Statement of the result

For an integer \(L\ge2\), let \(\Lambda_L=\{1,\ldots,L\}^2\), with the nearest-neighbor square-lattice edge set \(E(\Lambda_L)\) and open boundary conditions. Put \[\mathcal H_{\Lambda_L}=\bigotimes_{v\in\Lambda_L}\mathbb C^q.\] A PEPS on this grid is the contraction obtained by placing \(\sum_{a=1}^{D_e}|a\rangle\otimes|a\rangle\) on each edge \(e\) and applying at each vertex an arbitrary linear map \[A_v:\bigotimes_{e\ni v}\mathbb C^{D_e}\longrightarrow\mathbb C^q.\] Its maximum bond dimension is \(\max_eD_e\). The maps need not be isometries, and the resulting vector need not be normalized.

Theorem 1 gives an affirmative resolution of the polynomial PEPS approximation assertion for the stated open-square Hamiltonian class. The relation to other formulations of the PEPS conjecture is explained below. No gap for restrictions to subregions, frustration freeness, commutativity, translation invariance, or path from a product state is assumed. The error in (3) concerns the entire vector. The theorem makes no claim that its tensors can be found or contracted efficiently; their entries may depend arbitrarily on \(H\) and \(\Omega\).

History and significance

Projected entangled-pair states extend matrix product states to higher-dimensional lattices by contracting virtual pairs through local maps (Verstraete and Cirac 2004). Their local tensor structure makes them natural candidates for describing ground states whose entanglement is concentrated near region boundaries. In one dimension, quantitative Schmidt-tail truncation estimates connect entanglement spectra with global MPS approximation (Verstraete and Cirac 2006). Hastings established the entropy area law for unique ground states of uniformly gapped spin chains with bounded finite-range interactions and obtained polynomial-dimension MPS approximations (Hastings 2007). The higher-dimensional question requires a representation compatible across many intersecting cuts.

There are several distinct PEPS approximation questions. The global formulation in (Cirac et al. 2019, sec. 2.2, Questions 7–8) asks whether a spectral gap ensures polynomial bond dimension at inverse-polynomial Hilbert-space norm error. That formulation uses translation-invariant finite-range Hamiltonians on a torus. Theorem 1 addresses the corresponding assertion for arbitrary bounded on-site and nearest-neighbor interactions on an open square. Local-observable formulations instead control expectations on fixed-size regions; the stronger structural formulation in (Schwarz et al. 2017, Conjecture 2) also requires an injective approximant with a gapped parent Hamiltonian. The theorem here concerns the full vector and imposes no such structural requirements on the approximant.

A positive local approximation result is due to Huang (Huang 2026, Theorem 3). For states satisfying a Rényi entropy area law of fixed order \(0<\alpha<1\) across every rectangle, he obtains PEPS of bond dimension \(e^{O(1/\delta)}\), independent of lattice size, that approximate norm-one local-observable expectations to error \(\delta\). The observable support scale is held fixed in this bound. The Rényi hypothesis is stronger than a von Neumann entropy area law, and the conclusion controls local expectations rather than global vector error.

An entropy bound and a representation theorem are different statements. For arbitrary chain-state families, bounded von Neumann block entropy does not ensure efficient global MPS approximation (Schuch et al. 2008). In higher dimensions, Ge and Eisert constructed families satisfying strong area laws that have no efficient classical tensor-network descriptions (Ge and Eisert 2016, Definition 7 and Corollary 9). Their description model includes a bound on the classical complexity of tensor entries; it should be distinguished from a bound on virtual dimension with arbitrary complex entries. These examples concern general states, rather than the uniformly gapped ground-state class in Theorem 1. They explain why one must control compatible operations and their accumulated error, in addition to the spectra of separate marginals.

Thermal tensor networks provide another approach. Hastings constructed local spectral filters and tensor-network descriptions of Gibbs operators (Hastings 2006). Molnár, Schuch, Verstraete, and Cirac refined the thermal approximation and derived quasipolynomial-dimension ground-state PEPS under a gap together with a low-energy density-of-states bound (Molnár et al. 2015, sec. III.C). At inverse-polynomial accuracy their bond bound is exponential in a squared logarithm of system size. The low-energy counting condition is an additional hypothesis. A different cooling and postselection construction approximates the unique ground state of a gapped \(d\)-dimensional local Hamiltonian at the boundary of a \((d+1)\)-dimensional PEPS (Schuch et al. 2007, sec. V); transferring that representation to the original lattice is a further problem.

Recent tensor-network results also distinguish the dimension and the approximation task. Arad, Firanko, and Jain obtained two-dimensional mutual-information bounds for locally gapped frustration-free systems and polynomial-bond matrix product operator approximations to maximally mixed ground states in one dimension (Arad et al. 2026, Corollaries 2.7 and 2.9). Such entropy or one-dimensional operator bounds do not provide the simultaneous original-grid PEPS conclusion needed here.

Our analytic inputs are the entropy area law and the stronger collar estimates of the companion (OpenAI 2026). The present proof adapts its nested-filter and two-family ideas, and then develops compatible changes of ownership, compression of a distributed density network, and routing on the prescribed grid. The representation theorem uses the full companion interface recorded in Appendix 9, rather than the cutwise entropy inequality alone. Its polynomial bond bound permits arbitrary tensor entries and is an existence statement. The distinction from efficient contraction is substantive: general PEPS contraction has worst-case complexity obstructions (Schuch et al. 2007), although that fact alone does not classify the particular approximants constructed here.

Proof strategy

The main intermediate object is a distribution of the physical registers among auxiliary parties. A party owns a collection of registers and may perform arbitrary linear contractions on them privately. At the start, one party holds the entire vector \(\Omega\); at the end, each physical site has its own party. Preparing the initial vector is permitted because we seek a representation, rather than an algorithm for finding it. The constraint is on communication: each party must participate in only boundedly many operations with other parties over its whole lifetime. Each such operation must also have a polynomial-size expansion built from private contractions and normalized bipartite vectors or covectors. The bipartite states themselves may have arbitrarily large Schmidt rank.

The imported area law

The following is the range-one square-domain specialization of the companion theorem. Stating its uniformity here is useful because later auxiliary regions are not themselves assigned gapped Hamiltonians.

The companion theorem allows any finite induced square-lattice domain, with one bounded Hermitian summand per designated nonempty support of bounded graph diameter. In (1) the diameter bound is one. Its only spectral assumption is the unique full-system ground vector and (2). These are precisely the hypotheses used here. For a physical sample \(X\) of a square of radius \(r\), this implies \(S_\Omega(X)\le C(1+r)\), including squares clipped by the outer boundary.

We also use the companion’s sublinear collar and amplification propositions. Their geometric and exponent requirements are checked explicitly in Proposition 4; neither is a PEPS approximation statement. Appendix 9 records their exact interfaces, including the positive constraints and two-family construction underlying the companion argument. The complete companion is (OpenAI 2026).

Finite-dimensional information facts

We use strong subadditivity, the chain rule, monotonicity of mutual information under discarding systems, and, for a pure state on \(ABCD\), the duality \[I(A:C\mid B)=I(A:C\mid D).\] These are standard entropy identities and inequalities; see (Lieb and Ruskai 1973) and the preliminary section of (OpenAI 2026). For example \(|S(A\mid B)|\le\log\dim\mathcal H_A\), so the entropy cost of a bounded set of individual sites is bounded by its cardinality times \(\log q\).

For normalized density operators define \(F(\rho,\sigma)=\lVert \sqrt\rho\sqrt\sigma\rVert_1\) and \(D(\rho\Vert\sigma)=\mathop{\mathrm{Tr}}\rho(\log\rho-\log\sigma)\), with the usual infinite value when the support condition fails.

We recall the relative-entropy estimate through quantum affinity (Carlen and Lieb 2014, Equation (2.9)) and Uhlmann’s purification-overlap theorem (Uhlmann 1976, sec. 2 and 5). The finite-dimensional proof below fixes our root-fidelity convention and includes nonminimal purifying spaces.

If \(\frac12\lVert \rho-\sigma\rVert_1\le\epsilon\le1\), conditional entropy obeys \[ |S(A\mid B)_\rho-S(A\mid B)_\sigma| \le2\epsilon\log\dim\mathcal H_A +(1+\epsilon)h\!\left(\frac{\epsilon}{1+\epsilon}\right), \tag{5}\] where \(h\) is binary entropy (Winter 2016, Lemma 2). The dimension is that of \(A\), regardless of the purifying or conditioning space. We may also use the weaker consequence \(C\sqrt\epsilon\log(e\dim\mathcal H_A)\). All retained physical logarithmic dimensions are \(O(L^2)\); hence increasing a fixed error power absorbs the continuity loss.

Finally, the gap directly implies \[ \min_\theta\lVert \psi-e^{i\theta}\Omega\rVert^2 \le\frac{2}{\Delta}\langle\psi,(H-E_0I)\psi\rangle \tag{6}\] for every normalized \(\psi\). Indeed the excitation energy bounds \(\Delta(1-|\langle\Omega,\psi\rangle|^2)\), while phase minimization gives \(2(1-|\langle\Omega,\psi\rangle|)\).

The coarse collar interface

We specify the additional companion statements used here, beyond the entropy area law in Proposition 2. The constants \(C_{\rm tpl}\) and \(D_0\) below are fixed template constants from the companion. In the terminology of (OpenAI 2026, sec. 9 and 10), a template is a nonempty ambient lattice set \[T=\bigcup_{a=1}^{N_0}(P_a\cap\mathbb Z^2), \qquad n\ge C_{\rm tpl}N_0(s_0+1), \qquad \mathop{\mathrm{diam}}_\infty T\le n,\] where the pieces are closed rectangles or triangles, have sup diameter at most \(s_0\), and have sides of slopes \(0,\infty,1,-1\). The companion also fixes a physical cut \(A\subseteq\Lambda\) and its set \(Z_A\) of crossing-edge endpoints. Its template-clearance assumption is \(\mathop{\mathrm{dist}}_\infty(T,Z_A)>4D_0s_0\). Put \[\varepsilon_{\rm col}=10^{-5},\qquad \ell=\lfloor n^{1-\varepsilon_{\rm col}}\rfloor, \qquad T_\ell=T+([-\ell,\ell]^2\cap\mathbb Z^2).\] The Proposition “A sublinear collar bound” in (OpenAI 2026, sec. 9) states that, when \(\ell\le s_0\) and the scale is sufficiently large, \[ \begin{split} X&=A\cap T,\qquad S_0=A\cap T_\ell,\\ I_\Omega(X:\Lambda\setminus S_0)&\le Cn^{1-\varepsilon_{\rm col}}. \end{split} \tag{7}\] It also supplies \(|S_0|\le Cn^2\) and \(Cn\) upper bounds for the numbers of physical crossing edges of all three cuts \(X,S_0,S_0\setminus X\). All constants and thresholds are uniform in the domain, cut, and admissible template.

The Proposition “Amplification” in (OpenAI 2026, sec. 10) uses precisely these size, three-cut, and information hypotheses, with any fixed common upper-bound constant. Its fixed exponents are \[\beta_0=1-2\cdot10^{-6},\qquad \alpha=\frac{k}{k+1}>1-10^{-6},\] where \(k\) is a sufficiently large fixed integer. For every fixed integer \(M\ge1\), it gives a graph radius \(r\) such that \[ \ell+2r=o(n^{\beta_0}),\qquad I_\Omega(X:E)\le C_Mn^{-M} \quad\text{whenever }E\subseteq\Lambda\setminus N_{2r}(S_0). \tag{8}\] Here \(N_{2r}\) is a physical graph neighborhood. The positive system required by that Proposition is furnished by the Proposition “Positive replacement of the Hamiltonian” in (OpenAI 2026, sec. 4), for each prescribed integer \(k\). That replacement assumes the original full-system eigenvector and gap inequality. Its constants may depend on \(k\), which is fixed here. It does not assume a gap for any restricted Hamiltonian. Our on-site and nearest-neighbor Hamiltonian has induced-graph range one and one bounded term per designated support, so it satisfies the bounded-interaction hypotheses of all these results.

A conditional information bound for a cell

The next construction adapts the two-family ordering in (OpenAI 2026, sec. 11) to a cell with arbitrary exterior conditioning. We prove the construction and its conditioning order here. We use ambient colors in the next construction. Missing physical sites are simply ignored when taking entropies. Boundaries of all chambers are assigned by one consistent infinitesimal generic displacement; equivalently, one works with open chambers, uses their closures for distance estimates, and then samples with a common tie convention. Only finitely many chamber boundaries occur at a fixed instance. No color is assigned exclusively to a lower-dimensional face.

Reducing a collar information cost with the global gap

We use Lemma 3 in the following form. For density operators with \(\mathop{\mathrm{supp}}\rho\subseteq\mathop{\mathrm{supp}}\sigma\), \[ \lVert \sqrt\rho\sqrt\sigma\rVert_1 \ge \exp\left(-\tfrac12\mathop{\mathrm{Tr}}\rho(\log\rho-\log\sigma)\right), \tag{11}\] and, after enlarging the purifying space if necessary, an isometry on that space realizes the corresponding overlap with a prescribed purification. In particular the original purifying buffer need not be minimal. All enlarged factors will be removed before a Hamiltonian locality estimate is applied.

Regions tapering toward finitely many points

Distributed constructions and pair effects

A party is a label for a collection of registers. At every time the memory is a tensor product of the parties’ current Hilbert spaces. A gate has specified input and output spaces for each participating party; these spaces may differ. Registers outside the participating parties are spectators. A private gate has one participating party. Gates are linear contractions. Registers to be discarded are retained as part of their owner’s memory until the end, so the entire construction is a pure linear calculation before its final partial trace.

An allowed monomial in a gate expansion is a finite composition of local contractions, preparations of normalized vectors shared between two participating parties, and contractions by normalized bras shared between two participating parties. A transferred register of dimension \(d\) can also be allowed when \(d\) is uniformly polynomial: expand its identity transfer as \[\sum_{a=1}^{d}|a\rangle_{\mathrm{receiver}} \langle a|_{\mathrm{sender}}.\] Each summand consists of local norm-one maps. A bounded number of such transfers therefore changes a polynomial gate expansion by only a polynomial factor. We make this expansion first whenever such messages occur.

The density argument will sample prepared pair sources. The following lemma converts pair effects to that form while approximating the original gate in operator norm. The ideal extra output is the same fixed garbage vector in every branch; this common comparison lets us sum the branchwise operator-norm errors.

The density-compression theorem

Sheets and adaptive encodings

For a physical set \(A\subseteq\Lambda\), write \(\mathcal H_A=\bigotimes_{x\in A}\mathbb C^q\). A sheet has one raw register \(\mathbb C^q\) at each \(x\in\Lambda\) and a specified owner for each register. Different sheets have distinct registers. Their canonical raw reference vectors are independent copies of \(\Omega\), with sites indexed by position. Thus ownership does not permute the positions in the definition of the reference vector.

Fix a constant \(B_0>0\). In this section all applications of Proposition 8 use accuracy \[ \varepsilon=L^{-60}. \tag{75}\] For a center \(c\in\mathbb R^2\) and \(0<h\le B_0\log L\), let \[Q^-(c,h)=\{x:\lvert x-c\rvert_\infty\le h\},\qquad Q^+(c,h)=\{x:\lvert x-c\rvert_\infty\le 2h\}\] be the inner and outer squares. Their register sets are their intersections with \(\Lambda\). Choose the projector from Proposition 8 in the form \[ P_{c,h} =\sum_{j=1}^{m}\sum_{\ell=1}^{d_j} \bigl(|v_{j\ell}\rangle\langle v_{j\ell}|\bigr)_{D_j} \otimes\mathop{\mathrm{id}}_{\Lambda\setminus D_j}, \qquad \lVert (\mathop{\mathrm{id}}-P_{c,h})\Omega\rVert\le\varepsilon. \tag{76}\] Here the \(D_j\) are nested square samples, every one containing \(Q^-(c,h)\cap\Lambda\) and contained in \(Q^+(c,h)\cap\Lambda\); the vectors \(v_{j\ell}\in\mathcal H_{D_j}\) are normalized; the displayed cylinder ranges are mutually orthogonal; and \(\sum_jd_j\) is bounded by one fixed power of \(L\). The exponent may depend on \(B_0\) and the fixed physical parameters. Fix these choices once for each center and scale used in the construction, and use the same choices on different sheets.

Let \(\mathcal J_{c,h}\) be a tag register with orthonormal basis \(\{|j,\ell\rangle\}\). With a fixed one-site unit vector \(|0\rangle\), define \[ K_{c,h} =\sum_{j,\ell}|j,\ell\rangle_{\mathcal J_{c,h}}\otimes \left[ \bigl(|0\rangle^{\otimes D_j}\langle v_{j\ell}|\bigr)_{D_j} \otimes\mathop{\mathrm{id}}_{\Lambda\setminus D_j} \right]. \tag{77}\] The notation \(|0\rangle^{\otimes D_j}\) denotes the product zero vector on the sites of \(D_j\). Orthogonality of the tag basis gives the exact identity \[ K_{c,h}^{\dagger}K_{c,h}=P_{c,h},\qquad \lVert K_{c,h}\rVert\le1. \tag{78}\] In particular, the encoding retains raw registers at every site, even inside the hole. Its tag records which radius and which inside vector were used; it does not assert a low-rank approximation at one common radius. If the outer square has empty physical sample, we use the identity encoding with a one-dimensional tag.

Encodings of distinct holes have disjoint raw supports and separate tags, so they commute after the canonical identification of tag orderings. If the frame has \(r\) holes, their projectors also commute, and therefore \[ \lVert \Omega_{\mathcal F}\rVert =\left\|\prod_{a=1}^{r}P_a\Omega\right\|, \qquad 1-r\varepsilon\le\lVert \Omega_{\mathcal F}\rVert\le1. \tag{80}\] For the lower bound, expand \(\prod_aP_a-\mathop{\mathrm{id}}\) by successive factors and use contractivity and (76). We always compare vectors in their canonical site and sheet coordinates while separately specifying the owners of their input and output registers. Auxiliary sheets may eventually be retained as private discard registers.

We shall call a change bounded if it uses a fixed number of parties and gates of the type admitted by Theorem 10. The number of normalized pair resources per monomial is fixed; their Hilbert-space dimensions need not be bounded. For every fixed \(a>0\), an operator-norm approximation of error \(O(L^{-a})\) is allowed, with a polynomial expansion whose exponent can depend on \(a\). Scaling such an approximation by a factor \(1+O(L^{-a})\) restores contractivity.

A tensor estimate for bounded patch networks

The following elementary tensor estimate permits us to replace a complicated bounded patch network by pair resources without a dependence on the dimensions of its open wires. A normalized vector shared by three or more parties is not itself one of the permitted pair resources. We therefore need products of vectors whose factors each belong to at most two parties. The estimate below first produces a controlled number of product terms, keeping all open legs of each original tensor together as one group. In the patch rewrite that follows, we will verify that each such group belongs to at most two owners. These groups are determined by the individual input bras and output kets, rather than by the parties.

For a matrix \(M\), write \(\lVert M\rVert_1\) for its sum of singular values and \(\lVert M\rVert_2\) for its Hilbert–Schmidt norm. The Hilbert-space norm of a tensor equals the Hilbert–Schmidt norm of every matrix reshaping of that tensor.

Changing a frame on small patches

For a frame \(\mathcal F\), let \(H^-(\mathcal F)\) and \(H^+(\mathcal F)\) be the unions of the physical samples of its inner and outer hole squares. A tag is unchanged only if its hole, its encoding, and its owner are unchanged.

Splitting a raw buffer and changing a homogeneous owner

The next two operations use small mutual information. We state the purification consequence with its dimension-independent error explicitly.

The isometry in Lemma 15 will always act privately: all of \(U\), and both of its enlarged output factors, will be at one party.

In the geometric construction the information hypothesis is supplied by Proposition 7, with its exponent set to \(60\). In particular, for finitely many marked points and a fixed clearance constant, the logarithmic floor \(t=D\log L\) can be chosen so that any eligible tapering pair \(T,E\) satisfies the hypothesis. The set \(E\) in (86) includes outer hole footprints even when all their raw owners happen to be \(P_\circ\); this keeps the splitting isometry off every encoding support.

Exchanging assignments between two sheets

Proposition 7 will again verify the one-copy information bound in Lemma 17. Including holes from both sheets in (88) ensures that the private correction commutes with every encoding. Exchanging whole outer squares ensures that all tag and radius branches use the same register identification. The second sheet may be discarded after the exchange and any subsequent point treatments; its encoded reference vector is retained in the error comparison until that discard is taken.

Ambient guides, holes, and the hierarchy

Translate the lattice so that its sites are the cell centers \((i+\tfrac12,j+\tfrac12)\), \(0\le i,j<L\). Let \(N\) be a power of two with \(L\le N<2L\). The dyadic hierarchy consists of the aligned squares in \([0,N]^2\) of side \(n=2^j\), including the root square. Unoccupied cells of this padded square carry no registers. A different party is assigned to each dyadic square.

A guide is a finite polygonal partition of the ambient plane, whose labels name the parties owning the raw registers at its sampled sites. All guides below are specified by their open polygonal chambers. Use one consistent tie convention for all boundaries in the finite schedule: move a sampled point by an arbitrarily small generic vector into an incident chamber, consistently across all descriptions, and take the resulting label. Equivalently, choose a sufficiently small generic displacement after all the finitely many guides are specified. No label is introduced solely on a boundary face. Separation estimates will be stated for closures, and consequently remain valid for these sampled labels. Squares used for the patch constraints retain their closed-square sampling convention.

We pass successively from the uniform assignment by \(2n\)-squares to the uniform assignment by \(n\)-squares, for \(n=N/2,N/4,\ldots,1\). At a given level, view each old \(2n\)-square label as constant on its four \(n\)-square children, and repaint one \(n\)-square at a time to its new party. At the beginning and end of a repainting, every other \(n\)-square has either its old \(2n\)-square label or its final \(n\)-square label. In particular, the guide along an open edge of the square being repainted has a single label on each side.

At the beginning and end of such a repainting, the exterior of \([0,N]^2\) has one placeholder label. During that repainting it is represented by a fresh additional party attached to the square being repainted. At the next repainting its identity may be replaced. This does not transfer any main raw register: at those transition times the placeholder owns no physical main-sheet site. We will also ensure that it holds no persistent main-sheet tag. Temporary operations may give it physical main-sheet registers within a treated neighborhood, but the repainting restores the prescribed exterior assignment before that party is retired.

Fix, in an order made precise below, constants \(D>0\) and \(\epsilon_0>0\), and write \[ t=D\log L, \qquad h_n=\epsilon_0\min(n,t). \tag{94}\] A true vertex of a guide is a point lying in the closures of at least three distinct actual labels. The guides we construct have only finitely many such points. At every true vertex use the encoded hole of Definition 12, with inner radius \(h_n\) and outer radius \(2h_n\), and with test exponent \(60\). We will choose \(\epsilon_0\) so that the outer squares are disjoint within each guide. For a given center and radius, make the same choice of the adaptive constraint on every sheet. Thus each branch of a hole uses a square with radius between \(h_n\) and \(2h_n\).

Choose the tag holder canonically among the actual incident labels other than the placeholder, for example by a fixed order on genuine block parties. The placeholder is omitted from this choice, regardless of its current identity. A true vertex has at least two other incident labels, so the rule is always possible. An empty physical outer square uses the trivial one-branch encoding. When a guide is exchanged between sheets, use the same tag rule; its tag then accompanies the whole hole.

Before entering a new hierarchy level, resize existing main-sheet holes to its value of \(h_n\), one hole at a time, without changing the guide. These resizings will also be small-patch rewrites. A fresh auxiliary sheet has a uniform guide on the whole plane and therefore no holes. An auxiliary sheet that has been retired is kept idle, with its registers and tags at their current parties, until the final discard. Its holes are not resized at subsequent levels.

The edge construction before point treatment

Choose a large fixed constant \(K_0\). For \(n\le K_0t\) we will repaint the square directly by a small-patch rewrite. We first describe the large-scale construction, for \(n>K_0t\), before treating its finitely many contact points.

Let \(S\) be the \(n\)-square being repainted, with old label \(A\) and final label \(B\). For any of its edges let \(s\in(0,n)\) be the coordinate parallel to that edge, measured from one endpoint, and let \(d\) be the normal coordinate, positive into \(S\). Set \[ w(s)=10^{-3}\min(s,n-s), \qquad x=\frac{d}{w(s)}. \tag{95}\] An interval in \(x\) always refers only to this parallel-coordinate span. These intervals define polygonal bands, with their only bends at \(s=n/2\).

First birth \(B\) in the central part of \(S\), leaving the four inward edge bands \(0<x<1\) with label \(A\). More explicitly, the changed open chambers are the points of the interior of \(S\) lying beyond the \(x=1\) boundary for every edge. Their closure meets the exterior of \(S\) only at the four corners. These four corners are the marked contact points for this birth. Now process the four edges one at a time. At the start of an edge operation its normal word is \[C\quad A\quad B, \qquad x<0,\quad 0<x<1,\quad x>1,\] in the neighborhood of the edge. Here \(C\) is the label on the unique exterior neighboring \(n\)-square, or the placeholder at an exterior edge. Nominal names in all the words below are permitted to coincide; adjacent equal labels are simply one actual region. The marked points for an edge operation are its two endpoints.

The reason for using an auxiliary sheet is already visible when \(A,B,C\) are distinct. A direct death of the \(A\) strip into \(B\) would leave the surrounding word \(C\mid B\). The closure of the changed strip meets \(C\) along the entire original edge, so it has zero separation from the non-\(B\) owners. The angular-information estimate therefore does not supply the hypothesis needed for Lemma 16. We instead prepare additional \(A\) and \(B\) strips on an auxiliary sheet and exchange them into the exterior. The resulting \(C\) strip is surrounded by \(A\) and can die into it; the combined \(A\) strip is then surrounded by \(B\) and can die in turn. A final birth and death restore the original exterior assignment. The construction below supplies these buffers away from the endpoints; the subsequent point treatment supplies the logarithmic clearance there.

Perform the following finite sequence.

1. Initialize a fresh auxiliary \(\Omega\) sheet uniformly at \(C\). On this sheet birth \(B\) in \(-6<x<-1\), then \(A\) in \(-5<x<-2\), then \(C\) in \(-4<x<-3\). Each insertion lies strictly inside its surrounding band. The resulting auxiliary word is \[ C\mid B\mid A\mid C\mid A\mid B\mid C, \qquad -6,-5,-4,-3,-2,-1 \tag{96}\] where the numbers list successive interfaces.

2. Exchange the two sheet assignments on the lens \[ Y=\{0<s<n:\ -7<x<-7/2\}. \tag{97}\] Both of its open sides lie in common-\(C\) corridors: the side \(x=-7\) lies in the auxiliary \(C\) sector \(x<-6\), and the other passes through the auxiliary \(C\) band \((-4,-3)\). The main sheet was \(C\) throughout this exterior strip. After the exchange its word is \[ C\mid B\mid A\mid C\mid A\mid B, \qquad -6,-5,-4,0,1. \tag{98}\] The auxiliary sheet is then retired, after completing the point treatments for this exchange as described below.

3. On the main sheet, kill the \(C\) band \((-4,0)\) into \(A\), and then kill the resulting \(A\) band \((-5,1)\) into \(B\). The word is now \(C\mid B\) with interface at \(-6\). Birth \(C\) in \((-3,0)\) inside \(B\), and kill the remaining \(B\) band \((-6,-3)\) into \(C\). The final word is \(C\mid B\) with interface at zero.

Figure 1 records the complete normal-word sequence at a fixed parallel coordinate.

Every band used in this sequence is contained in \(-8<x<2\). Such bands along distinct edges of \(S\) have disjoint interiors. Opposite edges are separated by \(n\), whereas their bands have width at most \(0.004n\) on either relevant side. For adjacent edges, a common point in the parallel-coordinate spans would have to lie inside \(S\) near the shared corner. If its two distances from those edges are \(r_1,r_2\), membership in both enlarged bands would imply \(r_1<0.008r_2\) and \(r_2<0.008r_1\), which is impossible. The negative part of an edge band lies in its one exterior neighboring square, and cannot enter the parallel-coordinate span of an adjacent edge. Consequently an already completed edge does not alter the normal word needed at the next one. At the end of the four edge operations \(S\) has label \(B\) and every other square has its original label. The same description is valid along the placeholder exterior: all temporary plane changes there are restored at the end.

Uniform angular separation

We record precisely the geometry needed to invoke the information estimate. For a given elementary operation let \(\mathcal V\) be its set of marked corners or endpoints and put \(d_{\mathcal V}(y)=\min_{v\in\mathcal V}\lvert y-v\rvert_\infty\). For a birth, its surrounding guide is the guide before the insertion; for a death, it is the guide after the removal. In that surrounding guide the changed region lies within a single label \(P_\circ\).

All true vertices of an unmodified guide are separated on scale \(n\). Indeed the baseline vertices are grid corners. The new band interfaces are disjoint except at their indicated endpoints. Their bends at \(s=n/2\) have only two incident labels and so are not true vertices. In a fixed-radius neighborhood of a marked point, in units of \(n\), the only possible true vertex is the marked point itself.

Point treatment and the actual bulk operations

Before implementing one elementary birth or death, homogenize each involved sheet inside the radius-\(t\) square around every mark to its surrounding label \(P_\circ\). Make the same modification in the target guide of that elementary operation. Thus its actual bulk recoloring is truncated by those squares. For an exchange, homogenize both sheets there to \(C\); the lens \(Y\) itself need not be altered because the assignments coincide throughout the homogenized portions. After the bulk operation, remove the point treatment by changing to the specified target guide. We prove below that every homogenization and every removal is a small-patch rewrite.

Choose \(K_0\) large enough that, when \(n>K_0t\), the tenfold enlargements of the treated squares are disjoint, contain no other unmodified true vertices, and lie inside the straight-ray charts used above. In a treated neighborhood, any newly created true vertex is at the intersection of a ray with the square rim. The set of possible rim intersection points is finite after scaling by \(t\). Coincident points count as one point. There is therefore a positive minimum separation between distinct possible centers, uniformly over the finite pattern family. Take \(\epsilon_0\) small compared with these separations, with the ordinary grid separations, and with \(a_0\). This makes the outer holes disjoint, including in the temporary guides.

Figure 2 illustrates how a true vertex can move to the rim during a point treatment.

For a birth or death, a point of the remaining closed changed region lies outside the interiors of the treated squares. Consequently \(d_{\mathcal V}(y)\ge t\), and homogenization to the surrounding label can only remove its other-label obstructions. By (99), after decreasing a fixed constant if necessary, its separation from the other labels is bounded below by \[ a_1\max\{t,\min(n,d_{\mathcal V}(y))\}, \qquad a_1>0. \tag{101}\] Every true vertex of the surrounding temporary guide belongs to the closure of positions of labels different from \(P_\circ\). Enlarging such a center to its outer hole square subtracts at most \(2h_n\) from this separation. Since \(h_n=\epsilon_0t\) in the present large-scale case, a further fixed decrease of \(a_1\) preserves (101) for all outer hole sites too.

There are no changed true vertices in the bulk recoloring. To see this also at the treated rims, the closure of the truncated changed region has a positive neighborhood containing only \(P_\circ\) in the surrounding guide. A birth or reversed death introduces only one additional label in that neighborhood. Thus any newly drawn interface, including where it meets a treated-square rim, has at most two incident labels. Existing true vertices are separated from this closed change and retain their guides, holes, and tag holders. The raw target \(T\) is therefore outside all holes, compactly located in diameter \(O(n)\), and satisfies the tapering hypothesis against the set \(E\) consisting of other owners and all outer hole sites. The remaining raw buffer has a single owner. Lemma 16, in its forward or reverse form, implements the bulk change.

For the exchange, homogenization removes all noncommon-\(C\) positions inside the treated squares. By (100), every remaining such position has the same floor at \(t\) in its distance from \(\partial Y\). This also holds for new rim vertices: a rim true vertex belongs to the closure of an outside sector that is not \(C\), and the original angular estimate applies at that point. Thus \(\partial Y\), including each of its intersections with a treated-square rim, lies in an open common-\(C\) corridor. No new true vertex is created by exchanging the guides along that boundary.

Every true vertex on either sheet belongs to the closure of a noncommon-\(C\) region. Its distance from \(\partial Y\) is at least a fixed multiple of \(t\) in a treated neighborhood, and obeys the larger angular bound elsewhere. The full outer hole square, of radius \(2\epsilon_0t\), therefore lies strictly on one side of the lens boundary. The same tapered bound holds at every point of an outer hole, not only at its center: both \(y\mapsto\mathop{\mathrm{dist}}(y,\partial Y)\) and \(y\mapsto d_{\mathcal V}(y)\) are \(1\)-Lipschitz. If \(c\) is the center and \(|x-c|_\infty\le2\epsilon_0t\), put \(Q_x=\max\{t,\min(n,d_{\mathcal V}(x))\}\). A center clearance \(a_*\max\{t,\min(n,d_{\mathcal V}(c))\}\) gives \[\mathop{\mathrm{dist}}(x,\partial Y)\ge a_*Q_x-2\epsilon_0(a_*+1)t \ge \tfrac12a_*Q_x\] after fixing \(\epsilon_0\) small enough. In particular, every selected variable-radius square in that hole lies on the same side, since its radius is at most \(2h_n\). Exchanging a neighborhood of such a vertex substitutes one complete sheet guide and carries its entire hole and tag. It does not splice sectors of two guides at a vertex.

For Lemma 17, let \(T\) be the positions inside \(Y\) that are not common \(C\), together with all inside outer hole sites, and let \(E\) be the analogous set outside \(Y\). The set \(T\) has diameter \(O(n)\). Any segment from a point of \(T\) to a point of \(E\) crosses \(\partial Y\), so its length is at least the distance of that point of \(T\) from the boundary. The preceding clearance bounds, including the outer-hole enlargement, therefore give the required tapering separation of \(T\) from \(E\). On the remaining positions both raw copies are held by \(C\). This proves all hypotheses of the exchange lemma, including its condition on whole holes and their tags.

All statements use actual labels. Coalescing nominal labels cannot create a meeting of three actual owners: the closure of an identified region is the union of its finitely many former closures. If a nominal vertex disappears under such an identification, at most two actual owners meet there and no vertex encoding is required. The same argument covers the exterior placeholder. Empty physical targets need no information estimate.

Choose one fixed \(a>0\) smaller than all the surviving constants in these tapering bounds. Proposition 7, at accuracy \(L^{-60}\), then supplies a sufficiently large fixed \(D\) in (94). The birth and exchange errors are \(O(L^{-30})\), as required by their lemmas. The constants used to choose \(a\) depend only on the fixed geometric templates, not on \(D\).

The small-patch covers

We verify every invocation of Lemma 14. The following compactness observation makes the two-owner condition uniform, including along the boundary of an inner hole.

Fix a compact working neighborhood and one scaled guide. For each actual label \(c\), let \(F_c\) be the closure of its region in that neighborhood, with all open inner hole squares removed. Each \(F_c\) is compact. No three \(F_c\) with distinct actual labels have a common point, because every true vertex lies strictly inside its own inner hole. It follows that there is a positive footprint diameter such that no smaller footprint meets three of these sets. Otherwise take a sequence of shrinking footprints, pass to constant label choices and convergent points in the compact working neighborhood, and obtain a point common to three distinct \(F_c\).

This remains uniform for our small-scale and resizing radius parameters. After scaling, these parameters range over compact intervals bounded away from zero. The joint sets of a parameter and a point belonging to \(F_c\) are closed: being outside an open square is a non-strict distance inequality. The same convergent-subsequence argument applies to the joint sets. There are only finitely many label-equality types, so take the minimum positive footprint bound over those types. Apply the test separately to the old and new guides, then take the smaller bound. Different old and new hole centers do not mix the two tests.

Notice that the center of a footprint may lie inside a hole. The condition counts labels only at footprint points outside the relevant inner holes. Thus this observation permits a net of small patches to cover the whole outer hole footprints, as required by Lemma 14.

Party lifetimes and the protocol bounds

Each elementary birth, death, or exchange has a fixed number of marks. The point treatments have a fixed number of patches and parties. The edge word uses a fixed number of elementary operations and four auxiliary sheets at most per repainting. Consequently each repainting, and each junction resize, uses a uniformly bounded number of linked gates and parties. Small-patch rewrites can be replaced by arbitrarily accurate polynomial expansions and rescaled near one to remain contractions. Together with the \(L^{-60}\) patch tests and the \(O(L^{-30})\) splitting errors, choose their precision so that each change has reference-vector error at most \(C L^{-20}\).

At a level-\(n\) repainting, every active main owner is an \(n\)- or \(2n\)-block owner. On the affected footprints those blocks lie at distance \(O(n)\) from the repainted square. The same is true of their tag holders, since tags are held by incident labels. An auxiliary uses only the labels of the local edge and point diagrams; its uniform initial owner \(C\) is the actual neighboring-block owner, even though it privately holds a whole copy of \(\Omega\). The placeholder party is attached to the repainted block at scale \(n\). Resizing uses the current block labels near its junction. Private buffer isometries may act on registers at much more distant physical positions, but create no additional party or link.

The following stronger invariant concerns active main owners; the inactive exterior placeholder is omitted at the start and end of a level. It proves the lifetime claim, rather than just locality at an individual step: \[ \begin{aligned} \text{start of level }n &: \text{ only }2n\text{-block main labels};\\ \text{during level }n &: \text{ only }2n\text{- and }n\text{-block main labels, plus its temporary placeholder};\\ \text{end of level }n &: \text{ only }n\text{-block main labels}. \end{aligned} \tag{103}\] Main tags always use incident non-placeholder labels. Hence a party of side \(m\) disappears from every main tag as well as every main guide region by the end of level \(m/2\). In particular, when the last incident child square repaints, its corner rewrite updates the tag at that junction. Resizing introduces no obsolete owner. Thus this party can occur in linked gates only at levels \(m\) and \(m/2\) and their intervening transition. At these scales there are only a fixed number of repaintings or junctions within distance \(O(m)\) of its block.

Frozen auxiliary frames do not invalidate this invariant. Each is initialized, used in its one edge construction, returned from its point treatment to its specified final frame, and permanently retired before the repainting ends. Its remaining entangled registers and tags are kept idle at their owners. No subsequent gate undoes that exchange, and no later auxiliary initialization consults a frozen label: its \(C\) is always selected from the current main guide. Therefore frozen memory contributes no further linked incidence. The same applies to a retired placeholder party. A fresh placeholder identity at the next repainting has no raw main register or persistent main tag to inherit. This proves a uniform bound on every party’s total linked participation.

For later routing, a square of side \(n=2^j\) and block indices \((r,s)\) has the integer address \[ a(S)= \begin{cases} (nr+n/2,\,ns+n/2),&j\ge1,\\ (r,s),&j=0. \end{cases} \tag{104}\] Use the same address for its additional parties. The preceding locality statement gives \(O(n)\) address distances for parties sharing a gate, and their scales are equal or adjacent. The addresses lie in \(\{0,\ldots,N-1\}^2\); unit-square physical outputs have their actual site indices as addresses.

There are \(O((N/n)^2)\) repaintings and junctions at scale \(n\). Summing over dyadic scales gives \(O(N^2)=O(L^2)\) in total. The number of block parties, additional parties, auxiliary sheets, and changes is therefore \(O(L^2)\), and in particular bounded by \(C L^3\). Each frozen auxiliary has only a fixed number of holes: its guide differs from its uniform guide only in one bounded edge template and its endpoint treatments. Even the weaker total bound \(C L^3\) on frozen auxiliary holes will suffice.

Initialize all required copies of \(\Omega\) privately at their future owners, including the main copy at the root. This is a pure product initialization across parties, with no restriction on private state complexity. At the beginning the main guide has only the root label and the exterior placeholder, so it has no true vertex and no hole. At every stage use the tensor product of the designated main frame, active auxiliary frames, idle initial copies, and frozen final auxiliary frames as its reference vector. Untouched factors have norm at most one, so the local reference-error bounds remain valid with all these spectators.

At level \(n=1\), choose \(\epsilon_0<1/4\). A grid-corner outer hole then has radius \(2\epsilon_0<1/2\) and contains no cell-center physical site. Its encoding is trivial. Thus all remaining main registers are raw at their unit-square parties and the main reference is exactly \(\Omega\), apart from trivial tags. The other reference factors comprise \(\Xi\). Each encoded auxiliary frame has norm at least one minus its number of holes times \(L^{-60}\), by the frame constraint and contraction bounds. Multiplying these lower bounds and using the total frozen-hole count gives \(1-C L^{3-60}\le\lVert \Xi\rVert\le1\). All auxiliary registers and all nonphysical outputs can be discarded party by party.

This proves every assertion of Proposition 18.

Error and normalization

Apply Proposition 18 with patch and information power \(60\). The initial state is a product by party: the root privately holds the main \(\Omega\), and every future auxiliary copy of \(\Omega\) is held privately by its designated initializing party. Private dimensions and initial vectors are unrestricted in Theorem 10. The main initial guide has one label and needs no holes.

There are at most \(CL^3\) changes and at most \(CL^3\) holes across all frozen auxiliary reference frames. Each encoding is contractive, and its failure on \(\Omega\) is at most the number of its holes times \(L^{-60}\). Thus the final reference vector has the form \[ \Omega\otimes\Xi, \qquad 1-CL^{-57}\le\lVert \Xi\rVert\le1. \tag{105}\] At unit scale all physical outputs are raw at their unit parties; any remaining main hole has empty physical support and a trivial tag.

Instantiate the protocol using the contractive polynomial approximants of Lemma 14, with operator-norm error at most \(L^{-25}\), for every small-patch rewrite. The birth, death, and exchange maps already have the required form. Each selected change therefore takes its reference vector within \(O(L^{-20})\) of the next one, and the actual protocol uses these selected contractions throughout. Telescoping a product of contractions bounds the accumulated vector error by the sum of the individual errors. Consequently the actual final vector \(\Psi\) satisfies \[\lVert \Psi-\Omega\otimes\Xi\rVert\le CL^{-17},\qquad \lVert \Psi\rVert\le1.\] For vectors of norm at most one, \(\lVert |u\rangle\langle u|-|v\rangle\langle v|\rVert_1 \le2\lVert u-v\rVert\). Tracing the private outputs and using (105) gives \[ \left\|\mathop{\mathrm{Tr}}_{\rm private}|\Psi\rangle\langle\Psi| -|\Omega\rangle\langle\Omega|\right\|_1 \le CL^{-17}+CL^{-57}\le CL^{-10}. \tag{106}\] In particular the protocol does not rely on renormalizing an exponentially small success amplitude.

Every party has physical output dimension at most \(q\), every linked gate has the expansion required in Theorem 10, and linked incidence is bounded over the whole lifetime of each party. That theorem, with accuracy \(L^{-5}\), therefore gives a party operator network \(\sigma\) with polynomial link dimensions and \[ \lVert \sigma-|\Omega\rangle\langle\Omega|\rVert_1\le CL^{-5}. \tag{107}\] The network has only boundedly many links per party, between parties participating in the same local change. The sampling need not make \(\sigma\) positive or Hermitian.

Extracting one ket column

For sufficiently large \(L\), the bound \(CL^{-5}\) is less than one, so Lemma 20 applies to (107). We obtain a nonzero ket network on the party graph with normalized error at most \(CL^{-5}\). It remains to realize that graph on the prescribed grid.

Constant congestion on dyadic anchors

Let \(N\) be the least power of two with \(N\ge L\), so \(N<2L\). Shift genuine lattice coordinates to \(\{0,\ldots,L-1\}^2\) and work first on the padded grid \(\{0,\ldots,N-1\}^2\). A dyadic block of side \(n=2^j\) with block indices \((r,s)\) has anchor \[ a_{j,r,s}=\begin{cases} (2^jr+2^{j-1},\,2^js+2^{j-1}),&j\ge1,\\ (r,s),&j=0. \end{cases} \tag{108}\] Temporary parties use the anchor of their creating block; only boundedly many are attached to any block. All anchors lie on the padded grid.

The scale-separated network-to-PEPS embedding has a precedent in Barthel–Kliesch–Eisert (Barthel et al. 2010, sec. III.C, Eqs. (7)–(12)). The proof below gives the routing for the present bounded-lifetime network with polynomial link dimensions and treats finite-square boundary folding explicitly.

Each continued path carries an identity tensor on its virtual index. At a crossing, take the product of the continuing index factors and any party tensors located there. Group all indices traversing a grid edge into a tensor-product index. If each original link has dimension at most \(C_1L^r\) and congestion is at most \(\chi\), the new edge dimensions are at most \[ (C_1L^r)^\chi. \tag{109}\] The single tensor at a genuine vertex, including its physical output, is exactly an arbitrary map of the PEPS convention in Theorem 1. Vertices without additional operations contribute the requisite identity factors. Thus the ket network is a PEPS on the same physical square.

Order of constants and small sizes

For completeness, the following order makes every uniformity assertion explicit. Fix \(q,J,\Delta\), the numerical error powers, and the exponents used by the rectangular recursion. The finite band templates determine their conical clearances. Choose the large point-treatment factor, then the small hole fraction and the finite patch mesh. Choose a residual clearance \(a>0\), and then choose \(D\) from Proposition 7 at information power \(60\). The largest patch scale constant \(B_0\) is selected afterwards to cover all patches of scale at most a fixed multiple of \(D\log L\).

Proposition 8 then determines a fixed rank exponent. Only boundedly many such cylinder expansions occur in a change, so their total branch count is polynomial. Choose the singular-subspace dimensions in Lemma 14 next, to obtain operator error \(L^{-25}\). Choose the effect-inventory stack lengths from the original weighted expansion error in Lemma 9; only afterwards expand the permutation averages. Finally choose the Gaussian sample count in Theorem 10. None of these algebraic choices changes an earlier geometric support or clearance. The resulting \(C_1,r\) and the congestion \(\chi\) in (109) depend only on the fixed parameters and templates.

Increase a fixed threshold \(L_0\) so that every preceding asymptotic inequality holds and the error \(CL^{-5}\) is at most \(L^{-1}\) for \(L\ge L_0\). For the finitely many smaller sizes, every vector has an exact PEPS on a spanning tree of the square grid. One explicit construction uses a common index for the \(q^{L^2}\) product-basis configurations on all tree edges, equality tensors at tree vertices that also output the local basis state of that common configuration, and puts each expansion coefficient at one chosen root. Non-tree edges have dimension one. This bond bound is finite and independent of the Hamiltonian at each fixed \(L\). Enlarging the single prefactor \(C\) handles all \(2\le L<L_0\). Equation (109) and Lemma 20 now prove Theorem 1 for every \(L\).

Hamiltonians, metrics, and entropy.

The companion permits any finite induced domain \(\Gamma\subset\mathbb Z^2\) with local dimension \(q\) and one Hermitian term \(h_X\) of norm at most \(J\) for each nonempty support of induced-graph diameter at most a fixed \(R\). It assumes a unique unit ground vector and \(H-E_0I\ge\Delta(I-|\Omega\rangle\langle\Omega|)\). Constants depend on \(q,R,J,\Delta\) and explicitly fixed auxiliary parameters, uniformly in the domain and Hamiltonian. All entropy logarithms are natural. Its Theorem 1.1 and Corollary 1.2 give the area inequality stated in Proposition 2. Here \(\Gamma=\Lambda_L\) and \(R=1\), exactly covering (1).

Positive replacement.

The Proposition “Positive replacement of the Hamiltonian” in (OpenAI 2026, sec. 4) gives, for every fixed integer \(k\ge1\) and \(\alpha=k/(k+1)\), positive contractions \(k_i\) at the original interaction anchors, of uniformly bounded anchor multiplicity, satisfying \[k_i\Omega=0,\qquad \sum_i k_i\ge c_*^{-1}(H-E_0I)\ge(\Delta/c_*)(I-|\Omega\rangle\langle\Omega|), \qquad \|k_i-\mathbb E_{N_r(a_i)}k_i\|\le C e^{-cr^\alpha}.\] Here \(\mathbb E_{N_r(a_i)}\) is conditional expectation onto the induced-graph ball, for every integer \(r\ge0\); \(c_*,C,c\) may depend on \(k\) and the fixed physical parameters. This input underlies the companion amplification. It does not replace the original Hamiltonian by a finite-range frustration-free Hamiltonian.

Templates, collars, and amplification.

Definition “A template” and Proposition “A sublinear collar bound” in (OpenAI 2026, sec. 9) use a cut \(A\subseteq\Gamma\), its crossing-edge endpoints \(Z_A\), and a nonempty ambient lattice set \[T=\bigcup_{a=1}^{N_0}(P_a\cap\mathbb Z^2),\qquad n\ge C_{\rm tpl}N_0(s_0+1),\quad \mathop{\mathrm{diam}}_\infty T\le n, \quad \mathop{\mathrm{dist}}_\infty(T,Z_A)>4D_0s_0.\] The \(P_a\) are closed convex rectangles or triangles of sup diameter at most \(s_0\), with side slopes \(0,\infty,1,-1\); \(n,s_0\) are positive integers and \(s_0\ge1\). With \(\epsilon=10^{-5}\), set \(\ell=\lfloor n^{1-\epsilon}\rfloor\le s_0\), \(T_\ell=T+[-\ell,\ell]^2_{\mathbb Z}\), \(X=A\cap T\), \(S_0=A\cap T_\ell\), and \(Y=S_0\setminus X\). At all sufficiently large scales the input guarantees \[I_\Omega(X:\Gamma\setminus S_0)\le Cn^{1-\epsilon},\qquad |S_0|\le Cn^2,\qquad |\partial_\Gamma X|,|\partial_\Gamma Y|,|\partial_\Gamma S_0|\le Cn.\]

The Proposition “Amplification” in (OpenAI 2026, sec. 10) requires these three bounds with any fixed common constant \(C_0\) and \(\alpha=k/(k+1)>1-10^{-6}\). Put \(\beta_0=1-2\cdot10^{-6}\). For every fixed integer \(M\ge1\) it supplies a radius \[r\le C_M n^{(1-\epsilon)/\alpha},\qquad \ell+2r=o(n^{\beta_0}),\qquad I_\Omega(X':E)\le C_M n^{-M}\] simultaneously for every \(X'\subseteq X\) and \(E\subseteq\Gamma\setminus N_{2r}(S_0)\). The radius and constants are independent of the remote set and total volume. If also \(\mathop{\mathrm{dist}}_\infty(T,Z_A)>\ell+n^{\beta_0}\), then \(N_{2r}(S_0)\subseteq A\). In Proposition 4 we take \(A=\Lambda\), so \(Z_A=\varnothing\), choose \(s_0,n\) comparable to the target scale with fixed ratios, and use \(M=1\). A graph neighborhood lies in the corresponding ambient enlargement; this is the precise conversion used to obtain the ambient-separation statement.

The two-family interface and its adaptation.

The Lemma “Two-family entropy cancellation” in (OpenAI 2026, sec. 11) states that, for a pure state and a partition \(A=D\mathbin{\dot\cup}\bigcup_iX_i\) into two ordered families, bounds \[I\left(X_i:A^c\cup\bigcup_{j<i,\,f_j=f_i}X_j\right)\le\delta_i \quad(f_i\in\{0,1\}) \quad\Longrightarrow\quad S(A)\le S(D)+\tfrac12\sum_i\delta_i.\] Its Proposition “Two families with separated birth templates” constructs such regions for every cut with \(b=|\partial_\Gamma A|>0\) and any fixed lower scale \(n_*\). It gives \(|D|\le Cb\), \(X_i\subseteq A\cap T_i\), admissible separated templates with \(n_i\ge n_*\) and \(\lfloor n_i^{1-\epsilon}\rfloor\le s_i\), and \[\mathop{\mathrm{dist}}_\infty(T_i,Z_A)>2n_i^{\beta_0},\qquad \mathop{\mathrm{dist}}_\infty\left(T_i,\bigcup_{j<i,\,f_j=f_i}X_j\right)>2n_i^{\beta_0}, \qquad \sum_i n_i^{-100}\le Cb.\] The constant may depend on \(n_*\) and the template constants but not on the cut or domain. These statements are used within the companion’s area-law proof. Lemma 5 adapts their ordering principle to a cell with arbitrary exterior conditioning and proves that local variant in full. In particular, we do not infer a conditional-information bound merely by conditioning an ordinary mutual-information inequality.

The companion’s replica limit is taken at each fixed finite physical instance and schedule, before the geometric scale tends to infinity. Its exported bounds concern the original one-copy state with the uniform constants above. The later tapering construction, adaptive patch constraints, density compression, encoded protocol, and routing are proved in this article.