \int_0^\infty h(v)\,dv =D(\rho_f\|\sigma_f)-D(\rho_s\|\sigma_s)=\eta, \tag{60}\] where \(D(\rho\|\sigma)=\mathop{\mathrm{Tr}}\rho(\log\rho-\log\sigma)\). Indeed \(-\langle\sqrt\rho,\log(L_\sigma R_{\rho^{-1}}) \sqrt\rho\rangle=D(\rho\|\sigma)\), and the identity terms in the log integral cancel. The last equality in (60) follows by expanding the four entropies.

The local domination proved in (49) now reads \(\rho_f\le d^2\sigma_f\). Therefore \[\langle a_0,A^{-1}a_0\rangle =\mathop{\mathrm{Tr}}\rho_f^2\sigma_f^{-1}\le d^2, \qquad \langle b_0,Bb_0\rangle=\mathop{\mathrm{Tr}}\sigma_s=1.\] The first resolvent quadratic form in (59) is therefore at most \(d^2\). For another bound use \((A+vI)^{-1}\le v^{-1}I\) and \((B+vI)^{-1}\ge v^{-1}I-v^{-2}B\). We obtain \[ 0\le h(v)\le\min(d^2,v^{-2}),\qquad \|\delta_v\|_2\le\sqrt{h(v)/v}. \tag{61}\]

For \(-1<\Re z<0\), scalar integration and spectral calculus give \[A^za_0-VB^zb_0 =-\frac{\sin\pi z}{\pi} \int_0^\infty v^z\delta_v\,dv.\] One may approach \(z=-iu\) from \(-1/4\le\Re z<0\) under this integral: (61) gives an integrable majorant \(d\,v^{-3/4}\) near zero and \(v^{-3/2}\) near infinity. Hence \[\|A^{-iu}a_0-VB^{-iu}b_0\|_2 \le C|\sinh\pi u|\int_0^\infty\sqrt{h(v)/v}\,dv.\] For \(\eta>0\), split the integral at \(y/d^2\) and \(1/y\). The two ends cost at most \(C\sqrt y\) by (61); Cauchy–Schwarz and (60) bound the middle by \[\left(\int h(v)\,dv\right)^{1/2} \left(\int_{y/d^2}^{1/y}\frac{dv}{v}\right)^{1/2} \le C\sqrt{\eta\log(ed/y)}.\] If \(\eta=0\), nonnegativity and continuity imply \(h(v)=0\) for all \(v>0\), so the discrepancy is zero. To identify the resulting vectors, use the definition of \(V\): \[\begin{split} A^{-iu}a_0 &=\sigma_f^{-iu}\rho_f^{iu}\sqrt{\rho_f},\\ VB^{-iu}b_0 &=\bigl(\sigma_s^{-iu}\rho_s^{iu}\otimes I_F\bigr) \sqrt{\rho_f}. \end{split}\] After cancelling the common scalar phase \(d^{iu}\) and reversing the sign of the difference, its Hilbert–Schmidt norm is \[\big\|\big(\widehat\rho_U^{-iu}\widehat\rho_{xU}^{iu} -\widehat\rho_{UF}^{-iu}\widehat\rho_{xUF}^{iu}\big) \sqrt{\rho_f}\big\|_2.\] For any operator \(T\) on \(xUF\) one has \(\|T\theta\|=\|T\sqrt{\rho_f}\|_2\). This proves (58) in the faithful case.

For a possibly singular \(\rho_f\), replace it by \((1-\epsilon)\rho_f+\epsilon I/\dim(xUF)\) and use its reductions in the preceding proof. Conditional mutual information converges in finite dimension. The phase products on the square root converge in Hilbert–Schmidt norm in the required order. For example, the innermost lifted phase of \(\rho_{xU,\epsilon}\) acts on \(\sqrt{\rho_{f,\epsilon}}\). Its limit lies in \(\mathop{\mathrm{supp}}\rho_{xU}\otimes F\), because \(\mathop{\mathrm{supp}}\rho_f\subseteq\mathop{\mathrm{supp}}\rho_{xU}\otimes F\) and the square-root mass outside this space tends to zero. On this space the inner phase converges, since the mollified marginal commutes with its limit. Its output lies in \(x\otimes\mathop{\mathrm{supp}}\rho_U\otimes F\), so the outer \(\rho_{U,\epsilon}\) phase also converges there. The same argument applies to the \(xUF,UF\) pair. All phases have norm one, which controls the vanishing complementary pieces. Thus the limit is the product of support phases, equivalently the unitary extensions appearing in (58). No operator-norm convergence of phases on a limiting kernel is being used. The faithful proof was on Hilbert–Schmidt spaces, so it also applies when a faithful approximation would require a larger purifying system than \(P\). This establishes (58) in full generality. ◻

Proof. Put \(p=\langle\theta,h\theta\rangle\in[0,1]\), so \(f(0)=p\). We first bound \(f\) on a strip using the support dimension of \(h\), then compare its imaginary-axis values with \(p\), and finally estimate its real-axis deviation from \([0,\infty)\). Expand \(h=\sum c\otimes d'\) across \(P_0,x\) using matrix units on \(P_0\). There are at most \(d_h^2\) terms, with \(\|c\|\|d'\|\le1\) for each. For \(0\le\Re z\le1/4\) the contribution is the inner product of \[\rho_Y^{[\bar z]}(d')^*\rho_Y^{[-\bar z]}\theta \quad\hbox{and}\quad \rho_P^{[z]}c\rho_P^{[-z]}\theta.\] Each norm is bounded by the norm of its local operator: for a marginal \(\rho\) and \(0\le\Re z\le1/2\), \[\|\rho^{[z]}c\rho^{[1/2-z]}\|_2\le\|c\|\] by Schatten Hölder. This is also valid at the endpoints using support powers. Reflection \(f(-\bar z)=\overline{f(z)}\) treats negative real parts. Thus \(|f(z)|\le d_h^2\) on \(|\Re z|\le1/4\). On the imaginary axis it is an expectation in a unit vector and has absolute value at most one. Three-lines on the two half-strips gives \[ |f(z)|\le C\qquad (|\Re z|\le b), \qquad b=c_0/\ell_h, \tag{63}\] for a sufficiently small universal \(c_0>0\).

On the imaginary axis Schmidt mirroring across \(P,YF\) gives \[f(iu)=\langle w_u,hw_u\rangle, \qquad w_u=\widehat\rho_Y^{iu}\widehat\rho_{YF}^{-iu}\theta.\] Compare this with \(w_u^0=\widehat\rho_U^{iu}\widehat\rho_{UF}^{-iu}\theta\). The latter is a unitary acting only on \(UF\), so \(\langle w_u^0,hw_u^0\rangle=p\). Multiplication of both vectors first by \(\widehat\rho_P^{iu}\) and then by \(\widehat\rho_U^{-iu}\), and a second use of Schmidt mirroring, show that \[ \|w_u-w_u^0\| =\| (\widehat\rho_U^{-iu}\widehat\rho_Y^{iu} -\widehat\rho_{UF}^{-iu}\widehat\rho_{YF}^{iu})\theta\|. \tag{64}\] Lemma 14 therefore gives \(\|w_u-w_u^0\|\le C|\sinh(\pi u)|\mathcal R_\eta\), where \(\mathcal R_\eta\) is defined there with \(d=\dim x\).

It remains to pass from phases to the real-power skew. Positivity of \(h\) and the identity \(\langle w_u^0,hw_u^0\rangle=p\) imply \[|f(iu)-p|\le2\sqrt p\,\|w_u-w_u^0\| +\|w_u-w_u^0\|^2.\] In particular, differentiating at zero and using (58) gives \[ f'(0)\in i\mathbb R,\qquad |f'(0)|\le C\sqrt p\,\mathcal R_\eta. \tag{65}\] The imaginary character also follows from \(f(-\bar z)=\overline{f(z)}\).

Set \(m=\min(1,\mathcal R_\eta)\). The function \(F(z)=(f(z)-p)e^{z^2}\) has absolute value at most \(Cm\) on the imaginary axis: use (58), the Gaussian factor \(e^{-u^2}\), and also \(|f(iu)-p|\le2\). On the two outer lines \(\Re z=\pm b\) it is bounded by a universal constant by (63). Three-lines on each half-strip therefore bounds \(|F|\) by \(Cm^{1/2}\) on \(|\Re z|\le b/2\). If \(m=0\), the identity theorem already gives \(f\equiv p\). Otherwise Cauchy’s estimates on a circle of radius \(b/2\) centered at zero, applied also to the bounded factor \(e^{-z^2}\) on this circle, give \[|f(t)-p-tf'(0)|\le C(t/b)^2m^{1/2},\qquad 0\le t\le b/8.\] Write \(f'(0)=ib_1\) with \(b_1\in\mathbb R\). If \(p>0\), then \[|p+itb_1|-p\le t^2b_1^2/p\le C t^2\mathcal R_\eta^2;\] if \(p=0\), (65) makes \(b_1=0\). Adding the remainder changes \(|f|-\Re f\) by at most twice its absolute value. Thus \[|f(t)|-\Re f(t) \le C a^2\big(\ell_h^2m^{1/2}+\mathcal R_\eta^2\big).\] Since \(d=\dim x\le d_h\), the quantity supplied by Lemma 14 satisfies, for \(0<\eta\le1\), \(\mathcal R_\eta\le C\sqrt{\eta\log(ed_h/\eta)}\); for \(\eta\ge1\), the bound \(\eta\le2\log d\le2\ell_h\) gives \(\mathcal R_\eta\le C\ell_h\). These two cases imply \[\ell_h^2m^{1/2}+\mathcal R_\eta^2 \le C\ell_h^4\eta^{1/8},\] absorbing the logarithm of \(1/\eta\) by the smaller power of \(\eta\). Taking the universal \(c\) in the lemma small enough to ensure \(t=a/2\le b/8\) proves (62). ◻

Proof. The first formula in (68) is the value of the Weyl character at the identity. To derive the second, multiply the character decomposition of \((x_1+\cdots+x_q)^k\) by \(\det[x_i^{q-j}]\) and extract the coefficient of \(x_1^{l_1}\cdots x_q^{l_q}\). The strict decrease of the \(l_i\) selects only label \(\lambda\) and the identity term in its numerator alternant. On the other side the coefficient is \[k!\det\left[\frac1{(l_i-q+j)!}\right]_{i,j=1}^q =\frac{k!}{\prod_i l_i!} \det\bigl[(l_i)_{q-j}\bigr]_{i,j=1}^q =\frac{k!\prod_{i<j}(l_i-l_j)}{\prod_i l_i!}.\] Here a reciprocal negative factorial is zero and \((u)_m\) is the falling factorial. The last determinant is a Vandermonde because its columns are monic polynomials of the indicated degrees. The estimate \(\log n!=n\log n-n+O(\log(n+2))\) gives the entropy formula, including zero parts, with \(0\log0=0\). The fixed shifts from \(\lambda_i\) to \(l_i\) and the Vandermonde contribute only \(O_q(\log(k+2))\). There are at most \((k+1)^q\) labels, and the dimension formula bounds each \(\dim V^{(q)}_\lambda\) by \((k+q)^{q(q-1)/2}\).

Each label projector is a central character idempotent in the permutation algebra: \[\pi_{Q,k}^{\lambda} =\frac{d_\lambda}{k!}\sum_{\pi\in S_k} \chi_\lambda(\pi^{-1})U_Q(\pi).\] This follows from character orthogonality, or directly from Schur’s lemma applied to the central sum and its trace on each irreducible. Conjugation by \(U_{QE}(\pi)\) sends a central function of the \(Q\) permutations to itself. This proves nested commutation; disjoint commutation is immediate. On \(\mathcal S_k\), \(U_Q(\pi)z=U_{Q^c}(\pi^{-1})z\). A permutation is conjugate to its inverse, so \(\chi_\lambda(\pi)=\chi_\lambda(\pi^{-1})\) and the two central-projector sums agree there. This also shows that all Specht modules are self-dual: their dual characters are the same.

For disjoint subsystems, \([\nu]\) is a constituent of \([\lambda]\otimes[\mu]\) under diagonal permutations. This proves the first inequality of (69). The identity \[\operatorname{Hom}_{S_k}([\nu],[\lambda]\otimes[\mu]) \simeq \operatorname{Hom}_{S_k}([\lambda],[\nu]\otimes[\mu]^*)\] and self-duality prove the second. Conditional on \(\lambda,\mu\), a product of invariant states is the identity divided by \(d_\lambda d_\mu\) on the two Specht factors. If \(g_{\lambda\mu\nu}\) is the multiplicity of \([\nu]\) in their tensor product, its conditional label probability is exactly \(g_{\lambda\mu\nu}d_\nu/(d_\lambda d_\mu)\). The ambient decomposition of \((Q\otimes E)^{\otimes k}\) implies \[g_{\lambda\mu\nu}\, \dim V^{(d_Q)}_\lambda\dim V^{(d_E)}_\mu \le \dim V^{(d_Qd_E)}_\nu,\] so \(g_{\lambda\mu\nu}\) is polynomially bounded. With \(R=d_\lambda d_\mu/d_\nu\ge1\), the conditional expectation in (70) is \(\sum_\nu g_{\lambda\mu\nu}R^{b-1}\le\sum_\nu g_{\lambda\mu\nu}\), again polynomial.

For the copy-group assertion, choose a copy of \([\alpha]\otimes[\beta]\) inside the restriction of \([\lambda]\) to \(S_{k-r}\times S_r\). Its dimension gives the lower bound. The translates of this subspace span \([\lambda]\), by irreducibility. Choosing one representative of each subgroup coset bounds the span dimension by \(\binom{k}{r}d_\alpha d_\beta\). Central elements of \(S_k\) commute with its subgroup algebra, proving the commutation claim.

An invariant density has blocks \(\rho=\bigoplus_\lambda I_{[\lambda]}\otimes M_\lambda\). Every positive eigenvalue \(u\) of \(M_\lambda\) is repeated \(d_\lambda\) times, so \(u d_\lambda\le1\). This gives \(F_Q\le L_\rho\) and \[\mathop{\mathrm{Tr}}\rho e^{b(L_\rho-F_Q)} =\sum_{\lambda}\sum_{u>0}(u d_\lambda)^{1-b} \le\sum_\lambda\dim V^{(d_Q)}_\lambda \qquad(0\le b\le1),\] where the inner sum counts eigenvalues with multiplicity in \(M_\lambda\).

It remains to identify the marked-copy branches. Multiplying a Weyl numerator alternant by \(\sum_i x_i\) gives the sum of alternants with one exponent increased by one. Terms with equal exponents vanish; the remaining terms correspond exactly to adding a legal box to a partition. Thus \(V^{(q)}_\nu\otimes\mathbb C^q=\bigoplus_{\lambda=\nu+e_i} V^{(q)}_\lambda\), with multiplicity one. Comparing the two Schur–Weyl decompositions of the first \(k-1\) copies and the last copy shows that the \(\nu\) branch in the \(\lambda\) block has rank \(d_\nu\) on \([\lambda]\), and acts identically on \(V^{(q)}_\lambda\). The invariant state is uniform on \([\lambda]\), giving the probability in (73). The dimension formula gives the displayed quotient. In its Vandermonde ratio, a factor involving an earlier row is at most two, and a factor involving a later row is at most one. Positivity holds on a legal removable branch.

For the star eigenvalue, let \(E_{ab}^{\rm tot}\) be the sum over copies of the matrix unit \(E_{ab}\) on \(\mathbb C^q\). Direct multiplication gives \[\sum_{a,b}E_{ab}^{\rm tot}E_{ba}^{\rm tot} =kq I+2\sum_{j<j'}U_Q((j\ j')).\] The right side is scalar on a Schur block. The lexicographically leading weight in its Weyl character is \(\lambda\); a vector of that weight is annihilated by \(E_{ab}^{\rm tot}\) for \(a<b\), since those operators would increase the weight. On this vector the diagonal terms contribute \(\sum_i\lambda_i^2\), and the two terms for \(a<b\) contribute \(\lambda_a-\lambda_b\), using \([E_{ab}^{\rm tot},E_{ba}^{\rm tot}]=E_{aa}^{\rm tot}-E_{bb}^{\rm tot}\). The scalar is therefore \(\sum_i\lambda_i(\lambda_i+q+1-2i)\). Subtract the formula for \(k-1\) copies with label \(\lambda-e_i\). The difference of the transposition sums is \(kJ_{Q,k}\), and its value is \(\lambda_i-i\). ◻

Proof. Begin in dimension \(d\). Write \(\Delta(s)=\prod_{i<j}(s_i-s_j)\) and choose \(s\) on the simplex \(s_i\ge0\), \(\sum_i s_i=1\), with density proportional to \(\Delta(s)\Delta(s^t)\). This is a nonnegative, integrable density, positive off the collision hyperplanes in the simplex interior. It is the simplex normalization of the Laguerre Muttalib–Borodin weight; the gamma integral below is the corresponding form of (Forrester and Ipsen 2018, sec. 2.3, equation (26)). We include the calculation before verifying the common-subsystem and ratio properties. For \(d=1\) use the point mass at \(s_1=1\). Choose an independent Haar-distributed unitary eigenbasis and let \(\sigma\) have eigenvalues \(s\). The integral of \((\sigma^t)^{\otimes k}\) commutes both with permutations and with the diagonal unitary group; hence it is central in the Schur decomposition. Its eigenvalue on \(\lambda\) is the mean of \(\chi_\lambda^{(d)}(s^t)/\dim V^{(d)}_\lambda\).

Put \(l_i=\lambda_i+d-i\). The Weyl denominator cancels \(\Delta(s^t)\) in the density. Expanding the two remaining determinants and using the simplex integral \[\int_{\substack{s_i\ge0\\\sum s_i=1}} \prod_i s_i^{u_i}\,ds =\frac{\prod_i\Gamma(1+u_i)}{\Gamma(d+\sum_i u_i)} \qquad(u_i>-1)\] gives, before dividing by the Weyl dimension and the density’s normalization, \[\frac{d!}{\Gamma(c+tk)} \det\bigl[\Gamma(1+t l_i+d-j)\bigr]_{i,j=1}^d.\] The simplex identity follows, for example, by changing variables \(x_i=rs_i\) in the product of the defining gamma integrals; the Jacobian contributes \(r^{d-1}\). Factor \(\Gamma(1+t l_i)\) from each row. The remaining entries are rising-factorial monic polynomials in \(t l_i\) of degrees \(d-j\), so their determinant is \(t^{d(d-1)/2}\Delta(l)\). The Weyl dimension cancels \(\Delta(l)\). Normalizing at \(k=0\) gives exactly (75).

For \(d_Q\le d\), choose an isometry \(E:\mathcal H_Q\to\mathbb C^d\). Compression of a permutation on \((\mathbb C^d)^{\otimes k}\) by \(E^{\otimes k}\) is the same permutation on \(Q^{\otimes k}\). Compressing its central label function therefore gives the identical function \(w_k(\lambda)\) on all labels that occur in dimension \(d_Q\). On each sample, write \(B=E^\dagger\sigma^t E\) and \(\tau=B^{1/t}\). The min–max principle bounds the decreasing eigenvalues of \(B\) by the corresponding eigenvalues of \(\sigma^t\). Consequently \(\mathop{\mathrm{Tr}}\tau\le\mathop{\mathrm{Tr}}\sigma=1\), while \(B>0\) almost surely. Thus the compressed integral has the form (77), with the distribution of \(\tau\) as \(\zeta_Q\). This proves the asserted representation without changing the common label function. Exact complement equality follows from Lemma 16.

The gamma estimate \(\log\Gamma(1+u)=u\log u-u+O(\log(u+2))\), with \(0\log0=0\), applied to the fixed number of factors in (75), gives \(\log w_k(\lambda)=-tkH(\lambda/k)+O_{d,t}(\log(k+2))\). The terms containing \(\log t\) cancel. Formula (68) proves (76).

For the marked ratio, now put \(l_i=\lambda_i+d-i\) using the full dimension. On a removable branch its exact eigenvalue is \[ r_{\lambda,i}= \frac{\Gamma(c+t(k-1))}{\Gamma(c+tk)} \frac{\Gamma(1+t l_i)}{\Gamma(1+t(l_i-1))}. \tag{80}\] Gamma ratios give \(r_{\lambda,i}\asymp_{d,t}((l_i+1)/k)^t\). When \(l_i/k\) is bounded away from zero, the same ratios give uniform convergence to \((l_i/k)^t\). When \(l_i/k\) is small, both \(r_{\lambda,i}\) and \(\max((l_i-d)/k,0)^t\) are uniformly small. First separate at \(l_i/k=\varepsilon\), then let \(k\to\infty\) and \(\varepsilon\downarrow0\). The star eigenvalue formula proves (78) and a uniform bound for \(\lVert R_{Q,k}\rVert\).

The \(Q\) marginal of a symmetric vector is permutation invariant. Its conditional branch probabilities are bounded by \(C_d l_i/k\): Lemma 16 first uses \(\lambda_i+d_Q-i\), which is no larger than \(l_i\). Hence, conditional on any \(\lambda\), \[\sum_i p_{\lambda,i}r_{\lambda,i}^{-2} \le C_{d,t}\sum_i(l_i/k)^{1-2t}\le C_{d,t}.\] This proves the inverse vector bound. On the spectral part \(J_{Q,k}\ge\delta\), the inverse approximation follows in operator norm from (78). On the remaining part, \(l_i/k\le\delta+d/k\). The true inverse’s squared norm there is at most \(C_{d,t}(\delta+d/k)^{1-2t}\); the clipped inverse’s squared norm is at most \(C_d(\delta+d/k)\delta^{-2t}\). Taking the limit superior and the square root proves (79). All bounds were conditional on an arbitrary label, so they are uniform in the original symmetric vector. ◻

Proof. For a positive definite \(C\) and unit vector \(z\), Cauchy–Schwarz gives \[|\langle z,u\rangle|^2 \le\langle z,C^{-1}z\rangle\langle u,Cu\rangle, \qquad C\ge\langle z,C^{-1}z\rangle^{-1}\lvert z\rangle\langle z\rvert.\] It therefore suffices to bound the inverse quadratic form of \(C_{x,k}\) at \(\theta^{\otimes k}\). Since \(C_{x,k}^{-1}=A_k^{1/2}B_k^{-1}A_k^{1/2}\), \[ \begin{split} \langle\theta^{\otimes k},C_{x,k}^{-1}\theta^{\otimes k}\rangle &=\lVert B_k^{-1/2}A_k^{1/2}\theta^{\otimes k}\rVert^2\\ &=\lVert W_{Px,k}W_{xY_0,k}W_{P,k}^{-1}W_{Y_0,k}^{-1} \theta^{\otimes k}\rVert^2. \end{split} \tag{85}\] To obtain the second line, cancel the disjoint \(F\) factors and commute the remaining nested or disjoint central factors. This is an identity of the displayed norms; no commutation of \(A_k\) with \(B_k\) is asserted.

Let \(\rho_R\) be the marginal of \(\theta\) for \(R=P,Y_0\), and set \[T_R=W_{R,k}^{-1}(\rho_R^t)^{\otimes k},\] with the positive power zero on the kernel. The invariant density \(\rho_R^{\otimes k}\) has every eigenvalue in its \(\lambda\) block at most \(d_\lambda^{-1}\) by eigenvalue repetition, as in the proof of (72). Thus (76) implies \[ \lVert T_R\rVert\le\mathop{\mathrm{poly}}(k), \tag{86}\] uniformly in \(\theta\), including marginals of deficient rank. Furthermore \(T_P\) commutes with both numerator factors in (85), and so does \(T_{Y_0}\). For the overlapping numerator this follows because \(W_{R,k}\) is central on the nested subsystem and \((\rho_R^t)^{\otimes k}\) commutes with every simultaneous permutation on \(xR\); the other numerator is disjoint from \(R\).

Extend \(\rho_R\) by eigenvalue one on its kernel and denote this extension by \(\widetilde\rho_R\). The product of the two marginal support projectors fixes \(\theta\), since \(P\) and \(Y_0\) are disjoint. We can therefore insert \(T_P,T_{Y_0}\) and move them to the left in (85), obtaining the upper bound \[\mathop{\mathrm{poly}}(k)\, \lVert W_{Px,k}W_{xY_0,k} (\widetilde\rho_P^{-t}\widetilde\rho_{Y_0}^{-t})^{\otimes k} \theta^{\otimes k}\rVert^2.\] Here and below the polynomial can be enlarged to absorb its square. Expand the two numerator integrals (77) in their stated order. The triangle inequality and the tensor-power norm identity bound this by \[ \mathop{\mathrm{poly}}(k)\sup_{\sigma,\tau} \lVert \sigma_{Px}^t\tau_{xY_0}^t \widetilde\rho_P^{-t}\widetilde\rho_{Y_0}^{-t}\theta\rVert^{2k}, \tag{87}\] where \(\sigma,\tau\) are positive definite subdensities on the indicated subsystems. Normalizing a subdensity multiplies its \(t\)-power by a scalar at least one, so it suffices to take density matrices. Since \(P\) is disjoint from \(xY_0\), its inverse power commutes with \(\tau_{xY_0}^t\). The one-copy word is therefore \[\sigma_{Px}^{a/2}\widetilde\rho_P^{-a/2} \tau_{xY_0}^{a/2}\widetilde\rho_{Y_0}^{-a/2}\theta,\] exactly the word in Lemma 13. That lemma bounds its log norm by \(-a\eta_\theta/2+C a^{5/4}\log^C(e\dim x)\). Substituting in (87), and then using the inverse Cauchy–Schwarz inequality at the start of the proof, gives (84). Every polynomial bound used in this argument was uniform in \(\theta\). ◻

Proof. We first replace the marked ratios by bounded polynomial words, then prove a product calculus for those words, and finally pass to absolute values. The inverse approximation must be applied on symmetric input vectors; no full-space bound for \(R_{Q,k}^{-1}\) is needed.

Proof. We first prove the assertions for one binary vertex. For an invertible \(S\), put \(\widehat A=S^*AS\) and \(U=A^{1/2}S\widehat A^{-1/2}\). Then \(U\) is unitary and \[\widehat A^{-1/2}S^*BS\widehat A^{-1/2} =U^*(A^{-1/2}BA^{-1/2})U.\] Functional calculus proves \((S^*AS)\#_p(S^*BS)=S^*(A\#_pB)S\). After congruence to \(A=I\), the identity \(A\#_pB=B\#_{1-p}A\) reduces to \(C^p=C^{1/2}(C^{-1})^{1-p}C^{1/2}\). For \(0<p<1\), the scalar formula \[ x^p=\frac{\sin(\pi p)}{\pi} \int_0^\infty \lambda^{p-1}x(x+\lambda)^{-1}\,d\lambda \tag{103}\] follows by substituting \(\lambda=xt\) in the beta integral. The inverse reverses the order of positive definite operators, so \(C(C+\lambda I)^{-1}=I-\lambda(C+\lambda I)^{-1}\) is increasing in \(C\). Thus \(C\mapsto C^p\) is operator monotone. This proves monotonicity in \(B\), and symmetry proves it in \(A\). The endpoint cases are immediate. Scalar homogeneity at a vertex is \((cA)\#_p(dB)=c^{1-p}d^p(A\#_pB)\).

For the vector inequality let \(r=\langle z,Az\rangle\) and \(y=A^{1/2}z/\sqrt r\). Scalar Jensen applied to the spectral measure of \(C=A^{-1/2}BA^{-1/2}\) in the unit vector \(y\) gives \[\langle z,(A\#_pB)z\rangle =r\langle y,C^py\rangle \leq r\langle y,Cy\rangle^p =\langle z,Az\rangle^{1-p}\langle z,Bz\rangle^p.\] Induction proves the congruence, monotonicity, homogeneity, and vector inequality for the full tree.

To prove (102), fix \(0<\varepsilon<1\). For every leaf, \[A_j-(1-\varepsilon)c_jP =(1-\varepsilon)(A_j-c_jP)+\varepsilon A_j \geq\varepsilon\lambda_{\min}(A_j)I.\] Choose a common \(\delta>0\) so small that \((1-\varepsilon)c_j\delta\leq\varepsilon\lambda_{\min}(A_j)\) for all leaves. Monotonicity and homogeneity applied to the strictly positive inputs \((1-\varepsilon)c_j(P+\delta I)\) give \[M\geq(1-\varepsilon)\left(\prod_jc_j^{w_j}\right)(P+\delta I) \geq(1-\varepsilon)\left(\prod_jc_j^{w_j}\right)P.\] Let \(\varepsilon\) decrease to zero. This argument needs no definition of the mean on singular inputs.

For the last assertion, cross-band commutation implies \[\left(\prod_gD_g\right)\#_p\left(\prod_gE_g\right) =\prod_g(D_g\#_pE_g).\] Indeed the inverse square root of the first product factors, its whitened second input is the product of the commuting positive operators \(D_g^{-1/2}E_gD_g^{-1/2}\), and the \(p\)th power factors. Induction over the tree completes the proof. ◻

Proof. Differentiating (103) gives \[ D(C^p)[H]=\frac{\sin(\pi p)}{\pi} \int_0^\infty\lambda^p(C+\lambda I)^{-1}H(C+\lambda I)^{-1}\,d\lambda. \tag{106}\] The integral converges in operator norm: its integrand is \(O(\lambda^p)\) at zero and \(O(\lambda^{p-2})\) at infinity. Each integrand is a completely positive sandwich map. Formula (106), together with the congruences in (99), proves complete positivity for the second child. Exchange symmetry proves it for the first child. Scalar homogeneity gives \(\partial_D M[D]=wM\), proving unitality. The chain rule, with the square-root normalizations cancelling at successive vertices, proves (105). Composition preserves both properties; the map along an empty path is the identity. In the derivative integral all sandwich factors are block diagonal when the inputs are block diagonal; their first blocks depend only on the first blocks of the inputs. The same is true after exchanging the children and composing the maps. ◻

Proof. For fixed real \(z\ne0\), integrate \(e^{iuz}m_s(u)\) around a rectangle between the lines \(\mathbb R\) and \(\mathbb R+i\). The vertical integrals tend to zero because the denominator grows exponentially with \(|\Re u|\). The two poles in the strip are \(i(1/2-s)\) and \(i(1/2+s)\), with residues of \(m_s\) equal to \(1/(2\pi i)\) and \(-1/(2\pi i)\), respectively. Since \(m_s(u+i)=m_s(u)\), the residue theorem gives \[(1-e^{-z})\int_{\mathbb R}e^{iuz}m_s(u)\,du =e^{-(1/2-s)z}-e^{-(1/2+s)z}.\] Division gives (108); dominated convergence gives its value at zero. Positivity follows from the displayed denominator.

The algebraic identities in (109) follow by factoring \(e^{\pm sz}-1\). The nearest poles of \(m_{s/2}\) have imaginary parts of absolute value \(1/2-s/2\), larger than \(s/2\). Shifting its Fourier contour by \(\pm is/2\) therefore crosses no pole and gives the densities in (110). To check the bound without losing a constant, write \(c=\cosh(2\pi u)\). At \(s=1/4\), \[m_s(u)=c^{-1},\qquad m_{s/2}(u\pm is/2)=\frac1{c+1\pm i\sinh(2\pi u)}.\] Consequently \[\frac{|m_{s/2}(u\pm is/2)|}{m_s(u)} =\sqrt{\frac{c}{2(c+1)}}\leq\frac1{\sqrt2}.\] ◻

Proof. We first identify the derivative states and prove the entropy estimate. We then construct an auxiliary block tree for each physical term and derive (120) by a matrix Schwarz inequality.

Proof. Deterministic geometry and nesting. A truncated support meeting \(S_0\) has radius \(r_0\): when its anchor is at distance \(d>2r_0\), the alternative radius \(\lfloor d/2\rfloor\) cannot reach \(S_0\). Every splitting support meets a compact part contained in \(S_0\), so this observation applies. If an \(r_0\)-ball meeting \(S_0\) also contained a site of \(\Lambda\setminus A\), a graph path inside the ball would contain a cut edge with an endpoint at ambient distance at most \(L+2r_0\) from \(T\). This contradicts (170).

At a charge the largest newly assigned oriented depth is at most \(j+D+r_0\); subsequent movement of the nominal front only improves this bound. Each side receives at most \(\lceil nm/2\rceil\) fills, so advances through at most \(m/2+O(1)\) deterministic rows. Its charged lead is \(D+O(r_0)=o(m)\). The two fronts in a window remain separated by order \(m\), as do the active regions of successive windows. This proves both that an opposite side cannot have consumed a scheduled slot and that a support cannot split two fronts. All nonblank slots strictly behind a nominal front have been assigned to that side.

A near set contains its completed deterministic prefix and is contained in the prefix ending at its maximal allowed lead. The difference has at most \(CnD\) sites. For \(UY\), take the complement of the assigned far suffix within the original shell prefix: it is contained in the deterministic prefix ending at the nominal far front and differs from it only in \(O(D)\) rows. These deterministic comparison prefixes are all prefixes of one fixed ordered list of positive-depth slots and are consequently nested, even across histories and bands.

A full depth prefix has \(O(n)\) boundary edges. For its outer boundary at positive depth, use the last included row, since depth changes by at most one along an edge. For the boundary at depth zero, use the first positive row. Partial rows cost at most \(4n\) more edges. There are no compact-to-opposite-color edges, by the clearance. Changing at most \(CnD\) sites then gives the claimed \(O(nD)\) bound for all status cuts and their differences. Split anchors themselves lie in \(O(D)\) rows around the nominal fronts, containing \(O(nD)\) anchors per band. Summing gives \(CKnD\).

For two bands \(g'<g''\), the entire near side together with the middle of the first band is contained in the near side of the second band, for every pair of histories. After adjoining \(C\), this says \(P_{g'}Y_{g'}\subset P_{g''}\). Mirror the central function on \(F_{g'}\) to its complement \(P_{g'}Y_{g'}\), using Lemma 16. The resulting functions from the first band commute with those of the second: they are disjoint from its \(Y,F\) parts and invariant under the joint permutations whose center acts on its \(P\) part. This proves the asserted cross-band commutation.

Dilution by the initial offsets. For a fixed anchor to split, its oriented depth must lie between \(j-O(r_0)\) and \(j+D+O(r_0)\). Both formulas in (171) are affine, with coefficient \(1\) or \(-1\), in the initial uniform offset. Thus only \(O(D)\) of the \(m\) offsets can permit this event, and only \(O(1)\) bands can reach the fixed anchor. This proves \(CD/m\) separately for the old and new leaf distributions, and hence also for their common-\(p\) mixture. It does not require conditional independence of the split event and the later charge choices.

Charge ancestry and availability. We finally bound the probability that charged sites get too far ahead of a nominal front. This will ensure that every old split lies in the charge list. Suppose a side has an assigned site of \(A\) leading its nominal front by more than \(D/2\). Trace the site’s first assignment backwards through the charge ball that assigned it, choosing a previously assigned same-side site in that ball, until an initial or deterministic assignment is reached. An ancestry with \(j'\) charges changes oriented depth by at most \(2r_0j'\). If its first charge is at time \(t_1\) and the history is tested at time \(t\), then \[2r_0j'>D/2+j(t)-j(t_1)-O(1).\] The fixed fill schedule gives \(j(t)-j(t_1)\ge (t-t_1)/(4n)-O(1)\). Substitution in the preceding inequality yields both \[j'\ge cD/r_0,\qquad t-t_1\le Cnr_0j'.\] Thus all ancestry times lie in this last interval of rounds. All sites in the ancestry are in \(A\), including for the far side, because the charging balls are wholly in \(A\).

For a fixed endpoint, test time, and length \(j'\), the time choices and spatial choices have at most \[\binom{\lceil Cnr_0j'\rceil}{j'}(Cr_0^4)^{j'}\] possibilities. The spatial bound permits both an anchor and a predecessor within \(O(r_0)\) graph distance at each step, and includes the bounded anchor multiplicity. Conditional on earlier choices, a prescribed hit has probability at most \((cnD)^{-1}\). The bound \(\binom{v}{j'}\le(ev/j')^{j'}\) therefore makes the total probability at most \[\mathop{\mathrm{poly}}(n)\sum_{j'\ge cD/r_0}(Cr_0^5/D)^{j'}.\] The endpoints being counted are charged sites in \(S_0\), of which there are \(O(n^2)\); the numbers of test times and bands are also polynomial in \(n\). Thus no far-domain volume enters this factor. Since \(r_0\) is polylogarithmic and \(D=n^{\kappa+o(1)}\), the bound is smaller than every fixed inverse power of \(n\).

On a good old history, a split anchor has oriented depth at least \(j-r_0\), because its ball meets an unassigned site, and at most \(j+D/2+r_0\), because it meets the side. For large \(n\) this interval lies in the sampling interval \([j-r_0,j+D]\). Choosing the relevant side and its labelled slot has probability \(1/(2M_n)\ge c/(nD)\), and the resulting move is exactly the \(Y\)-part of that designated support. ◻

Proof. The transfer from \(\Omega\) to \(\widetilde\Omega\) costs \(O(n^{-248})\) in any entropy in (172): the trace distance is \(O(n^{-500})\), and the compact systems have \(O(n^2)\) sites, so Lemma 3 applies. A partial last depth row adds at most \(n\log q\). Lemma 25 then gives the matched-prefix and mismatch bounds required by the comparators. It also gives the \(O(nD)\) status boundaries. A crossing support has Hilbert dimension at most \(q^{C(1+r_0)^2}\), and its anchor lies within distance \(r_0\) of a cut edge. Consequently the marginal-tail parameter of Lemma 5 is at most \(CnD(\log n)^C\). At the target itself it is at most \(Cn(\log n)^C\). The latter bound shows that the width \(w=n^{3/5}\) has failure mass at most \(n^{-100}\) for all sufficiently large \(n\). In fact its tail is bounded by \(C\exp[-c n^{1/10}/(\log n)^C]\).

At a charge round write \(h\) for an old joint history, of probability \(w_h\), and \(\sigma_{h,u}\) for its transported state in Proposition 23. Let \(\mu_\sigma\) denote the coherent probability measure of Lemma 19. The entropy derivative uses the old-history weights \(w_h\); the same transported states occur in the energy estimate with terminal weights \((1-p)w_h\). We therefore measure the entropy cost of the old splits by \(\mathcal Q(p)\), defined as \[ \sum_{h\ {\mathrm{good}}}w_h \int_{\mathbb R}m_s(u) \int\sum_{i\ {\mathrm{split\ in}}\ h} \eta_{i,h}(\theta)\,d\mu_{\sigma_{h,u}}(\theta)\,du. \tag{175}\] Each split term in this sum has a unique splitting band, and its partition \(P,Y,F\) is understood to be that band’s partition. If term \(i\) splits on the near side, its moved set is \(x=Y\cap\widetilde X_i\), the unassigned part of its designated \(r_0\)-ball, and \(\eta_{i,h}=S_\theta(x\mid P)+S_\theta(x\mid Y\setminus x)\); on the far side exchange \(P,F\). These quantities are nonnegative. They are also bounded by \(C(\log n)^C\), because the moved sites lie in an \(r_0\)-ball. The measure \(m_s(u)\,du\) has fixed finite mass \(2s\).

On every good old history each split term is sampled with probability at least \(c/(nD)\), and its sampled move has exactly the corresponding \(\eta_{i,h}\). Equation [transport:entropy-gain] therefore gives, at a charge round, \[ -\partial_p\log N^2 \ge \frac{cka}{nD}\mathcal Q(p) -CkaK a^{1/4}(\log n)^C-O_{\rm fixed}(\log(k+2)). \tag{176}\] At fill rounds retain just the same negative error; the entropy gains are nonnegative. Here and below “fixed” means that the complete single-copy instance and the scanner have been fixed. No such constant survives the replica limit.

Integrate over every full round, including fills. Adjacent endpoint metrics agree by construction, so the left sides telescope. Initially \(-\log N^2\ge-O_{\rm fixed}(\log(k+2))\): the floor (83) passes to the initial mean tree by Lemma 20, and \(\|\mathrm{pre}\|\le1\). At the last endpoint the rough upper comparison (144) gives \[-k^{-1}\log N^2 \le 2\log d_*-\log z+C_eW(nL^e+nD+n)+o_k(1) \le C_eW n^{1+e}+o_k(1).\] For the last inequality use \(\kappa\le e\), \(L\le n\), and \(\log d_*=S_{\widetilde\Omega}(X)+O(w+|\log z|)\). Restrict the positive charge integrals in (176) to \([\epsilon/2,\epsilon]\), but pay the negative errors on all \(2nm\) full rounds. Since \(aK=W\), the integrated inequality, normalized by the number of charge rounds, is \[\frac1{nm}\sum_{\text{charge rounds}} \int_{\epsilon/2}^{\epsilon}\frac{\mathcal Q(p)}{KnD}\,dp \le C_e\left(\frac{n^e}{m}+a^{1/4}(\log n)^C\right)+o_k(1).\] The interval has length \(\epsilon/2\), and \(m\asymp n^\mu\). Averaging therefore gives a charge round and a \(p_k\) in this interval such that \[ \frac{\mathcal Q(p_k)}{KnD}\le\delta_n+o_k(1). \tag{177}\] The fixed factor two in the interval length is absorbed into constants. All integrands are bounded at fixed \(k\), and continuous in the interior interpolation parameter, so ordinary averaging suffices for this selection.

Apply the energy estimate (120) at this same point. For each term its split weight is at most \(CD/m\). There are at most \(CKnD\) split term occurrences in each joint leaf. On old good leaves the transported states are exactly those in (175); their full terminal weight differs only by the factor \(1-p_k\le1\). Hölder’s inequality, applied to the positive measure over histories, terms, \(u\), and \(\theta\), bounds their sum of \(\eta^{1/8}\) by \[CKnD\bigl(\delta_n+o_k(1)\bigr)^{1/8}.\] On bad old histories use the uniform local dimension bound and Lemma 25 with an arbitrarily large fixed tail power. On all new leaves use the same dimension bound and their total terminal weight \(p_k\le\epsilon\). After increasing the fixed logarithmic power, the energy estimate is therefore \[\langle v_k,\overline{\widetilde H}v_k\rangle \le 2\widetilde E_0+ C a^2\frac Dm KnD(\log n)^C \bigl(\delta_n^{1/8}+\epsilon+n^{-100}\bigr)+o_k(1).\] The factors in this expression have distinct origins: \(D/m\) is the split probability of one fixed term, \(KnD\) counts split occurrences, and \(a^2\) is the local skew cost. Their product simplifies to \[a^2\frac Dm KnD =\frac{W^2nD^2}{mK} \le CW^2n^\ell D^2.\] Subtracting \(\widetilde E_0\) and dividing by \(\widetilde g\ge g/2\) proves (173).

Only splitting terms contribute to this energy error; all others have zero skew exactly. Thus its coefficient contains no total-volume factor. The symbol remainders can be summed over all terms before taking \(k\to\infty\), because that complete term set is finite and fixed. Their uniformity in replica states and in the interpolation parameter also justifies the possibly \(k\)-dependent selection in (177). ◻

Proof. Consider first one sampled piece. On an integer horizontal row its sites form an integer interval. Its left endpoint is a maximum of affine integer-valued functions with slopes in \(\{-1,0,1\}\), and its right endpoint is a minimum of such functions; horizontal sides impose an interval constraint on the row index. To obtain these formulas, round the intercepts of the defining line inequalities. Because the slopes are integers, this rounding is independent of the integer row index. The difference of the lower and upper bounds is convex, so the nonempty rows are consecutive. On consecutive nonempty rows each endpoint moves by at most one. Consecutive row intervals overlap or are adjacent as integer intervals.

After integer sup-norm dilation by \(j\), the interval on a row is the union of the original intervals in the window of rows at distance at most \(j\), each extended by \(j\) on both ends. This union is still an integer interval. Its endpoints are the sliding minimum and maximum of the old endpoints, minus or plus \(j\). They again move by at most one between consecutive nonempty rows: the nonempty index windows at two consecutive rows have Hausdorff distance at most one, and the old endpoint functions are 1-Lipschitz. One more dilation changes each endpoint on an old row by at most two and adds at most two rows, each of length \(O(s_0+j+1)\). This proves a bound \(C(s_0+j+1)\) for both the new layer and the edge boundary of one piece. Summing over the pieces proves (178), with the layer constant absorbed in \(C_{\rm tpl}\). The area bound follows from \(|P_a\cap\mathbb Z^2|\le C(s_0+1)^2\).

We also need a count at intermediate dyadic scales. In the usual dyadic partition of the lattice into integer squares, the number of side-\(u\) squares containing both a site of a dilated piece and a site outside it is \[ O\bigl((s_0+j+1)/u+1\bigr),\qquad u\ge1. \tag{180}\] Indeed, in a block of \(u\) consecutive nonempty rows the two endpoints vary by at most \(u\). Only constantly many square columns near each endpoint can be mixed. At the first and last blocks count instead the whole width, which costs \(O((s_0+j+1)/u+1)\) in total. A square mixed for a union, or for a difference of two unions, must be mixed for at least one of the constituent sets. Thus the corresponding count for \(T\), \(T_j\), or \(T_j\setminus T\), with \(j\le s_0\), is at most \(C(N_0+n/u)\).

Cover \(T_j\setminus T\) by maximal contained dyadic squares, with side capped at the largest power of two not exceeding \(L\). These squares partition the set: a single site is an admissible square, and the dyadic nesting makes distinct maximal squares disjoint. At the cap there are at most \(Cn/L\) squares because \(|T_j\setminus T|\le nj\le nL\). Below the cap, the parent of each maximal square is mixed. Equation (180) therefore bounds the number of squares of side \(u\) by \(C(N_0+n/u)\le Cn/u\), since \(u\le L\le s_0\) and \(N_0s_0\le n\). All these squares are safe: their sites lie in \(T_j\), their sides are at most \(s_0\), and \[\mathop{\mathrm{dist}}_\infty(T_j,Z)\ge\mathop{\mathrm{dist}}_\infty(T,Z)-j >4D_0s_0-s_0>D_0s_0.\] Subadditivity and the assumed box bound now give \[S_\Omega(Q_j) \le C_e n\sum_{\substack{u\le L\\u\ \mathrm{dyadic}}}u^e \le C'_e nL^e.\] Covering \(T\) in the same way, with cap comparable to \(s_0\), uses \(O(n/s_0)\) squares at the cap and \(O(n/u)\) below it, and gives the first bound in (179).

Adding an arbitrary part of one depth row changes entropy by at most \(n\log q\). Finally, near these sets there is no edge joining \(A\) to its complement: such an edge would have an endpoint in \(Z\) within distance \(s_0+1\) of \(T\). Thus their physical boundaries are bounded by the ambient template and shell boundaries just counted. This argument does not count missing lattice edges as physical boundary edges. ◻

Proof. Let \(e_0\in(0,1)\) be supplied by Proposition 7. If \(e_0\le e_*\), the assertion already follows. Otherwise fix, once for the entire bootstrap, \[ g_0=(1-e_0)/2, \quad \mu=1-g_0, \quad \nu=g_0/1000, \quad \ell=\nu/100, \quad \kappa=\min(\nu/100,e_*/4), \quad W=4. \tag{181}\] These parameters will not change when the current entropy exponent improves.

Suppose \(F_{\rm box}(r)\le C_e r^{1+e}\), where \(e_*<e\le e_0\). Take any safe integer rectangle \(T\) of actual maximal side length \(s_0\), and set \(n=\lceil C_2s_0\rceil\) for a sufficiently large fixed \(C_2\). If \(A\cap T\) is empty, its entropy is zero, so assume otherwise. For large \(s_0\), \(L=o(s_0)\), so (170) holds. In particular the weaker safe-box clearance \(D_0s_0\), rather than the stronger template clearance of Definition 27, is enough for this application.

Here is the shell entropy input explicitly. The shell \(T_j\setminus T\), \(j\le L\), has area \(O(nL)\) and consists of a bounded number of rectangular strips. Its mixed dyadic squares of side \(u\) number \(O(n/u+1)\), by the boundary-line count. The maximal-contained-square argument of Lemma 28, capped at side comparable to \(L\), therefore gives \(S_\Omega(Q_j)\le C_e nL^e\). These covering squares are safe: their sides are at most \(L\), and their distance from \(Z\) is at least \(D_0s_0-L>D_0L\) once \(s_0\) is large. This also covers thin rectangles. The target entropy is at most \(C_e s_0^{1+e}\). Adding a partial row costs \(O(n)\), so all the hypotheses of Proposition 26 hold.

We check its quantitative output. Since \(K\asymp n^{g_0-\ell}\), \(a\asymp n^{-g_0+\ell}\). In (174) the two powers before logarithms are at most \[e_0-\mu+\nu=-0.999g_0, \qquad -(g_0-\ell)/4+\nu=-0.2489975g_0.\] Thus \(\delta_n\le n^{-0.24g_0}\) for sufficiently large \(n\), after constants and fixed logarithmic powers have been absorbed. The energy prefactor has power \(\ell+2\kappa\le0.00003g_0\). Its slower decaying input is \(\epsilon=n^{-0.001g_0}\), rather than \(\delta_n^{1/8}\le n^{-0.03g_0}\). Hence \[ E_{{\rm def},k}\le Cn^{-\nu/2}+o_k(1). \tag{182}\] All scale conditions and the smallness of \(ar_0^2\) hold at large \(n\). No sharp marginal-moment condition has been used.

Put \(\tau=n^{-\nu/4}\). Apply the rough upper and lower comparisons (144) and (145) at the selected interpolation. Write \(S_X=S_{\widetilde\Omega}(X)\). The upper pin cost has leading coefficient \(2S_X\); the lower coefficient is \(4sWS_X=4S_X\). Replacing \(\log d_*\) and \(S'\) by \(S_X\) costs \(O(w+|\log z|)\). The remaining relative loss in this coefficient is at most \(C(\tau+E_{{\rm def},k}/\tau)\), which is small after taking \(k\to\infty\) and then \(n\) sufficiently large. The two comparisons consequently imply \[ S_X\le C_e\bigl(nL^e+nD+w+h(\tau)/a+1\bigr). \tag{183}\] To justify the order of limits in this absorption, keep \(n\) fixed in both norm comparisons and use (182). The resulting scalar inequality has the same \(S_X\) for every \(k\); all other remainders are uniform at the selected points and vanish. Only then choose the large-scale threshold making the coefficient loss smaller than the fixed margin between four and two.

For \(0<\tau<1/2\), \(h(\tau)\le C\tau\log(e/\tau)\). Therefore \[h(\tau)/a \le C n^{g_0-\ell-\nu/4}\log n=O(n),\] because \(g_0<1/2\). Also \(w=n^{3/5}\le n\), and \(\kappa\le e_*/4<(1-\ell)e\) while \(e>e_*\). Equation (183), followed by the negligible transfer back to \(\Omega\), gives \[S_\Omega(A\cap T)\le C'_e n^{1+(1-\ell)e} \le C''_e s_0^{1+(1-\ell)e}.\] Bounded \(s_0\) are included by increasing the constant, using the volume bound. Taking the supremum over safe boxes thus replaces the current exponent \(e\) by \((1-\ell)e\).

Starting from \(e_0\), finitely many repetitions reach an exponent at most \(e_*\), which can be loosened to \(e_*\). Every iteration uses only the current safe-box estimate on the shell covering squares. The parameters in (181), and the finite number of iterations, depend only on the fixed Hamiltonian parameters. Thus the final constant is uniform over the domain and cut. ◻

Proof. If \(X\) is empty there is nothing to prove. Otherwise Proposition 29 and Lemma 28 supply (172) with \(e=e_*=2\cdot10^{-6}\). Since \(s_0\le n\), the core entropy is at most \(Cn^{1+e}\). The template clearance and \(L\le s_0\) imply (170) at large \(n\); the logarithmic truncation radius is negligible compared with \(L\).

Use the following fixed parameter choices: \[ \mu=\frac14,\quad \nu=\frac1{1000},\quad \ell=\frac1{100000},\quad \omega=\frac1{20000},\quad \kappa=\frac1{1000000},\quad W=n^\omega. \tag{185}\] Then \(a\asymp n^{-0.74994}\). The two terms in (174) have powers \(-0.248998\) and \(-0.186485\), before the fixed logarithmic factors. Hence \(\delta_n\le n^{-0.18}\) at sufficiently large scales. The power of the energy prefactor is \[\ell+2\kappa+2\omega=0.000112.\] The \(\epsilon=n^{-0.001}\) term is again the limiting one, and Proposition 26 gives \[ E_{{\rm def},k}\le Cn^{-0.0005}+o_k(1). \tag{186}\] Set \(\tau=n^{-0.00025}\). All fixed multiples of \(a\) lie in the marginal-moment interval required for the sharp comparisons, since \[a\sqrt{\mathcal B} \le C n^{-0.2499395}(\log n)^C\longrightarrow0.\]

At each leaf write, as before, the physical partition as \(X,U,Y,V\). All entropies in the next calculation refer to \(\widetilde\Omega\). Let \[g_{\rm pre}=S(XU)+S(V)-S(Y),\qquad g_{\rm post}=S(U)+S(UY)-S(Y).\] These are the quantities in the sharp norm comparisons. Their upper bounds are uniform in the status. In particular \(S(XU)\le S(X)+S(U)\le Cn^{1+e}\), and by purity \(S(V)=S(XUY)\le S(X)+S(UY)\le Cn^{1+e}\). Here the matched-prefix error \(CnD\) is absorbed because \(\kappa<e\). Thus \(0\le g_{\rm pre}\le Cn^{1+e}\).

Denote the full terminal path and uniform band average by \(\langle\cdot\rangle_{\rm leaf}\). Subtracting (146) from (147), and then dividing by \(2sW\), bounds \(2S(X)+\langle g_{\rm pre}-g_{\rm post}\rangle_{\rm leaf}\) by a constant times \[ \frac{S(X)+1}{W}+w +(\tau+E_{{\rm def},k}/\tau)n^{1+e} +\frac{h(\tau)}a+a\mathcal B +\frac{|\log z|}a+o_k(1). \tag{187}\] This calculation uses \(S'=S(X)+O(w+|\log z|)\) and the analogous estimate for \(\log d_*\). It also restores the lost \(\tau\langle g_{\rm pre}\rangle_{\rm leaf}\) from the lower comparison using the preceding uniform bound.

For each leaf separately, purity gives the exact identity \[ 2S(X)+g_{\rm pre}-g_{\rm post} =I(X:V)+I(X:YV). \tag{188}\] Both \(V\) and \(YV\) contain the fixed subsystem \(\Lambda\setminus S_0\). Monotonicity of mutual information therefore makes the left side of (188) at least \(I_{\widetilde\Omega}(X:\Lambda\setminus S_0)\), independently of the history. This remains true after the leaf average, even though the selected round and its weights can vary with \(k\).

For completeness, after \(k\to\infty\) the terms in (187) have the following upper bounds on their powers of \(n\), apart from fixed logarithms.

All are strictly below \(1-\varepsilon=0.99999\). Equation (186) controls the mass-loss term uniformly before the limit. This proves \(I_{\widetilde\Omega}(X:\Lambda\setminus S_0) \le Cn^{1-\varepsilon}\).

Finally write mutual information as \(S(X)-S(X\mid\Lambda\setminus S_0)\) and use Lemma 3. Only \(\log\dim\mathcal H_X=O(n^2)\) enters, even if the far subsystem is very large. The trace distance \(O(n^{-500})\) between the original and truncated ground states therefore changes this mutual information by \(O(n^{-248})\). This proves (184). The final size and boundary assertions were established in Lemma 28. ◻

Proof. We first prove an operator estimate for unital channels. Define \[\mathcal E_i(B)=G_iBG_i+K_iBK_i, \qquad \mathcal E_{i,l}(B)=G_{i,l}BG_{i,l}+K_{i,l}BK_{i,l},\] where \(K_{i,l}=(E_{i,l}k_i)^{1/2}\). When \(k_i\Omega=0\), functional calculus gives \(G_i\Omega=\Omega\) and \(K_i\Omega=0\), so \[ \mathcal E_i(C)(\Omega\otimes\xi)=G_iC(\Omega\otimes\xi) \tag{193}\] for every operator \(C\) and spectator vector \(\xi\). Products of these channels therefore recover the contraction products when evaluated on the ground vector. We localize the channels first: their unitality makes each local channel fix the algebra outside its supporting ball. They are also completely positive, hence contractions in operator norm, including after adjoining spectators. Lemma 11 gives \[ \|\mathcal E_i-\mathcal E_{i,l}\|_{\rm cb} \le C e^{-c l^\alpha}. \tag{194}\] The completely bounded norm here is the supremum of the operator norms after adjoining identity maps on finite spectator systems. In this case the estimate follows directly from the norm bounds for the two square roots, so it has no dimension factor.

Put \(\mathcal E_{i,-1}=\mathop{\mathrm{id}}\) and \(\Delta_{i,l}=\mathcal E_{i,l}-\mathcal E_{i,l-1}\) for \(l\ge0\). By increasing \(C\) to include \(l=0\), \[ \|\Delta_{i,l}\|_{\rm cb}\le C e^{-c l^\alpha}. \tag{195}\] Both channels in this difference fix operators commuting with the algebra on \(N_l(a_i)\). Thus the difference vanishes on that commutant. For an operator \(B\) define its oscillation at a physical site \(y\) by \[d_y(B)=\sup_{U_y\ {\rm unitary}}\|[U_y,B]\|.\] Average over the on-site unitaries in a set \(Q\), one site at a time. The total norm change is at most \(\sum_{z\in Q}d_z(B)\), and the result commutes with the algebra on \(Q\). Consequently \[ \|\Delta_{i,l}(B)\| \le C e^{-c l^\alpha}\sum_{z\in N_l(a_i)}d_z(B). \tag{196}\] This argument also applies to operators acting on spectators.

To bound the oscillation after one event, suppose first that \(m=d_\Lambda(a_i,y)<\infty\). The channel \(\mathcal E_{i,m-1}\) does not act at \(y\) and commutes with conjugation by \(U_y\); hence it cannot increase \(d_y\). Telescope the shells from radius \(m\) onward and use \(d_y(C)\le2\|C\|\) to obtain \[ d_y(\mathcal E_i B)\le d_y(B)+ C\sum_{l\ge d_\Lambda(a_i,y)}e^{-c l^\alpha} \sum_{z\in N_l(a_i)}d_z(B). \tag{197}\] For \(m=0\) the initial channel is the identity. If \(m=\infty\), \(\mathcal E_i\) acts in a different component and cannot increase \(d_y\). Exact component support follows either from the construction or by letting \(l\) exceed the component diameter in the approximation assumption.

Sum the increment in (197) over the retained anchor labels. Define the nonnegative matrix \[A_{yz}=C\sum_{i\ {\rm retained}}\sum_{l\ge0} e^{-cl^\alpha}\mathbf1_{\{y,z\in N_l(a_i)\}},\] with the constant from (197). It has a uniformly bounded weighted row sum: \[ \sup_y\sum_{z:\,d_\Lambda(y,z)<\infty} A_{yz}e^{a d_\Lambda(y,z)^\alpha}\le v \tag{198}\] for some \(a>0\). Indeed, at radius \(l\) there are at most \(C(1+l)^2\) anchor labels whose balls contain \(y\), and at most \(C(1+l)^2\) sites in each ball. Their distances from \(y\) are at most \(2l\). The left side is therefore bounded by \[C\sum_{l\ge0}(1+l)^4e^{-c l^\alpha+a(2l)^\alpha},\] which is finite for sufficiently small \(a\). There are no kernel entries between different components.

Let \(B_u\) be the result of the retained channel events up to time \(u\), starting from \(B\). The Poisson jump formula and (197) imply, in integral form, the differential upper bound \[\frac{d}{du}\mathbb E d_y(B_u) \le\sum_z A_{yz}\mathbb E d_z(B_u).\] The site and anchor sums are finite for a fixed domain, and the radius sums converge absolutely with the uniform bounds in (198). Initially \(d_y(B)\le2\|B\|1_S(y)\). On components meeting \(S\), weight the last inequality by \(e^{a d_\Lambda(y,S)^\alpha}\). The triangle inequality and \(0<\alpha<1\) give \[d_\Lambda(y,S)^\alpha-d_\Lambda(z,S)^\alpha \le d_\Lambda(y,z)^\alpha.\] The weighted supremum thus grows at rate at most \(v\). Gronwall’s inequality yields \[ \mathbb E d_y(B_u) \le2\|B\|e^{vu-a d_\Lambda(y,S)^\alpha}. \tag{199}\] Oscillations on components disjoint from \(S\) remain zero.

We can now remove the distant events. For an omitted anchor at finite distance \(d=d_\Lambda(a_i,S)>r\), sum (196) over all \(l\ge0\) and apply (199). When \(l\le d/2\), every \(z\in N_l(a_i)\) has \(d_\Lambda(z,S)\ge d/2\). The sum over these radii is bounded by \(C\|B\|e^{vu-a(d/2)^\alpha}\). For \(l>d/2\) use the shell decay and the ball cardinality instead. Absorbing its polynomial factor gives, in both cases, \[ \mathbb E\|(\mathcal E_i-\mathop{\mathrm{id}})(B_u)\| \le C\|B\|e^{vu-c'd^\alpha}. \tag{200}\] At infinite distance the left side is zero.

Couple full and retained processes by using the same clocks. The difference of their products telescopes with earlier retained channels and later full channels. All the later channels contract the operator norm. The earlier retained process is independent of the omitted clocks, so their expected contributions are obtained by integrating the rate-one bound (200). There are at most \(C|S|(d+1)^2\) anchor labels at distance \(d\) from \(S\). Summing the stretched-exponential tail and integrating time therefore prove the coupled expected-norm estimate \[ \mathbb E\|\mathcal E_{\rm full}(B)-\mathcal E_{\rm ret}(B)\| \le Ct|S|\|B\|e^{vt-c''r^\alpha}. \tag{201}\] In particular this is a bound on the expectation of the norm, not merely on the norm of a difference of averaged operators.

If \(k_i\Omega=0\), evaluating both channel products on \(\Omega\otimes\xi\) and iterating (193) converts (201) into (192). Unitality is essential in the channel argument: the single map \(C\mapsto G_iCG_i\) need not fix operators on a disjoint system.

Finally there are at most \(C|S|(1+r)^2\) retained labels, and hence their expected number of events is at most \(Ct|S|(1+r)^2\). Each event’s contraction changes by at most \(Ce^{-c r^\alpha}\) under radius-\(r\) approximation. Telescoping the contraction products proves (191). A retained anchor is within \(r\) of \(S\), so every approximating factor is supported on \(N_{2r}(S)\). This completes the proof. ◻

Proof. The case \(X=\varnothing\) is immediate; take \(r=1\). Suppose that \(X\) is nonempty. We keep the entire physical system fixed while constructing the auxiliary operators; the estimates below are uniform in its volume.

Set \(w=n^{3/5}\). Lemma 5, applied in the original ground state to the three cuts, provides spectral projections \(P_X,T_Y,Q_{S_0}\) onto their positive eigenvalues with surprisal in the intervals \[[S(X)-w,S(X)+w],\quad [S(Y)-w,S(Y)+w],\quad [S(S_0)-w,S(S_0)+w],\] respectively. The original interactions have uniformly bounded support dimensions and at most \(Cn\) terms cross each cut. Thus each failed projection has probability at most \(C\exp(-cn^{1/10})\). For any fixed integer \(K\ge1\), increasing the scale threshold makes these three probabilities at most \[ \delta=n^{-2K}. \tag{207}\] For a trivial subsystem its projection is the identity. All inverse eigenvalues used below are restricted to the selected positive spectrum, so no invertibility assumption is imposed on a marginal.

Choose a full eigenbasis \(\{|x\rangle\}\) for \(\rho_X\), with eigenvalues \(p_x\ge0\), and write a Schmidt expansion \[\Omega=\sum_{y:\,p_y>0}\sqrt{p_y}\, |y\rangle_X|\omega_y\rangle_{YZ'}.\] Append two systems \(s',e'\), each a copy of the Hilbert space on \(X\), with fixed unit blank vectors denoted \(|0\rangle\). Consider the unit vectors \[\begin{align*} \phi&=|0\rangle_X|0\rangle_{s'} \sum_{y:\,p_y>0}\sqrt{p_y}|y\rangle_{e'} |\omega_y\rangle_{YZ'}, \tag{208}\\ \eta&=\Omega\otimes \sum_{x:\,p_x>0}\sqrt{p_x}|x\rangle_{s'}|x\rangle_{e'}. \tag{209}\end{align*}\] In \(\phi\), the original target has moved to \(e'\), so its joint reduction with a remote physical system is unchanged after relabeling. In \(\eta\), \(e'\) is purified by \(s'\) and is independent of all physical systems. The unitary we construct will leave \(e'\) untouched and act only on \(s'\) and a graph neighborhood of \(S_0\).

For readability write \(Q=Q_{S_0}\) and \(T_0=T_Y\). On \(s',X,Y\) define the square operator \[ R_0=\sum_{x\ {\rm selected}}p_x^{-1/2} |x\rangle\langle0|_{s'}\otimes Q\bigl(|x\rangle\langle0|_X\otimes T_0\bigr). \tag{210}\] It acts trivially on \(e'\) and \(Z'\). The localization lemma applies to vectors of the form \(B(\Omega\otimes\xi)\) with a local operator \(B\). To put \(R_0\phi\) in this form, define an operator supported on \(S_0,s',e'\) by \[ B_0=\sum_{\substack{x\ {\rm selected}\\y\ {\rm in\ the\ full\ basis}}} p_x^{-1/2}|x\rangle\langle0|_{s'}\otimes |y\rangle\langle0|_{e'}\otimes Q\bigl(|x\rangle\langle y|_X\otimes T_0\bigr). \tag{211}\] The summation over \(y\) includes zero-probability eigenvectors. Direct substitution gives \[ R_0\phi=B_0(\Omega\otimes|00\rangle_{s'e'}). \tag{212}\]

We establish three estimates for these operators before introducing the constraint product. The first is a norm bound controlled by \(\mathcal I\). Let \[B_Y=\sum_{x\ {\rm selected}}p_x^{-1} T_0\langle x|Q|x\rangle T_0\ge0.\] Orthogonality of the output vectors in (210) gives \(R_0^\dagger R_0=|0\rangle\langle0|_{s'}\otimes |0\rangle\langle0|_X\otimes B_Y\). Similarly, \(B_0^\dagger B_0=|00\rangle\langle00|_{s'e'}\otimes I_X\otimes B_Y\). The typical spectral inequalities are \[p_x^{-1}\le e^{S(X)+w},\qquad Q\le e^{S(S_0)+w}\rho_{S_0},\qquad T_0\rho_Y T_0\le e^{-S(Y)+w}T_0.\] They imply \[ \|R_0\|,\ \|B_0\|\le M_0,\qquad M_0=\exp\bigl((\mathcal I+3w)/2\bigr), \tag{213}\] because purity gives \(S(X)+S(S_0)-S(Y)=I(X:Z')=\mathcal I\).

The second estimate is an adjoint identity with no factor \(M_0\). Let \(\mathcal R_X\) be the isometry that relabels \(X\) as \(e'\) and inserts the blank vectors on \(X,s'\); thus \(\mathcal R_X\Omega=\phi\). For every physical operator \(L'\) the coefficients \(\sqrt{p_x}\) in (209) cancel the selected inverse square roots, giving exactly \[ R_0^\dagger (L')^\dagger\eta =\mathcal R_X(P_X\otimes T_0)Q(L')^\dagger\Omega. \tag{214}\] The three projections are contractions, whether or not they commute. Telescoping them on \(\Omega\) and using (207) therefore gives \[ \|R_0^\dagger(L')^\dagger\eta-\phi\| \le3\sqrt\delta+\|(L')^\dagger\Omega-\Omega\|. \tag{215}\]

Third, although \(R_0\phi\) may have large norm, its physical ground component has norm at most one. Define \[M_X=\mathop{\mathrm{Tr}}_Y\bigl[(I_X\otimes T_0)\rho_{S_0}Q\bigr].\] Since \(Q\) commutes with \(\rho_{S_0}\), one has \(0\le\rho_{S_0}Q\le\rho_{S_0}\). Cyclicity of the partial trace for operators on \(Y\) allows a second \(T_0\) on the right. Hence \[0\le M_X\le\mathop{\mathrm{Tr}}_Y[(I\otimes T_0)\rho_{S_0}(I\otimes T_0)] \le\rho_X.\] The last inequality follows by adding the analogous positive term with \(I-T_0\); the two cross terms have zero partial trace. In particular \(M_X\) vanishes on \(\ker\rho_X\). On its support put \(A_X=\rho_X^{-1/2}M_X\rho_X^{-1/2}\), so \(0\le A_X\le I\). With \(\rho_X^+\) denoting the inverse on the support and zero on the kernel, \[M_X\rho_X^+M_X =\rho_X^{1/2}A_X^2\rho_X^{1/2}\le M_X.\] The ancillary coefficient of \(|x\rangle_{s'}|y\rangle_{e'}\) in \(P_\Omega R_0\phi\) is \((M_X)_{yx}/\sqrt{p_x}\) for selected \(x\). It follows that \[ \|P_\Omega R_0\phi\|^2 =\sum_{x\ {\rm selected},y}\frac{|(M_X)_{yx}|^2}{p_x} \le\mathop{\mathrm{Tr}}(M_X\rho_X^+M_X)\le\mathop{\mathrm{Tr}}M_X\le1. \tag{216}\]

We have now constructed a restoring operator whose norm is \(e^{O(\mathcal I+w)}\), whose adjoint is accurate on the desired state, and whose ground component is bounded. Positive constraints will suppress the remaining component while preserving the last two properties.

Run the clocks of Lemma 31 for time \[ t_*=C(\mathcal I+w+K\log n),\qquad r=\left\lceil\bigl(C'(t_*+K\log n)\bigr)^{1/\alpha}\right\rceil, \tag{217}\] where \(C\) and then \(C'\) are chosen sufficiently large. Use \(S=S_0\), and put \(\zeta=R_0\phi\). The full, retained, and truncated products are denoted \(D,D_{\rm ret},L'\) respectively. Equations (202) and (213) imply \[\mathbb E\|D(I-P_\Omega)\zeta\|^2 \le M_0^2e^{-gt_*}.\] Apply Lemma 31 with \(B=B_0\), \(\xi=|00\rangle_{s'e'}\), and \(S=S_0\). The column representation (212) then gives \[\begin{align*} \mathbb E\|(D-D_{\rm ret})\zeta\| &\le Ct_*|S_0|M_0e^{vt_*-cr^\alpha},\\ M_0\mathbb E\|D_{\rm ret}-L'\| &\le Ct_*|S_0|(1+r)^2M_0e^{-cr^\alpha}. \end{align*}\] All of these estimates are on the same clock space.

Here is explicit control of the scale choices. First choose \(C\) so that \(gt_*\ge\mathcal I+3w+6K\log n\). The first expectation is then at most \(n^{-6K}\). By (203), both \(t_*\) and \(r\) are bounded by fixed powers of \(n\), with constants depending on \(K\). Their polynomial prefactors and \(|S_0|\) therefore cost only \(O_K(\log n)\) in the logarithm. Since \(\log M_0=(\mathcal I+3w)/2\le Ct_*\), choosing \(C'\) larger makes each of the last two expectations at most \(n^{-3K}\) for large \(n\). Markov’s inequality and a union bound consequently provide a realization satisfying \[ \|D(I-P_\Omega)\zeta\|\le n^{-K},\quad \|(D-D_{\rm ret})\zeta\|\le n^{-K},\quad M_0\|D_{\rm ret}-L'\|\le n^{-K}. \tag{218}\] Indeed the three failure probabilities sum to at most \(n^{-4K}+2n^{-2K}<1\) at sufficiently large scales.

Every exact factor fixes the ground vector in both directions. Thus \(D\) preserves the ground component in (216), and \(D_{\rm ret}^\dagger\Omega=\Omega\). Since \(M_0\ge1\), (218) gives \[ \|L'R_0\phi\|\le1+3n^{-K},\qquad \|(L')^\dagger\Omega-\Omega\|\le n^{-K}. \tag{219}\] Combining the latter bound with (215), and writing \(R'=L'R_0\), we obtain \[\|(R')^\dagger\eta-\phi\|\le4n^{-K}.\] Because \(\phi,\eta\) are unit vectors, the real overlap satisfies \(\Re\langle\eta,R'\phi\rangle\ge1-4n^{-K}\). Expanding the square using (219) gives \[ \|R'\phi-\eta\|\le Cn^{-K/2}. \tag{220}\]

The square operator \(R'\) is supported on \(s'\) and \(N_{2r}(S_0)\), and acts trivially on \(e'\). Its polar decomposition extends on this finite local space to \(R'=U'H_+\), where \(U'\) is unitary and \(H_+=(R'^\dagger R')^{1/2}\). Such a unitary extension exists even when \(R'\) is singular: its two polar support spaces have equal dimension, as do their orthogonal complements. Tensor it with the identity elsewhere. For \(\xi=(U')^\dagger\eta\) we have \[\|H_+\xi-\phi\|\le4n^{-K},\qquad \|H_+\phi-\xi\|\le Cn^{-K/2}.\] Subtract these two errors in the identity \[(I+H_+)(\xi-\phi) =(H_+\xi-\phi)-(H_+\phi-\xi).\] Since \(\|(I+H_+)^{-1}\|\le1\), this proves \[ \|U'\phi-\eta\|\le Cn^{-K/2}. \tag{221}\] In particular no bound on \(\|H_+\|\) is needed.

Let \(J\subseteq\Lambda\setminus N_{2r}(S_0)\) be arbitrary. The unitary \(U'\) changes neither \(e'\) nor \(J\), so their joint reduction in \(U'\phi\) is exactly the original \(XJ\) reduction, with \(X\) relabeled as \(e'\). In \(\eta\) that joint reduction is a product. Apply Lemma 3 to \(I(e':J)=S(e')-S(e'|J)\) and (221). Only the dimension of \(e'\) appears. It has \(\log\dim e'=|X|\log q\le Cn^2\), and therefore \[I_\Omega(X:J)\le Cn^{2-K/4}.\] Choose the fixed integer \(K\ge4(M+3)\). This proves (205), uniformly for every such \(J\). Discarding part of \(X\) proves the assertion for its subsets.

Finally (217) and (203) give \(r\le C_M n^{(1-\varepsilon)/\alpha}\). The choices in (189) imply \((1-\varepsilon)/\alpha<\beta\) and \(1-\varepsilon<\beta\), which prove (204). If a graph path of length at most \(2r\) from \(S_0\) reached \(\Lambda\setminus A\), it would contain an endpoint of the original cut. A starting point of this path has ambient distance at most \(L\) from \(T\), and each graph step changes ambient distance by at most one. That endpoint would be within \(L+2r\) of \(T\), contradicting (206) for large \(n\). Thus the graph neighborhood lies in \(A\), as claimed. ◻

Proof. Write \(A_f=\bigcup_{i\in\mathcal I_f}X_i\) and \(E_f=\sum_{i\in\mathcal I_f}\varepsilon_i\). The entropy chain rule and purity give \[S_\psi(BA_f) \ge S_\psi(B)+\sum_{i\in\mathcal I_f}S_\psi(X_i)-E_f =S_\psi(A)+\sum_{i\in\mathcal I_f}S_\psi(X_i)-E_f.\] On the other hand, purity and subadditivity give \[S_\psi(BA_f)=S_\psi(DA_{1-f}) \le S_\psi(D)+\sum_{i\in\mathcal I_{1-f}}S_\psi(X_i).\] Adding the inequalities for \(f=0\) and \(f=1\) cancels every individual tile entropy and proves the claim. ◻

Proof. Consider a belt cell of side \(t=t_k\). In its \(10t\)-neighborhood only its own and adjacent layers can occur, by (227) and \(t_k/2^k\to0\). All fine-cell sides there are \(t/2,t\), or \(2t\). Cell corners, fan centers, and side midpoints therefore lie on the affine mesh \(o+(t/4)\mathbb Z^2\). All active lines have one of the four allowed slopes and intercepts on this mesh. Marks have local scales comparable to \(t\). Distinct marks have distance at least \(t/4\), and a supporting line that does not contain a given mark is at sup distance at least \(t/8\) from it. The same bounds apply to the nearest other endpoint of an incident segment. Choosing \(a_0\) small gives both assertions locally. Apply this argument at the larger scale for a pair of nearby marks; pairs outside the local neighborhood already satisfy the asserted separation.

The preceding coloring shows that two sectors separated by an active ray have opposite labels. Coincident boundary segments or inactive seams are counted only once, so this describes the actual tile regions near the mark. ◻

Proof. Suppose the assertions hold at one generation. By (231), the squares of radius \(6d(S)\) are contained in their parent stars. They are also disjoint from all other current removal squares. Indeed for parent marks \((v,S),(w,T)\), \[\lVert v-w\rVert_\infty\ge a_0\max(S,T),\qquad 6d(S)+d(T)\le \frac{7a_0}{1000}\max(S,T).\] Thus each modification sees only its parent’s radial boundaries.

Across a new perimeter segment away from ray hits, the new interior label was chosen opposite to the exterior label. Across an interior ray, the two labels remain opposite because the two exterior labels were opposite. All other positive-length interfaces are unchanged. Contacts not lying in the relative interior of one of these segments are among the center, corners, or side midpoints of the new square. This proves the coloring and contact assertions.

Normalize one removal square to \([-1,1]^2\). Its post-replacement boundaries are subsets of the original eight radial rays together with the four lines supporting this square. The possible new marks are the nine points of \(\{-1,0,1\}^2\). Every nonincident supporting line stays at a fixed positive distance from a mark, and every other endpoint stays at distance at least one. Thus a fixed sufficiently small \(a_0\) gives an isolated radial star at each child, on the child’s scale \(d(S)\). This is the same finite line configuration at every generation, so no iteration of a shrinking constant is involved.

Two children of the same parent are at least \(d(S)\) apart. For children of different parents their separation is at least \[a_0\max(S,T)-d(S)-d(T) \ge \frac{a_0}{2}\max(S,T),\] which is larger than \(a_0\max(d(S),d(T))\) by (231). Their isolated-star squares therefore satisfy the original separation invariant as well.

The child stars lie inside the parent star. Consequently they avoid all terminal squares from earlier generations. A terminal square created at another parent in the current generation is excluded by the same cross-parent distance estimate. For a direct statement covering all later generations, each successive child displacement is at most the preceding removal radius, and subsequent radii shrink by the fixed factor at most \(a_0/1000\). Their sum, including the radius of the final descendant birth square, is less than \(2d(S)\). Subtrees of distinct parents are therefore separated by the preceding inequalities, even if one subtree has already terminated. This proves the asserted terminal-square avoidance and descendant bound.

Finally, as long as a branch continues, its successive scales satisfy \(S_{j+1}=d(S_j)\le S_j^{\rho_0}\) with \(\rho_0<1\). After finitely many steps the computed next halfside is below \(S_*\), and that removal is terminal. This completes the induction. ◻

Proof. At the birth of an initial belt tile, compare its closure with the primary and belt closures at that time. At the birth of a replacement, compare it with the old remnants after that generation’s removal squares have been cut out, and with the peer replacements. Each final actual region is a subset of the corresponding compared region. In particular an ancestor is compared after its square has been removed; its original birth region need not be disjoint from the new one.

Restrict attention to a closed box obtained by enlarging the tile’s cell or removal square by \(2u\) in each direction. Points outside this box are at distance at least \(2u\) from the birth region. Translate by its cell corner, or by its removal center, and rescale lengths by \(u\). Only finitely many local pairs of closed regions can arise:

• For belts, the nearby active lines and endpoints lie on the affine quarter mesh, with their four prescribed slopes. In a bounded box there are only finitely many such lines and endpoints. The local closures are unions of faces of their arrangement; allowing all possible identifier groupings still leaves only finitely many pairs of unions. We classify only the restrictions to this box: disconnected pieces of an identifier outside it do not change the local pattern.

• For a replacement, the enlarged box lies in the isolated star of its parent, by (231). It sees only the old radial layout and the new square and sectors. These lie on a fixed finite arrangement in the rescaled coordinates. Other parents’ simultaneous modifications do not enter this box.

Thus there is a common positive scale for the isolation neighborhoods used in all these comparisons.

The two compared closures of the same label have no positive-length interface. Their intersections, if any, are contact marks by the belt construction or Lemma 36. At a belt contact the mark scale is comparable with \(u\) (neighboring belt scales differ by at most two); at a replacement contact it is \(u\). Choose disjoint open contact squares of radius \(\lambda u\) inside the isolated stars, for a fixed sufficiently small \(\lambda>0\).

First consider pairs of points \((x,y)\) in the two compared closures that do not both belong to the same open contact square. This condition specifies a closed subset of the compact product of the two local closures. On it the distance \(\lVert x-y\rVert_\infty\) has a positive minimum: if the minimum were zero, \(x=y\) would be a common contact, and both points would belong to its open contact square. Since the list of rescaled patterns is finite, the minimum is uniformly at least \(c_2u\) whenever this subset is nonempty.

It remains to consider \(x,y\) in one contact star centered at \(v\). Inside that star each closure is a union of closed radial sectors. Their direction sets are disjoint. A shared direction would give a shared ray segment, contrary to the absence of positive-length same-label contacts. Their boundary directions are among the eight axis and diagonal directions, so the angle between their direction sets is at least \(\pi/4\). Elementary Euclidean geometry, followed by comparison of Euclidean and sup norms, gives \[ \lVert x-y\rVert_\infty \ge c_3\max\{\lVert x-v\rVert_\infty,\lVert y-v\rVert_\infty\}. \tag{232}\] For example, the squared Euclidean distance between radial vectors of lengths \(a,b\) and angle \(\theta\ge\pi/4\) is \((a-b)^2+2ab(1-\cos\theta)\), which bounds a fixed multiple of \(\max(a,b)^2\).

At the immediately following trim, the compared old or peer identifier loses a square about \(v\) of radius at least \(c_4u^{\rho_0}\), by (230) and comparability of mark scales. This removal also occurs if the square is terminal. That identifier only shrinks afterward. Thus a point \(y\) in its final actual region has \(\lVert y-v\rVert_\infty\ge c_4u^{\rho_0}\), and (232) supplies the required lower bound for pairs inside the contact star. Together with the preceding compactness estimate and the exterior-of-box estimate this proves the lemma. ◻

Proof of Theorem 1. The empty domain and the empty or full region have zero entropy. Put \(b=|\partial_\Lambda A|\). If \(b=0\), Lemma 4 gives \(S_\Omega(A)=0\).

Assume \(b>0\). Choose \(n_*\) large enough for Proposition 30 and Proposition 32 with \(M=100\) to apply to the templates of Proposition 34, uniformly with the stated template bounds. Increase it so that every radius \(r\) supplied by amplification at scale \(n\ge n_*\) satisfies \(L+2r<n^\beta\). Let \(D,X_i,T_i,n_i,f_i\) be the resulting geometric decomposition, and put \[S_i=A\cap\bigl(T_i+[-L_i,L_i]^2_{\mathbb Z}\bigr),\qquad J_i=(\Lambda\setminus A)\cup \bigcup_{\substack{j<i\\f_j=f_i}}X_j.\] Here \([-L_i,L_i]^2_{\mathbb Z}\) denotes the integer square. For each \(i\), let \(r_i\) be the radius supplied by amplification, using the size, boundary, and information bounds from Proposition 30. The resulting mutual-information estimate applies to \(A\cap T_i\) against any sites outside the graph neighborhood \(N_{2r_i}^{\Lambda}(S_i)\).

We verify this condition for the whole set \(J_i\), rather than for its individual pieces. A graph path of length at most \(2r_i\) from \(S_i\) stays within ambient distance \(L_i+2r_i<n_i^\beta\) of \(T_i\). Equation (225) excludes every earlier same-family \(X_j\) jointly. Such a path also cannot reach \(\Lambda\setminus A\): its first crossing of the cut would put an endpoint of a cut edge within that ambient distance of \(T_i\), contradicting (224). Thus \[J_i\subseteq\Lambda\setminus N_{2r_i}^{\Lambda}(S_i).\] Proposition 32 and monotonicity under discarding part of its first system yield \[I_\Omega(X_i:J_i) \le I_\Omega(A\cap T_i:J_i)\le C n_i^{-100}.\] These estimates all concern the same original state \(\Omega\), although the auxiliary construction used to prove each estimate may differ.

Apply Lemma 33 to the two labels. Since each site has dimension \(q\), \[S_\Omega(A) \le |D|\log q+\frac C2\sum_i n_i^{-100} \le C' b.\] Every choice of threshold and every constant in this proof depends only on the fixed local dimension, interaction range and norm bound, and spectral gap. None depends on the size, shape, connectedness, or holes of \(\Lambda\), or on the choice of \(A\). This proves the theorem. ◻