An invariant map satisfies \(\mathcal{P}(A)\Pi _U=\mathcal{P}(A)\), since \(\Pi _U\) is the average of the \(U_g\). If \(\mathcal{P}(A)\) is injective on the invariant subspace, choose a linear map \(L\) into the invariant subspace with \(L\, \mathcal{P}(A)x=x\) for invariant \(x\); then \(L\, \mathcal{P}(A)=L\, \mathcal{P}(A)\Pi _U=\Pi _U\). Conversely, if \(L\, \mathcal{P}(A)=\Pi _U\) and \(\mathcal{P}(A)x=0\) for an invariant \(x\), then \(x=\Pi _Ux=L\, \mathcal{P}(A)x=0\).

Every vector is invariant under the trivial representation, and \(\mathcal{P}(A)\) is the linear map whose columns are the vectors \(A^{\bullet }_{\alpha \beta \gamma \delta }\).

The map \(\sigma \) is the average of the conjugation action \(X\mapsto U_gXU_g^{-1}\), so it is a projection onto the operators fixed by every conjugation, and these are the operators commuting with every \(U_g\). For a unitary representation, \(\operatorname{tr}[Y^\dagger U_gXU_g^\dagger ]=\operatorname{tr}[(U_g^\dagger YU_g)^\dagger X]\), and summing over \(g\), with \(g\) replaced by \(g^{-1}\), gives the symmetry of \(\sigma \).

The operator \(\sum _g\chi _i(g^{-1})D^j(g)\) commutes with every \(D^j(h)\), because characters are constant on conjugacy classes, so by Schur’s lemma it is a multiple of the identity; its trace is \(\sum _g\chi _i(g^{-1})\chi _j(g)\), which orthogonality of characters evaluates. By Maschke’s theorem \(U\) is a direct sum of irreducible subrepresentations. On each summand the projectors act as just described, which gives \(\sum _iP_i=\mathbb {1}\), and the trace of \(U_kP_i\) is the sum of \(\chi _i(k)\) over the summands with character \(\chi _i\). Orthogonality of characters counts these summands as \(m_i\).

Only the projector of \(\chi _i\) acts nontrivially on such a subrepresentation, and it acts as the identity.

By the trace formula for the projectors, \(\operatorname{tr}[U_k\Delta ]=\frac{1}{|G|}\sum _i d_i\chi _i(k)\), so \(\sum _k\operatorname{tr}[U_{k^{-1}}\Delta ]U_k=\sum _iP_i=\mathbb {1}\). Substituting \(h=kg\) gives \(\sum _h\operatorname{tr}[U_{gh^{-1}}\Delta ]U_h=\big(\sum _k\operatorname{tr}[U_{k^{-1}}\Delta ]U_k\big)U_g =U_g\), and the first expression equals the second by cyclicity of the trace.

If \(e\) intertwines \(\rho \) with \(\sigma \) and is invertible, every nonzero intertwiner \(f\) from an irreducible representation into \(\rho \) gives a nonzero intertwiner \(ef\) into \(\sigma \). Apply the same argument to \(e^{-1}\) for the converse.

The character of the left-regular representation is \(|G|\delta _{k,1}\), so every character \(\chi _i\) has multiplicity \(d_i\) in it, and \(\Delta =\frac{1}{|G|}\sum _iP_i=\frac{1}{|G|}\mathbb {1}\). Every irreducible representation occurs in the left-regular representation, and, since \(U\) is semi-regular, in \(U\); hence \(U\) and the left-regular representation have the same irreducible characters. Comparing the two expressions for \(\operatorname{tr}[U_k\Delta ]\) in the left-regular representation gives \(\sum _id_i\chi _i(k)=|G|\delta _{k,1}\), and therefore \(\operatorname{tr}[U_{g^{-1}}U_h\Delta ]=\frac{1}{|G|}\sum _id_i\chi _i(g^{-1}h) =\delta _{g,h}\).

The trace-dual identity gives independence for a semi-regular representation. Conversely, if an irreducible character \(\chi \) is absent, its character projector vanishes:

Indeed, the projector vanishes on every irreducible summand of \(V\). Independence in (1) would imply \(\chi (1)=0\), contrary to the positive dimension of the irreducible representation. The standard basis of the group algebra identifies the last assertion with independence of the group operators.

Inversion permutes the group indices, and transposition is injective on endomorphisms of a finite-dimensional vector space. It therefore preserves independence of the group operators. Conversely, independence of their transposes implies independence of the original operators.

A dependence among the tensor-product operators is a dependence among the tensors \(\rho (g)\otimes \sigma (g)\), by the natural identification of their endomorphism spaces. Applying the trace-dual functional at \(h\) to the first factor gives \(c_h\sigma (h)=0\). Since \(\sigma (h)\) is invertible on a nonzero vector space, \(c_h=0\). The group-algebra criterion proves semi-regularity. Interchanging the tensor factors gives the other order.

The annihilator of an invariant dual subspace is invariant in the original representation, since the group action on the dual uses the inverse element. Irreducibility makes this annihilator zero or the whole space. Finite-dimensional annihilator duality then makes the original dual subspace the whole space or zero, respectively.

A one-dimensional representation is irreducible, so the semi-regular representation \(V\) contains the trivial representation and has a nonzero invariant vector \(v\). If \(D\) is irreducible and \(f\) is a nonzero intertwining map from \(D\) to \(U\), then \(x\mapsto f(x)\otimes v\) is a nonzero intertwining map from \(D\) to \(U\otimes V\).

Since \(\operatorname{tr}[A^iU_gXU_g^{-1}]=\operatorname{tr}[U_g^{-1}A^iU_gX]\), invariance of \(\mathcal{P}(A)\) under conjugation by \(U_g\) says that \(U_g^{-1}A^iU_g\) and \(A^i\) have the same trace pairing with every operator \(X\); pairing with the rank-one operators \(|v\rangle \langle f|\) shows that they are equal. The operators fixed by conjugation are the commutant, and a linear map is injective on a subspace if and only if it has a left inverse there.

Invariance of \(C\) follows from \(U_gA^iB^jU_g^{-1}=(U_gA^iU_g^{-1})(U_gB^jU_g^{-1})\). For the left inverse, first \(\sum _i\operatorname{tr}[A^iY]\, L_A|i\rangle =\sigma (Y)\) for every \(Y\). Second, for every operator \(M\),

writing \(M\) in a basis \(\{ b_k\} \) with coordinate functionals \(b^*_k\), the left-hand side applied to \(w\) is \(\sum _k\sigma \big(|w\rangle \langle b^*_k|\big)Mb_k\), which is the right-hand side applied to \(w\). Taking \(Y=B^jX\) in the first identity and \(M=\Delta U_g\) in the second,

the last step by the expansion \(\sum _h\operatorname{tr}[U_{h^{-1}}\Delta U_g]U_h=U_g\).

Induction on \(L\): the blocked tensor on \(L+1\) sites is the concatenation of \(A\) with the blocked tensor on \(L\) sites, up to relabelling the physical index, and relabelling preserves \(G\)-injectivity.

Invariance of \(C\) is invariance of the map \(X\mapsto (\operatorname{tr}[C^iX])_i\) under conjugation, and an invariant map does not see the average over the group. For the second claim, \(U_gYZU_g^{-1}=(U_gYU_g^{-1})Z\).

Summing over \(j\) first, \(\sum _j\operatorname{tr}[B^jY_k]\, L_B|j\rangle =L_B\mathcal{P}(B)Y_k=\sigma (Y_k)\).

The right-hand side lies in both sets, with \(M^k=A^kX\) and \(N^i=XA^i\). Conversely, let \(\operatorname{tr}[A^iB^jM^k]=\operatorname{tr}[N^iB^jA^k]\) for all \(i,j,k\). Fix left inverses with \(L_A\mathcal{P}(A)=L_B\mathcal{P}(B)=\sigma \), and let \(L_C\) be the left inverse of \(C^{jk}=B^jA^k\), so that \(L_C\mathcal{P}(C)=\sigma \). For fixed \(i\) the vector \((\operatorname{tr}[B^jA^kN^i])_{j,k}\) equals \((\operatorname{tr}[B^jM^kA^i])_{j,k}\), and applying \(L_C\) gives

because \(A^i\) commutes with every \(U_g\). Since \(C\) is invariant, \(\operatorname{tr}[N^iB^jA^k]=\operatorname{tr}[B^jA^k\sigma (N^i)]=\operatorname{tr}[A^iB^jA^kX]\).

The right-hand side lies in both sets, with \(M=N=\sum _g\lambda _gU_g\), since \(U_g\) commutes with \(A^j\). Conversely, let \(\operatorname{tr}[B^iA^jM]=\operatorname{tr}[B^iNA^j]\) for all \(i,j\), and let \(L_C\) be the left inverse of \(C^{ij}=B^iA^j\). Applying \(L_C\) to the vector \((\operatorname{tr}[B^iNA^j])_{i,j}\), and then the identity \(\sum _j(L_A|j\rangle )\, Y\, A^j=\frac{1}{|G|}\sum _h\operatorname{tr}[U_{h^{-1}}Y]\, U_h\) with \(Y=\Delta \sigma (N)\),

Since \(C\) is invariant, \(\operatorname{tr}[B^iA^jM]=\operatorname{tr}[B^iA^j\sigma (M)]\), and \(\operatorname{tr}[U_{h^{-1}}\Delta \sigma (N)]=\operatorname{tr}[U_{h^{-1}}\sigma (N)\Delta ]\) because \(\sigma (N)\) commutes with \(U_{h^{-1}}\).

Fixing the last letter \(k\) writes the vector as \(\operatorname{tr}[A^iB^wM^k]\) and fixing the first letter \(i\) as \(\operatorname{tr}[N^iB^wA^k]\), where \(B\) is the tensor of \(K\) blocked sites, which is \(G\)-injective by Corollary 26.1.20. The intersection property gives the form \(\operatorname{tr}[A^iB^wA^kX]\). Iterating along the non-wrapping windows of the ring gives the second claim, as in the proof of Theorem 3.5 of the source.

Each \(U_g\) commutes with every \(A^i\), so it can be moved to any bond, and every window of \(|\mathcal M(A|U_g)\rangle \) lies in the local ground space. Conversely, by Lemma 26.1.25 a vector of \(\mathcal G_{N,L}(A)\) is \(\operatorname{tr}[A^{\sigma _1}\cdots A^{\sigma _N}X]\), and its translate by one site is \(\operatorname{tr}[A^{\sigma _1}\cdots A^{\sigma _N}Y]\). Hence \(\operatorname{tr}[A^aB^wX]=\operatorname{tr}[A^aYB^w]\), where \(B\) is the tensor of the last \(N-1\) blocked sites. The closure property, applied to \(A\) and \(B\), writes the vector as a combination of the states \(|\mathcal M(A|U_g)\rangle \).

On an irreducible subrepresentation with character \(\chi _i\) the operator \(\sum _hU_{hgh^{-1}}\) commutes with the representation, so by Schur’s lemma it is a scalar, and its trace \(|G|\chi _i(g)\) fixes the scalar as \(|G|\chi _i(g)/\chi _i(1)\); the projector \(P_i\) acts there as the identity and every other projector as zero. Evaluating \(\sum _i\mu _iP_i\) on a nonzero vector of such a subrepresentation gives \(\mu _i\) times the vector. For semi-regular \(U\), the functional \(X\mapsto \operatorname{tr}[U_{g^{-1}}X\Delta ]\) takes the value \(\delta _{g,h}\) on \(U_h\) by Theorem 26.1.11; on \(\sigma (U_k)\) it is \(\frac{1}{|G|}\) times the number of \(h\) with \(hkh^{-1}=g\), which vanishes unless \(k\) is conjugate to \(g\) and is positive for \(k=g\).

Since \(A\) is invariant, \(|\mathcal M(A|K)\rangle =|\mathcal M(A|\sigma (K))\rangle \) for every closure \(K\); in particular the closures of one conjugacy class give the same state. By Lemma 26.1.27 each \(|\mathcal M(A|U_g)\rangle \) is a combination of the states \(|\mathcal M(A|P_i)\rangle \), and conversely each \(P_i\) is a combination of the \(U_g\), so Theorem 26.1.26 gives the spanning claims. The tensor of \(N\) blocked sites is \(G\)-injective, so \(K\mapsto |\mathcal M(A|K)\rangle \) is injective on the operators commuting with every \(U_g\). These contain the projectors \(P_i\) and the twirls \(\sigma (U_g)\), and Lemma 26.1.27 gives the linear independence of both families. Computing the dimension of \(\mathcal G_{2,2}(A)\) in the two ways gives \(I\) equal to the number of conjugacy classes. The source instead quotes that the numbers of conjugacy classes and of irreducible representations agree, and passes from the \(U_g\) to the \(P_i\) through the completeness of the characters in the class functions.

Expanding \(\tau ^*(g)e_i^*\) in the dual basis, \((\tau ^*(g)\otimes \tau (g))\, \omega =\sum _j e_j^*\otimes \tau (g)\sum _ie_i^*\big(\tau (g^{-1})e_j\big)e_i =\sum _je_j^*\otimes e_j\). The pairing of \(\omega \) is \(\sum _ie_i^*(Me_i)=\operatorname{tr}M\), and the pairing of \((\tau ^*(g)\otimes \tau (h))\, \omega \) is \(\sum _ie_i^*\big(\tau (g^{-1})M\tau (h)e_i\big)\). For \(M=\Delta \) this is \(\operatorname{tr}[U_hU_{g^{-1}}\Delta ]=\delta _{g,h}\) by Theorem 26.1.11.

Invariance of \(A\) gives \(\mathcal{P}(A)(\sigma _A(g)x\otimes f)=\mathcal{P}(A)(x\otimes \tau ^*(g^{-1})f)\) and similarly for \(B\), so acting with \(\sigma _A(g)\otimes \sigma _B(g)\) on the virtual system of \(C\) acts with \(\tau ^*(g^{-1})\otimes \tau (g^{-1})\) on \(\omega \), which leaves it invariant by Lemma 26.1.29. For the left inverse, \(L_A\mathcal{P}(A)(x\otimes f) =\frac{1}{|G|}\sum _g\sigma _A(g)x\otimes \tau ^*(g)f\) and \(L_B\mathcal{P}(B)(e\otimes y) =\frac{1}{|G|}\sum _h\tau (h)e\otimes \sigma _B(h)y\). Contracting through \(\Delta \) gives

by Lemma 26.1.29. Theorem 26.1.2 turns invariance and the left inverse into \(G\)-injectivity, and Theorem 26.1.16 gives the last claim.

The group elements attached to the two contracted legs cancel, because \(\omega \) is invariant under \(\tau ^*(g)\otimes \tau (g)\). Hence \(\Pi _{\sigma \otimes \tau ^*\otimes \tau }(x\otimes \omega ) =\frac{1}{|G|}\sum _g\sigma (g)x\otimes \omega \), and pairing \(\omega \) through \(M\) gives \(\operatorname{tr}M\). By Theorem 26.1.11 with \(g=h=1\), \(\operatorname{tr}\Delta =1\) for semi-regular \(\tau \).

Invariance of \(\mathcal{P}(A)\) under conjugation is equivalent to \(L_gA^iL_g^{-1}=A^i\), and \(c^{-1}\mathcal{P}(A^\dagger )\mathcal{P}(A)=\sigma \); the left-inverse form of \(G\)-injectivity applies.

By Theorem 26.1.11, \(\Delta =\frac{1}{|G|}\mathbb {1}\) for the left-regular representation, that is, \(|G|\, \Delta =\mathbb {1}\). Substitute this in the two left inverses.

By Lemma 26.1.34 the left inverse built from \(c_A^{-1}\mathcal{P}(A^\dagger )\) and \(c_B^{-1}\mathcal{P}(B^\dagger )\) sends \(|i,j\rangle \) to \((c_Ac_B)^{-1}(B^j)^\dagger (A^i)^\dagger =(c_Ac_B)^{-1}(A^iB^j)^\dagger \), and by Theorem 26.1.19 it is a left inverse of \(\mathcal{P}(C)\). Invariance of \(C\) holds as for \(G\)-injective tensors.

Unitarity gives \(\langle \rho (g)x,y\rangle =\langle x,\rho (g^{-1})y\rangle \); averaging and replacing \(g\) by \(g^{-1}\) proves self-adjointness. Since \(T\Pi _\rho =T\), the restricted inner-product identity gives \(\langle Tx,Ty\rangle =c\langle x,\Pi _\rho y\rangle \), which is equivalent to the adjoint identity. Conversely, apply the identity to invariant vectors, on which \(\Pi _\rho \) is the identity.

The inverse change of coordinates is \((a,(c_i)_i)\mapsto (a,(a^{-1}c_i)_i)\). Thus simultaneous multiplication by \(g\) sends \((a,(ab_i)_i)\) to \((ga,(gab_i)_i)\), and in relative coordinates only the first label changes. The permutation matrices are unitary, and the matrix of a product permutation is the Kronecker product of the factor matrices.

Write \(T(a,b)=(a,ab)\), with multiplication coordinatewise on \(b\in G^K\). Cancellation in \(G\) gives

Expanding the actual graph contraction into vertex group labels gives

Substitution of (7) extracts the Bell factors independently of \(q\), proving (6). The coordinate map is a permutation, so it preserves inner products. Summing over the two independent endpoint registers on each edge gives \(\langle B_K,B_K\rangle =D^{|E(\Gamma )|}\), proving the stated normalization.

The average of the unitary representation is self-adjoint. Local \(G\)-isometry gives \(T^*T=cP_\rho \) for some \(c{\gt}0\). Consequently \(JT=\sqrt c\, P_\rho \). For \(x=Tu\) and \(y=Tv\), both \(\langle Jx,Jy\rangle \) and \(\langle x,y\rangle \) equal \(c\langle u,P_\rho v\rangle \). Restriction of the same linear map to the range, with its induced inner product, gives the stated isometry.

For each virtual edge assignment \(\eta \), form the basis vector whose incident label at \((v,e)\) is \(\eta _e\). Applying \(\mathcal T\) to the sum of these basis vectors gives exactly \(\Psi _T\), proving range membership. Interchanging finite sums applies \(F_v\) separately to each site column, proving \(\mathcal F\Psi _T=\Psi _{FT}\). The same finite product-of-sums identity gives composition of product matrices. The local Gram equalities therefore imply \((\mathcal F\mathcal T)^*(\mathcal F\mathcal T)=\mathcal T^*\mathcal T\), which gives the inner-product statement on the actual product range.

On an outgoing leg, the transposed inverse gives \((M_{g^{-1}})_{\eta ,\sigma }=\delta _{\sigma ,g\eta }\), exactly as on an incoming leg. Multiplying the indicators gives simultaneous translation of the full incident configuration. Translation permutes its basis and hence preserves all inner products.

Apply the normalized-adjoint construction at each vertex, using the derived unitarity of the bundled incident action. It gives \(J_vT_v=\sqrt{c_v}P_v\), where \(P_v\) is the canonical averaging tensor, and preserves inner products on the local range. Testing on site columns gives \((J_vT_v)^*(J_vT_v)=T_v^*T_v\), so the product operation is isometric on \(S\). Actual contraction transports it to \((\prod _v\sqrt{c_v})\Psi _K\). Physical bond separation then gives the asserted normalized Bell product and its scalar. Restricting this same explicit map to \(S\) gives \(I\); membership of \(\Psi _a\) in \(S\) was derived from the contraction and is not an additional assumption.

Reindex the fine horizontal and vertical sums by the displayed bijection. The remaining internal sum separates as \(\sum _{x:C\to X^4}\prod _v f_v(x_v)=\prod _v\sum _{x_v\in X^4}f_v(x_v)\). Its factor at \(v\) is precisely \(B_v(\alpha _v,\tau _v)\).

The physical label map is bijective, so its matrix and the inverse permutation matrix preserve inner products. Evaluating its action on each coefficient gives the exact reblocking equality.

Every coarse graph edge is uniquely a horizontal or vertical outgoing edge. Reindexing its label sum yields the bond-indexed contraction. Apply the exact fine-to-coarse reblocking equality.

The map \(E\) carries invariant vectors to invariant vectors. The identity \(FT'=TE\) transfers invariance and injectivity on the invariant subspace. Since \(E\) and \(F\) permute orthonormal bases, \(\langle T'x,T'y\rangle =\langle TEx,TEy\rangle =c\langle Ex,Ey\rangle =c\langle x,y\rangle \) for invariant \(x,y\).

The internal bond shared by corners \(0\) and \(1\) gives \(q_0y_1=x_1=q_1y_1\), hence \(q_0=q_1\) by right cancellation. The bonds shared by corners \(1,2\) and \(2,3\) similarly give \(q_1=q_2=q_3\). With this common translation, the sixteen fine-leg entries determine exactly the eight boundary labels and four internal labels. The converse follows by simultaneous translation of every label. For each \(y\in G^4\) and each \(g\) with \(\alpha =g\beta \), the unique remaining assignment is \(x=gy\) and \(q_i=g\). There are \(|G|^4\) choices of \(y\).

The local isometries give \(\sum _s\overline{a_i(\eta ;s)}a_i(\theta ;s) =(c_i/|G|)\sum _g\delta _{\eta ,g\theta }\). Expanding both internal contractions and summing the four independent physical indices gives

The translation-counting theorem cancels the factor \(|G|^{-4}\) and gives (18). Since \((P_8)_{\alpha ,\beta }=|G|^{-1}\sum _g\delta _{\alpha ,g\beta }\), the block factor is \(|G|\prod _i c_i\). The extra factor \(|G|\) is contributed by the internal cycle with unnormalized bonds. This retains the scalar omitted from the diagrams in [ 235 , Lemma 6.2 and Observations 6.5–6.6 ] .

Translate all four internal summation variables by \(g\). Local virtual invariance then gives \(B(g\alpha ;\sigma )=B(\alpha ;\sigma )\). The Gram identity is \(B^*B=c_BP_8\) with \(c_B{\gt}0\), so \(c_B^{-1}B^*\) is a left inverse on the invariant boundary subspace. For invariant \(x,y\), it also gives \(\langle Bx,By\rangle =c_B\langle x,y\rangle \). Separating each paired leg merely reindexes these formulas.

Reindex the four regular labels as an ordered quadruple. The coordinate bijection intertwines simultaneous translation, so it preserves \(G\)-isometry. The closure-sector nonvanishing theorem applies to this same tensor with identity closures. Its squared norm is positive because the common centralizer contains the identity. Identity closures put the identity matrix on every seam, giving the asserted native bond contraction. For \(w,h\geq 3\), its coefficient equals the incident-graph contraction coefficient under the four-leg enumeration.

The normalized adjoint is already isometric on \(\operatorname {ran}T\). Its image is contained in \(\operatorname {ran}P_\rho \) because \(JTu=\sqrt c\, P_\rho u\). Conversely, \(P_\rho u=JT(c^{-1/2}u)\), so it maps onto the canonical support.

Reindex every edge label by the displayed two-label bijection. Each boundary coordinate still has the same group value, so translation commutes with the bijection. Reindexing the local virtual sum gives the equality of physical maps.

The transposed inverse regular matrix on an outgoing leg has the same entries \(\delta _{a,gb}\) as the incoming matrix. The four-leg incident bijection therefore intertwines both representations and preserves their coordinate inner products. Apply the canonical support equivalence to the transported original tensor.

The explicit four-site Gram calculation gives G-isometry on all eight separate boundary labels. Transport it through the actual boundary coordinate equivalence.

The geometric bond bijection gives \(W\Psi _a=\Psi _B\). The graph contraction is the product of the local block maps applied to its virtual bond vector, hence \(\Psi _B\in \operatorname {ran}T_B\).

Apply physical support accessibility and regular bundle separation to the actual coarse graph tensors, whose G-isometry has been derived from the fine tensors. Compose its support isometry with the restricted physical regrouping \(W\). The exact state equality supplies the required state argument, and its support membership was derived separately. Each unnormalized surplus equality vector is \(\sqrt{|G|}\) times its normalized Bell vector, producing the displayed factor on all edges.

Translation changes \(u\) to \(gu\) and leaves \(r\) fixed, so invariance follows from the original tensor. Write \(E_v\eta =(u,r)\) and \(E_v\theta =(u',r')\). Summing the original and surplus physical indices separately gives

The last equality uses the bijection \(E_v\) and its translation rule. This is the averaging-projector Gram identity with factor \(c_v{\gt}0\), and hence proves the assertion on the invariant subspace.

On each virtual edge, make the bijective substitution \((u_e,z_e)=(u_e,u_er_e)\). The actual contraction then separates as

Split each vertex indicator into its individual incidences and regroup them by edges. The sum over \(r_e\) is \(\sum _{r_e}\delta _{\tau _{v,e},r_e}\delta _{\tau _{w,e},r_e} =\delta _{\tau _{v,e},\tau _{w,e}}\) independently at every edge.

Let \(J_A=c_A^{-1/2}A^*\) be the normalized adjoint and set \(F=c_B^{-1/2}BJ_A\). On the range of \(A\), the vector \(J_Ax\) is invariant, so G-isometry of \(B\) and isometry of \(J_A\) imply \(\langle Fx,Fy\rangle =\langle x,y\rangle \). Moreover, \(FA=\sqrt{c_A/c_B}\, BP_\rho =\sqrt{c_A/c_B}\, B\). The local column Gram equalities imply the same inner-product equality for the product physical map on the product range. Expanding its action on the graph contraction gives the displayed state equality.

The squared norm of the pair-indexed coefficients is the product of the two finite sums of squared absolute values. Thus \(\| x\| =\| Ix\| =r\| \psi \| \), and the positive scalar cancels in the normalized equality.

Every square root and every denominator is positive. Each surplus equality vector has its inverse-square-root dimension normalization, so their product has squared norm one and hence norm one.

The actual four-site block tensors are G-isometric from fine-site isometry. The target is the original tensor with its surplus identity registers, which is G-isometric for the same bundled representation. Physical support transport gives the first equality, with one ratio of local positive factors per block. The target graph contraction factors into the original coarse contraction and surplus equality vectors. Normalizing the latter contributes the Bell factor in \(R\). The identity closure sector proves nonvanishing of both original states. Their exact isometric norm relation then gives the second equality.

Choose the local normalized-adjoint maps before choosing \(K\). Their column Gram identities imply isometry on the product range. Expand each inserted contraction over its independent endpoint labels. Products of physical maps commute with this sum, and every summand belongs to the product range. The local tensor identities give the displayed equality with the same factors for every \(K\).

Reindex all fine bonds into four internal bonds per block and the two labels of every coarse crossing bond. The corner-by-corner endpoint identity retains the original heads of both orientations. Summing the internal labels gives exactly the four-site tensor. For a doubled periodic coordinate \(2x+b\), the outgoing bond crosses the seam precisely when \(b=1\) and \(x\) is the final coarse coordinate. Thus precisely the two crossing bonds, and no internal bond, receive each seam insertion.

Use the incidence-coordinate and endpoint bijections in the finite sums. A permutation matrix for left multiplication satisfies \(L_u^{\mathsf T}=L_{u^{-1}}\), also for simultaneous multiplication of the two labels. The resulting native and ordered contraction factors therefore agree at every endpoint, including periodic seam endpoints.

At a twisted head the relative label remains unchanged because \((ua)^{-1}(ub_k)=a^{-1}b_k\). The sum over distinguished labels gives the original inserted contraction, and the independent relative sums give the surplus equality vectors. Apply uniform physical support transport and retain their inverse-square-root normalizations.

Identify the four-coordinate tensor with its native four-leg form. Its closure self-overlap is a positive site factor times a positive power of \(|G|\) and the order of the common centralizer of the closure labels. This centralizer contains the identity. Support membership follows from the exact inserted coarse contraction and its product range membership.

Apply the uniform inserted support isometry between the actual four-site block tensor and the original tensor with surplus registers. Exact native seam coordinates identify its input and output. Their positive norm relation gives the normalized equalities. Linearity gives the coherent equality with the same \(I\) and \(R\); its norm relation is \(\| x\| =R\| y\| \), proving nonvanishing equivalence and normalization.

For a unitary representation, \(G\)-isometry with constant \(c\) means \(\mathcal{P}(A)^{\dagger }\mathcal{P}(A)=c\, \Pi \), since \(\mathcal{P}(A)\) is invariant and \(\Pi \) is self-adjoint. The Gram matrix of \(C\) is

Writing both projectors as averages over \(G\), the link indices contribute \(\sum _{g,g'}(L_{h})_{gg'}(L_{k})_{gg'}=|G|\, \delta _{h,k}\), which is the identity \(\Delta =|G|^{-1}\mathbb {1}\) for the left-regular representation, so the right-hand side is \(c_{A}c_{B}\, \Pi _{(a,b),(a',b')}\) for the projector \(\Pi \) of \(\sigma _{A}\otimes \sigma _{B}\). Hence \(\langle \mathcal{P}(C)x,\mathcal{P}(C)y\rangle =c_{A}c_{B}\langle x,y\rangle \) for invariant \(x,y\), which also gives injectivity on the invariant subspace. Entrywise, \(\mathcal{P}(C)^{\dagger }_{(a,b),(k,l)} =\sum _{g}\mathcal{P}(A)^{\dagger }_{(a,g),k}\mathcal{P}(B)^{\dagger }_{(g,b),l}\), so \(\mathcal{P}(C)^{\dagger }\) is the contraction of the adjoints. Invariance of \(C\) holds because the actions on the contracted link cancel, as in Theorem 26.1.30. Equivalently, Theorem 26.1.36 gives the left inverses \(c_A^{-1}\mathcal P(A)^\dagger \) and \(c_B^{-1}\mathcal P(B)^\dagger \). Lemma 26.1.34 identifies their contraction with \((c_Ac_B)^{-1}\mathcal P(C)^\dagger \), and Theorem 26.1.30 supplies its left-inverse equation.

Let \(L\) be a left inverse with \(L\mathcal{P}(A)=\Pi _U\). Since \(\mathcal{P}(A)\) is onto, every physical vector is \(\mathcal{P}(A)x\) with \(x\) invariant, \(Ly\) is invariant and \(\mathcal{P}(A)Ly=y\). For the first claim let \(W=\mathcal{P}(A)VL\). As \(V\) commutes with \(U_g\), it preserves the invariant subspace and commutes with \(\Pi _U\), so \(W\mathcal{P}(A)=\mathcal{P}(A)V\Pi _U=\mathcal{P}(A)\Pi _UV=\mathcal{P}(A)V\); and for invariant \(x,x'\), \(\langle W\mathcal{P}(A)x,W\mathcal{P}(A)x'\rangle =c\langle Vx,Vx'\rangle =c\langle x,x'\rangle =\langle \mathcal{P}(A)x,\mathcal{P}(A)x'\rangle \), so \(W\) is unitary. For the converse, \(V=LW\mathcal{P}(A)\) satisfies \(VU_g=V\) because \(\mathcal{P}(A)U_g=\mathcal{P}(A)\), and \(U_gV=V\) because \(V\) takes invariant values; \(\mathcal{P}(A)V=W\mathcal{P}(A)\) because \(\mathcal{P}(A)L\) is the identity, and the same computation as before shows that \(V\) preserves inner products of invariant vectors.

The entries satisfy \(\overline{\operatorname {tr}[A^{\tau }(A^{\sigma })^{\dagger }]} =\operatorname {tr}[A^{\sigma }(A^{\tau })^{\dagger }]\). From \(\Gamma _{n}\Gamma _{n}^{\dagger }\Gamma _{n}=c\, \Gamma _{n}\) one gets \(P_{n}^{2}=c\, P_{n}\), and \(P_{n}\Gamma _{n}X=c\, \Gamma _{n}X\) shows that the range of \(P_{n}\) is that of \(\Gamma _{n}\). A Hermitian idempotent is the orthogonal projector onto its range.

By Theorem 26.1.35 the two-site tensor \(C^{ab}=A^{a}A^{b}\) is \(G\)-isometric with some constant \(c\), so \(\mathcal{P}(C^{\dagger })\mathcal{P}(C)=c\, \sigma \). Since \(\mathcal{P}(C)\) is invariant, \(\mathcal{P}(C)\sigma =\mathcal{P}(C)\), and \(\Gamma _{2}\Gamma _{2}^{\dagger }\Gamma _{2}=\mathcal{P}(C)\mathcal{P}(C^{\dagger }) \mathcal{P}(C)=c\, \Gamma _{2}\). Theorem 26.1.86 applies.

Let \(\mathcal{P}(A^{\dagger })\mathcal{P}(A)=c\, \sigma \). For an operator \(N\) commuting with every \(L_{g}\) this gives \(\sum _{b}\operatorname {tr}[A^{b}N]\, (A^{b})^{\dagger }=c\, N\), hence \(\sum _{b}\operatorname {tr}[M(A^{b})^{\dagger }]\operatorname {tr}[A^{b}N] =c\operatorname {tr}[MN]\) for every \(M\). The matrices \(A^{i}\) commute with every \(L_{g}\), and so do their adjoints since \(L_{g}\) is unitary; so does every product of them. The matrix entry of the first product is

which by cyclicity of the trace is of this form with \(M=(A^{w_{0}})^{\dagger }A^{s_{0}}A^{s_{1}}\) and \(N=A^{s_{2}}(A^{w_{2}})^{\dagger }(A^{w_{1}})^{\dagger }\), so it equals \(c\operatorname {tr}[A^{s_{0}}A^{s_{1}}A^{s_{2}} (A^{w_{0}}A^{w_{1}}A^{w_{2}})^{\dagger }]\); the second product is treated in the same way. The sum over \(g\) in the twirl drops out because \(N\) commutes with every \(L_{g}\), which is the step in which the source moves the group action from one bond to the other. By Theorem 26.1.88 the local terms are \(\mathbb {1}-c_{2}^{-1}P_{2}\), so the two translated terms on three sites commute; the commutation transports to every three consecutive sites of a ring of \(N\geq 3\) sites, and terms with disjoint supports commute.

Associativity proves \((k_v\alpha _{v,e})r_e=k_v(\alpha _{v,e}r_e)\). Right cancellation preserves each flatness equality under \(T_r\). If \(q\) witnesses flatness of \(\alpha \), then \(kq\) witnesses flatness of \(L_k\alpha \); apply the inverse translation for the converse. A diagonal indicator commutes with a permutation preserving its support. Averaging gives the assertions involving \(P_R\) and \(B_{R,e}\). For \(e\ne f\), the right translations act on distinct half-edge coordinates; for \(e=f\), the two averages are the same operator.

On an internal edge the two labels become \(1\eta _e\) and \(u_e\eta _e\); the boundary labels do not change. Apply the assumed invariance.

Expand \(P_R\) as \(|G|^{-|R|}\sum _k U_k\). Each summand fixes \(f\), proving (39). Reindex its sum over \(\beta \) by the bijection \((u,\eta )\mapsto b(u,\eta )\). Theorem 26.1.1.5 replaces the coefficient by \(f(b(u,\eta |_{\partial E_R}))\). Grouping reference assignments with the same boundary restriction gives exactly the shared-bond sum defining \(\Gamma _R(u)|\theta \rangle \).

Change the incident bond variable to \(k_v^{-1}\eta _e\) when the tail belongs to \(R\), and to \(k_w^{-1}u_e\eta _e\) otherwise. This is a bijection and restricts to \(\tau _{k,u}\) on the boundary. Internally the transformed head label is \((k_w^{-1}u_ek_v)(k_v^{-1}\eta _e)=k_w^{-1}u_e\eta _e\). Left invariance of each local regular projector therefore preserves each product in the bond sum. If the residuals are \(1\), all new insertions are \(1\). For \(b(u,\theta )\), the flatness equations read \(k_v^{-1}=k_w^{-1}u_e\), which give \(k_w^{-1}u_ek_v=1\).

Apply the expansion (40). If a coefficient is nonzero, flat support supplies a vertex gauge removing its internal insertions. By Theorem 26.1.1.7, its column is \(\Gamma _R|\theta '\rangle \) for a transported boundary label \(\theta '\). Every summand therefore lies in \(\operatorname {range}\Gamma _R\).

Conversely, each original column is left invariant by the local regular average and right invariant by a change of its shared internal bond variables. Its coefficient can be nonzero only if \(\alpha _{v,e}=k_v\eta _e\) for some vertex and shared-edge labels. Those labels witness flatness. Linear combinations preserve all three conditions.

For a right-edge permutation matrix \(M_g\) one has \(M_gM_h=M_{hg}\) and \(M_g^*=M_{g^{-1}}\). Reindexing \(h\mapsto hg\) in the double average gives \(B_{R,e}^2=B_{R,e}\); reindexing \(g\mapsto g^{-1}\) gives \(B_{R,e}^*=B_{R,e}\). The vertex average is the finite-group average of the unitary representation \(\rho _v\). Each flatness matrix is diagonal with entries \(0\) or \(1\). Products of commuting orthogonal projections are orthogonal projections. If \(P\) and \(Q\) are commuting idempotents and \(PQx=x\), then \(Px=P^2Qx=PQx=x\) and \(Qx=PQ^2x=x\). Applying this identity successively gives the stated common fixed space.

Different vertex averages act on different coordinates. Different edge averages also act on different coordinates, even when their edges share a vertex. Coincident averages are identical. Every vertex translation commutes with every shared right translation by (36).

Flatness on \(R\) is preserved by arbitrary shared right translations: both sides of each internal equality acquire the same right factor. It is also preserved by arbitrary vertex translations, by changing its witnessing gauge. Thus \(D_R\) commutes with every \(A_v\) and every \(B_e\), including those outside \(R\). Two flatness projectors commute because they are diagonal. Every factor in \(Q_R\) therefore commutes with every factor in \(Q_S\).

Apply the fixed-vector identity to each column of \(Q\). An idempotent fixes its range. For orthogonal projectors with equal ranges, \(PQ=Q\) and \(QP=P\); taking adjoints in the first identity gives \(QP=Q\), hence \(P=Q\).

Restriction to \(X\) and \(C\) commutes with the sitewise bijections. Thus conjugating \(K\otimes I_C\) by \(E_X\otimes E_C\) gives \((E_XKE_X^*)\otimes I'_C\), proving (45). In the partial trace, \(E_C\) only permutes the summed complementary configuration, while \(E_X\) relabels the two retained indices. This gives (46) for the same vector \(\xi \). These are the coordinate identities for the identity-extension and partial-trace conventions used by the regularized patch problem.

Multiply the coefficient formula in Definition 26.1.2.3 and sum over the middle coordinates; the complementary Kronecker delta leaves one slice. For \(R\subseteq U\), the two delta conditions on \(U\setminus R\) and \(V\setminus U\) combine to equality on \(V\setminus R\). Extend both operators first to \(R\cup S\) and then take their product, difference, or commutator there.

Regroup the coordinates as \(\mathcal H_V=\mathcal H_R\otimes \mathcal H_{V\setminus R}\). For each complementary configuration \(\tau \in X_{V\setminus R}\), put \(\xi _\tau (\alpha )=\xi (\alpha ,\tau )\) for \(\alpha \in X_R\). The coordinate decomposition is an isometry, and each slice satisfies \((\widehat K_R\xi )_\tau =K\xi _\tau \). Therefore

Taking nonnegative square roots gives the bound. The coefficient formula \(\widehat K_R((\alpha ,\tau ),(\beta ,\rho )) =K(\alpha ,\beta )\delta _{\tau ,\rho }\) shows that the extension depends linearly, and hence continuously, on \(K\).

Split the product of entries over \(R\) and its complement to obtain (47). Tensoring the regional intertwining identity with \(F_{V\setminus R}\) proves its global form. Choose a nonzero entry \(L_{ij}\) and cancel it from each entry \(M_{ab}L_{ij}=0\). Choosing one nonzero entry of every \(F_v\) gives a nonzero product entry; the empty product is \(1\).

Adjoin coordinates one at a time, writing a function on an enlarged set as its new single-coordinate factor times the previous function. For a finite-group representation \(\rho \), the average \(|G|^{-1}\sum _g\rho (g)\) fixes exactly its invariant vectors. Inverse right multiplication is a representation because \((gh)^{-1}=h^{-1}g^{-1}\); reindexing by inversion gives the stated edge average. Apply single-coordinate generation to the vertex and edge assignments. Finally \(D_Rx=x\) means \(\mathbf1_{\mathrm{flat}}(\alpha )x(\alpha )=x(\alpha )\).

Apply the regional invariant-and-flat characterization to each slice. The actions in Definition 26.1.2.9 leave that slice’s exterior label unchanged. The fixed space of the commuting product \(Q_R\) is therefore exactly \(\widehat{\mathcal S}_R^0\). Both \(Q_R\) and \(\widehat{P_R^0}\) are orthogonal projections onto this subspace, so they are equal. Commutation of \(Q_R,Q_S\) gives the last identity.

Set \(B=AQA^*\). Under the first hypotheses, \(B^2=cAQEQ A^*=cB\) and \(B^*=B\). Under \(QE=Q\), \(c^{-1}AQA^*A=AQE=AQ\). Each value \(c^{-1}AQA^*x\) lies in \(A(\operatorname {ran}Q)\), while \((c^{-1}AQA^*)AQx=AQ^2x=AQx\) gives the reverse inclusion.

The physical product map takes each canonical open-region column to the original tensor’s column with the same boundary labels. Their spans give \(\mathcal S_R=T_R(\mathcal S_R^0)\). Every vector in \(\mathcal S_R^0\) is fixed by the product of local regular averages, proving \(E_RP_R^0=P_R^0\); adjoints give \(P_R^0E_R=P_R^0\). Tensor the local Gram identities and apply Theorem 26.1.2.11. The resulting orthogonal projector has range \(\mathcal S_R\), so uniqueness identifies it with \(P_R\) and gives (49).

The constant function \(1\) is a nonzero invariant vector, even when the incident-leg set is empty; hence \(E_v\ne 0\). The Gram identity gives \(L_vT_v=E_v\), and \(T_vE_v=T_v\) gives \(J_vT_v=T_v\) and \(J_v^2=J_v\). The scalar \(c_v\) is real, so \(J_v^*=J_v\). If \(J_v=0\), then \(T_v=J_vT_v=0\), contradicting \(L_vT_v=E_v\ne 0\). Finally \(J_RT_R=T_R\) fixes every vector of \(\mathcal S_R=T_R(\mathcal S_R^0)\) and therefore gives \(J_RP_R=P_R\).

Taking adjoints gives right support. Every partial site product on \(R\) fixes \(J_R\), hence fixes \(H_R\) on both sides. Extend that equality through the complementary product to obtain commutation with \(M_U\). Use \(M_{R\cup S}=M_RM_S\), commute \(M_R\) past \(\widehat H_S\), and absorb each regional factor.

Put \(U=R\cup S\) and \(D=[\widehat H_R,\widehat H_S]\). Extend both intertwining identities to obtain \(DA_V=A_V[\widehat K_R,\widehat K_S]=0\). Multiplying by \(L_V\) gives \(DJ_V=0\), while support gives \(DM_U=D\). Write \(D=\widehat F_U\). Splitting across \(U\) gives

Each \(J_v\ne 0\), so \(J_{V\setminus U}\ne 0\). Cancel this factor in (50) to get \(FJ_U=0\) and then \(D=0\).

By definition \(h_R\) projects orthogonally onto \(\mathcal S_R^\perp \). Thus \(I-h_R\) is an orthogonal projector with range \(\ker h_R=\mathcal S_R\), and uniqueness identifies it with \(\Pi _{\mathcal S_R}\). Both extension conventions have coefficient \(h_R(\alpha ,\beta )\delta _{\tau ,\rho }\). Extension preserves \(I\) and subtraction.

Local \(G\)-isometry supplies \(L_v,J_v\) with \(T_vL_v=J_v\ne 0\) and \(J_RP_R=P_R\). Equation (49) gives \(P_RT_R=T_RP_R^0\), and (48) gives commuting virtual extensions. Apply Theorem 26.1.2.16 with \(A_v=T_v\). Finally \([I-\widehat P_R,I-\widehat P_S]=[\widehat P_R,\widehat P_S]=0\). The complementary-factor argument applies in the full physical space, even when the site maps have proper images.

Restrict a flatness potential to obtain flatness on a subregion. Conversely, let \(k_R,k_S\) witness flatness on \(R,S\). Set \(t=k_S(b)^{-1}k_R(b)\) and replace \(k_S(v)\) by \(k_S(v)t\). The new potentials agree at \(b\), and for each internal edge \(e=(v,w)\) of \(S\) one has

Thus the potentials define one potential on \(R\cup S\). An edge internal to neither region would join their disjoint parts, contrary to the hypothesis, so every required equality holds.

Replace simultaneous vertex and edge invariance in the actual range characterization by the corresponding individual fixedness conditions. For \(R\subseteq S\), every vertex and internal edge of \(R\) belongs to \(S\), and flatness on \(S\) implies flatness on \(R\).

A vector in both ranges is fixed by all vertex averages on \(R\cup S\). Every internal edge belongs to one of the two regions, so all required edge averages also fix it. A nonzero coefficient must be flat on both regions and hence, by Theorem 26.1.3.1, on their union. The atomic characterization gives membership in the union range. The reverse inclusion follows from nesting.

A complementary slice of \(J_Tx\) depends only on its labels in \(T\setminus R\), and every assignment of those labels extends to the whole complement by setting exterior labels to the group identity. This proves the pullback formula. For \(R=T\) each slice is simply \(x\). Taking the inverse image of (53) under \(J_T\) and using that inverse images preserve intersections gives (54).

Count the edges in (55). The overlap is \(\{ 1\} \), the union is \(T\), and the only possible edge between the disjoint parts would be \(02\), which is absent. Apply Theorems 26.1.3.3 and 26.1.3.5.

The weight of a labelling vanishes unless the two ends of every bond carry the same index, and the labellings with equal ends correspond to the assignments of one index per bond.

Expand each product \(P\, O\, Q\) into its entries. The resulting sum runs over the labellings of the bond ends and over new intermediate indices; exchanging the roles of the two families, the intermediate indices become the labelling of the bond ends and the old labels become the summation indices of the absorbed matrices at the sites. The factors \(P_{v+(1,0)}\) and \(Q_{v+(0,1)}\) are reindexed by a unit shift of the torus.

Let \(\Phi _v=g^{-1}\) on the sites of the rectangle and \(\Phi _v=1\) elsewhere, and apply Lemma 26.1.4.3 with \(P_v=U_{\Phi _v}\) and \(Q_v=U_{\Phi _v}^{-1}\) to the second string. A bond with both ends inside or both outside the rectangle carries the identity in both strings and keeps it. A bond entering the rectangle across its top side is multiplied by \(U_g^{-1}\) at its head, which cancels the \(U_g\) of the second string; a bond leaving it across its right side is multiplied by \(U_g\) at its tail, which cancels its \(U_g^{-1}\). A bond entering across the left side acquires \(U_g^{-1}\) and a bond leaving across the bottom side acquires \(U_g\), which is the first string. By the invariance every site tensor absorbs \(U_{\Phi _v}\) on its incoming and \(U_{\Phi _v}^{-1}\) on its outgoing legs unchanged.

Split each edge assignment into its incident and exterior parts. Exterior labels do not occur in any tensor of \(R\), so their sum gives the stated multiplicity. Apply the existing coefficient identity for the globally indexed region tensors on the two sides. Their exterior multiplicities multiply to the product of all non-crossing bond dimensions; cancelling this nonzero product gives (56). Regular bond dimensions are \(|G|{\gt}0\).

Translate every incident-edge assignment by the same group element. This permutes the internal summation and translates the crossing labels; local invariance leaves the product of site coefficients unchanged. Averaging this symmetry gives \(AP=A\). The exact cut identity gives \(K=AB^{\mathsf T}\), and inserting \(P\) and then \(C=P/\sqrt r\) yields the remaining formulas.

Distribute the finite product over the finite sums. A product of zero-one indicators is the indicator of the conjunction of their conditions.

The local identities are \(c_v^{-1}A_v^\dagger A_v=P_v\) and \(A_vP_v=A_v\). The first follows from the local regular Gram kernel; the second follows from virtual invariance by averaging the group action. Apply the physical product maps inside the actual internal-bond contraction. Each local coefficient becomes the corresponding matrix product, giving (61). Expanding \(P_v\) and distributing the finite products gives the translation formula (60).

Expand the matrix products and interchange the finite sums over physical and virtual labels. The sum over independent physical labels factors at each vertex. Composition follows from the same finite distributivity; complex conjugation gives the adjoint identity. Applying these identities to \(F_v^\dagger F_v=\mathbb 1\) proves the isometry assertion.

The defining adjoint identity is \(A_v^\dagger A_v=c_vP_v\). Evaluating both sides in the group basis gives (65). The identity \(A_vU_g=A_v\) gives coefficient invariance.

Expand both copies of the region tensor. Interchange the finite sums, and distribute the independent physical sums over the vertices. Each factor is the corresponding site Gram kernel.

For an internal edge with endpoints \(u,v\), the two equations give \(q_u\theta _e=q_v\theta _e\), hence \(q_u=q_v\). Equality propagates along paths. Every incident edge has an endpoint in \(R\), so its label is translated by this common element. Nonemptiness makes the assignment \(g\mapsto (q_v=g)\) injective.

The second configuration has one free label on each internal edge, and fixes the first configuration by translation. An incident edge that is not a crossing edge has both endpoints in \(R\), which identifies these free indices with the internal bonds.

Expand the Gram matrix into local pairings. Their averages contribute \(|G|^{-V_R}\) and a sum over vertex group labels. Compatibility reduces this to one common translation. For each such translation, the internal labels contribute \(|G|^{I_R}\). The remaining boundary translation sum is \(|G|P_{\partial R}\).

Local isometry gives \(A_v^\dagger A_v=c_vP_v\), with \(c_v{\gt}0\). Products and adjoints of the product physical maps are computed site by site, so \(T^\dagger T=cP\), where \(c=\prod _{v\in R}c_v\). Hence

Equal Gram matrices define an isometry between the two ranges. Extend it to a unitary on the finite-dimensional physical space.

If two normalized vertex labellings have the same tree gradient, their relative label \(r_v^{-1}s_v\) is constant along every tree edge. Connectedness and the root normalization give \(r=s\). A tree has \(|R|-1\) edges, so the domain and codomain of the gradient each have \(|G|^{|R|-1}\) elements. Its injectivity therefore gives bijectivity.

Given \(\alpha \), recover \(r\) from the tree transports \(\alpha _{v,e}\alpha _{u,e}^{-1}\). Then set \(y_e=r_v^{-1}\alpha _{v,e}\) on the boundary, \(a_e=r_u^{-1}\alpha _{u,e}\) on each internal edge, and \(z_e=r_v^{-1}\alpha _{v,e}a_e^{-1}\) outside the tree. Substitution into (70)–(73) recovers every original label. Conversely, the reconstructed tree transports equal \(r_vr_u^{-1}\), so uniqueness recovers \(r\), and then the same formulas recover \(y\), \(a\), and \(z\).

Expand the original internal-bond contraction into a sum over bond labels \(\eta \) and vertex translations \(q\). On each tree edge \(e=(u,v)\), compatibility gives \(r_ua_e=q_u\eta _e\) and \(r_va_e=q_v\eta _e\). Thus \(q_vq_u^{-1}=r_vr_u^{-1}\). The uniqueness of rooted tree gradients gives \(q_v=r_vx\), where \(x=q_o\); this conclusion already follows from compatibility on the tree alone.

Compatibility on the remaining internal edges forces \(z_e=1\). The bond labels are then uniquely determined by \(x\), and boundary compatibility is precisely \(y=x\theta \). Consequently the coefficient is \(|G|^{-|R|}\sum _x\mathbf1_{\{ y=x\theta ,\ z=1\} }\), which is (74). There are exactly \(N\) nonzero ancillary coefficients, each of modulus \(N^{-1/2}\), proving the stated normalization.

Translating all virtual labels in the contraction proves boundary invariance. The proved connected-region Gram formula gives \(T_R^\dagger T_R=c_R\Pi _{\partial R}\) with \(c_R{\gt}0\). Dividing the adjoint by \(c_R\) gives a left inverse on the invariant boundary space. The same identity gives the required inner-product identity there.

Invert the tree-gradient bijection at \((u_e)_{e\in T}\) to obtain the normalized \(k\). On a tree edge, \(k_t^{-1}u_ek_s=k_t^{-1}(k_tk_s^{-1})k_s=1\). The reference identity follows from \((k_t^{-1}u_ek_s)(k_s^{-1}\eta _e)=k_t^{-1}u_e\eta _e\). Each transport is multiplication by a fixed group element at each coordinate, hence is reversible. At a head endpoint the native twisted-label identity is \((K_t^{-1}u_eK_s)(K_s^{-1}\eta _{t,e}) =K_t^{-1}u_e\eta _{t,e}\); at a tail endpoint both expressions are \(K_s^{-1}\eta _{s,e}\).

The first equality means that \(x-\bar z\) is an integer multiple of \(n\), so \(x\) is an integer with the stated residue. In the second assertion, \(f\) and the affine path have identical projections and agree initially; uniqueness of a covering lift on the connected interval makes them equal.

The finite disjoint union of copies of \(K\) is compact and locally path connected. Its natural continuous surjection onto the stated union is closed, hence is a quotient map. Local path connectedness passes through this quotient map.

The quotient projection is continuous, so each projected closed square is compact. The finite coproduct of the compact convex squares maps continuously and surjectively onto \(X_R\). This is a closed quotient map into a Hausdorff space, and local path connectedness therefore passes to \(X_R\). Split a native unit path at its midpoint: its first half lies in the initial cell and its second half in the final cell.

Apply covering-space lifting to the two coordinate maps into \(\mathbb R/w\mathbb Z\) and \(\mathbb R/H\mathbb Z\) separately. Every lift of a native site has integer coordinates. Along an internal unit path, uniqueness of the lift identifies the lifted path with the corresponding affine unit segment. Its endpoints thus satisfy the exact integer unit-step equations.

The radial segment from \(p_v\) to \(p_v+\pi (d)\) stays in the same closed cell. Uniqueness of its covering lift identifies the lifted segment with \(t\mapsto F(p_v)+td\). Two neighboring cells have a common side midpoint; evaluating its lift from the two cells shows that their lifted centers differ by the prescribed unit step. The fibers over native sites have integer coordinates. These identities identify the image cell by cell. Since \(\pi \circ F\) is the inclusion, \(F\) is injective. Compactness of \(X_R\) and the Hausdorff property of the plane make this continuous injection a closed embedding, giving the stated homeomorphism.

The additive-circle fibers are integer multiples of the two periods. Comparing a point with the lift of its projection gives the stated union of all period copies. The projection of \(\delta _j\) is zero. If \(F(q)+\delta _j=F(p)\), projecting to the torus gives \(q=p\), because the lift is a section of the projection. Subtracting \(F(p)\) then gives \(\delta _j=0\). Positivity of both periods forces \(j=0\), a contradiction. The exact image description in Theorem 26.1.4.27 gives the statement for \(P_L\), and restricting to individual cells gives the last separation assertion. Simple connectedness supplies the continuous lift and hence the required integer coordinates.

Local path connectedness follows from the finite compact-cell quotient theorem. Suppose \(K\) is such a finite candidate and choose a missing cell center in \(K\). Translate it to the origin. Simple connectedness gives a continuous real lift of the angular map on \(P_A\). Subtract this lift from the principal argument and divide by \(2\pi \). At cell corners this gives a real potential whose horizontal increments vanish. Its vertical increment is one precisely when the edge crosses the negative real ray from the missing center, and is zero otherwise.

The potential is defined along every exposed side of \(K\), since its neighboring cell belongs to \(A\). Summing its square curls cancels all interior edges and gives zero. The same boundary increments, evaluated by the branch-cut indicator, give total curl minus one. This contradiction excludes the finite candidate, and hence excludes every finite missing-cell component. Finally, the actual covering lift identifies \(X_R\) homeomorphically with its integer cell union, so the torus conclusion follows.