Each internal edge has exactly two incidences in \(R\), whereas each crossing edge has exactly one. Read these incidences in the coordinate bijection.

Partition incident edges into internal and crossing edges and reindex their labels along the resulting bijection. The prescribed crossing condition removes the crossing-label sum. For numbered coefficients, reindex each incident virtual label by the same finite enumeration; the site products and boundary condition are unchanged.

Extending the incidence factors by \(1\) outside \(R\), regrouping the ambient incidence product by edges, and separating internal edges from crossing edges yields the decomposition.

Fixing the crossing labels and summing only over internal virtual labels, expanding the independent site averages and regrouping their incidence products sums each internal label once, giving \((U(q_h)W)(WU(q_t^{-1}))=W^2U(q_hq_t^{-1})\). Crossing labels are fixed, so their single endpoint factors remain. The incidence bijection reads the physical endpoint labels directly, and finite enumerations reindex the same coefficient sum.

At \(a=j(b)\), the sum \(\sum _c\delta _{a,j(c)}x(c)\) has only the term \(c=b\); outside \(j(B)\) every summand is zero. Conversely, if \(\psi \) vanishes outside \(j(B)\), take \(x(b)=\psi (j(b))\) to prove (6). The product of coordinate indicators is the indicator of equality of the full configurations. For a mismatched bond, each summand in the product inclusion has a zero factor at that bond.

Regrouping the open contraction into crossing endpoints and internal head–tail pairs, the internal virtual sum gives \((U(q_h)W)(WU(q_t^{-1}))=W^2U(q_hq_t^{-1})\). Each crossing edge contributes only its retained endpoint factor from (10). Collecting these factors and the site-average normalization yields (11).

\(=\)

The block-scalar matrix \(W\) commutes with \(U(g)\), and \(W^2U(g)=\bigoplus _i\sqrt{d_i}\, D_i(g)\). Applying coherent multiplicity restoration to (11), with coherent coefficients \(c_R(\theta ,\sigma ;q)\) and bond labels \(q_hq_t^{-1}\), gives both product support and (13). Only internal bonds are acted on, so the crossing factors remain those in (10).

The open contractions \(\mathcal C_R(A)|\theta \rangle \) span \(\mathcal G_R(A)\). Reindexing their virtual and physical labels identifies their slices with those of Theorem 38.1.1.5. The slice is linear and the product inclusion range is a subspace, so support extends to every vector in \(\mathcal G_R(A)\). At a given physical configuration, choosing its own crossing labels for \(\sigma \) and regrouping the remaining endpoints shows that a mismatched internal sector forces a zero coefficient by Theorem 38.1.1.2.

Choosing a region containing both endpoints of a mismatched bond and fixing the exterior physical configuration places the resulting regional slice in \(\mathcal G_{R_i}(A)\) by the parent condition, so Theorem 38.1.1.6 makes its coefficient zero. Reassembling the regional and exterior indices gives the original coefficient of \(\psi \). The image of the coordinate inclusion \(j^E\) consists exactly of configurations with matching sectors on every bond, since each matched pair is determined by its shared sector and its two internal labels. Equation (6) therefore turns the vanishing condition into (15).

Conjugating each coefficient in (16) gives the adjoint formula, and interchanging the finite sum over \(a\) with the input-coordinate sums gives (17). For each fixed \(a\), the independent coordinate sums factor into the products defining \(F_bx_{b,a}\) and \(K_fy_{f,a}\).

A slice of \(F^{\otimes S}\chi \) is a linear combination of columns of \(F\), with coefficients obtained by contracting all other factors. For the converse choose a linear right inverse of \(F\) on its image and extend it to \(L:\mathbb {C}^Y\to \mathbb {C}^X\). Thus \(FL\) fixes every slice. Applying \(FL\) at one coordinate fixes \(\psi \). Induction over finite subsets of \(S\) shows that applying it at all coordinates also fixes \(\psi \). Consequently \(F^{\otimes S}(L^{\otimes S}\psi )=(FL)^{\otimes S}\psi =\psi \). The same induction proves the assertion about an arbitrary \(P\).

The adjoint of a coordinate inclusion selects its included coordinates. For equal sectors, the multiplicity map retains equal copy labels and multiplies by \(m_i^{-1/2}\); for unequal sectors its coefficient is zero. This proves the formula for \(F\phi \). Evaluating it on coordinate vectors gives (22); the product of its coordinate indicators selects the single input configuration in (24). Equal copy labels have equal sectors and hence equal normalizations, proving symmetry.

Since \(F^\dagger F=E_XE_X^\dagger \) and \(E_X^\dagger E_X=I\), the factorization \(F=E_YJE_X^\dagger \) gives \(FF^\dagger F=F\). Products respect multiplication. Write \(\psi =E_X^{\otimes E}\chi \) and \(\phi =\mathcal F\eta \), respectively, and cancel the corresponding local factors.

The block-diagonal and identity factors force the sector and copy coordinates in the sum. The remaining matrix entry is that of \(T_i\), multiplied by \(\lambda _i m_i^{-1/2}\). Transposing the block matrices exchanges the two contraction formulas. The action formula follows by summing the single surviving input coordinate in (27). Its unit-weight coefficient is real. Off-block entries vanish; on a block, the row and column sector labels agree.

Applying (29) with unit weights and using \(B_z^\dagger =B_z^{\mathsf T}\) gives the adjoint boundary formulas, since reading the block weight from the fixed \(y\) index permits cancellation of \(\lambda _{i(y)}\). Because the repeated matrix-entry vector in (33) is \(F\) applied to the square-root-weighted block vector and the block-diagonal input lies in the matching-sector range, \(F^\dagger F=E_XE_X^\dagger \) yields the internal adjoint identity. The fourth-root identities follow by multiplying block-scalar matrices on the left and on the right.

The product coefficients agree by definition, and reindexing the matrix-vector sum by the bijection \(j_{e^X}\) gives the intertwining identity. Conjugating each factor in (35) proves the adjoint identity.

Splitting the global input-coordinate sum into its regional and exterior coordinates gives (36). Taking the adjoint exchanges input and output configurations. If \(\psi \in \mathcal P_{\mathcal R}(A)\), every summand in (36) lies in \(\mathcal G_{R_i}(B)\) by the assumed block inclusion. This space is closed under sums, proving all target parent constraints.

Since every edge has zero, one, or two endpoints in \(R\), the product in (35) separates into these three cases. The first contributes \(c_{R,F}\), the second gives (40), and the third retains \(F\) on its head–tail pair. Regional regrouping places these factors in the two coordinate families of \(\mathcal M\).

Separating the indicator for the fixed endpoint from the inside endpoint in (22) and using symmetry of \(a(y,z)\) gives both fixed-endpoint identities; substituting (45) into (43) and collecting the crossing indicators into \(\widehat c_R\) yields the regional block formula.

Decoding the numbered labels and applying the open contraction formula shows that the inverse regional coordinate map puts each crossing, head, and tail label back at its original incidence, proving the column equality in (53). Open columns span \(\mathcal G_R\), and their boundary numberings are bijective, so their images span \(\mathcal H_R\). Injectivity of \(J_{e,R}\) gives the membership equivalence. The final assertion follows directly by decoding the virtual boundary labels in the finite contraction sum and then fixing the crossing physical coordinates; it needs no representation or commutation hypothesis.

Updating an internal edge changes only its two inside incidences; it leaves all crossing and exterior incidences fixed. This proves (55). For each edge \(f\), choose a covering region \(R_i\). The parent condition places \(S_{R_i,\tau }\psi \) in \(\mathcal G_{R_i}(A)\), so (54) and the forward implication of (18) put its \(f\)-slice in \(\operatorname {im}F\). By (55) this is the global \(f\)-slice, so the reverse implication of (18) across all edges places \(J_e\psi \) in \(\operatorname {im}F^{\otimes E}\).

Choose \(x_g\) with \(Fx_g=\operatorname {vec}(W^2U(g))\). For each internal slice use the preimage

where \(c_R\) retains all boundary factors and the averaging normalization. Applying \(F^{\otimes E_R}\) term by term gives the weighted slice formula. Numbered and incidence slices agree, and the regional range is spanned by open columns, so the inclusion extends by linearity. Theorem 38.2.4.4 then proves (56).

When each target slice is expanded with its own boundary coefficients \(c_R^{U',I}(\theta ,\sigma ;q)\), coherent multiplicity restoration sends the internal factors \(\operatorname {vec}(\bigoplus _i\sqrt{d_i}\, D_i(q_hq_t^{-1}))\) to \(\operatorname {vec}(U'(q_hq_t^{-1}))\), while retaining these same coefficients. This provides a preimage for each target open slice. Linear combinations give all regional vectors; edge coverage and Theorem 38.2.4.4 give (58).

Expanding the normalized group average separately at every vertex yields the factor \(|G|^{-|R|}\) and the sum over \(q:R\to G\), after which we regroup the incidence product by internal and crossing edges. Summation over one internal virtual index gives either \((U_{q_h}W)(WU_{q_t^{-1}})\) or \((U_{q_t}W)(WU_{q_h^{-1}})\) with the physical indices interchanged. Commutation with \(W\) gives (64). Each crossing edge retains its original \(\theta _e\), giving (65); commuting the scalar products also gives the internal-first form of the same sum. The bijections of incidence and numbered labels preserve every finite sum. Finally, the regional range is the span of all its boundary columns, proving (68).

Comparing (61) at \(W=1\) with (71) coefficient by coefficient identifies the local map with \(\Pi _v\). The group average satisfies \(\Pi _v\rho _v^o(g)=\Pi _v\) and fixes every invariant vector. Hence its restriction to the invariant subspace is injective. The invertible coordinate changes \(P_v,E_v\) preserve both assertions.

Each orientation selects one of the two assumed matrix orders. Choosing a preimage under \(F\) for every \(I_e(q)\) and taking their product with the scalar coefficient in (72) produces a vector whose image under \(F^{\otimes E_R}\) is the displayed sum. Span closure proves the assertion for every regional vector. For a global parent vector, choose a region containing the endpoints of a prescribed edge. Every one-edge slice then belongs to \(\operatorname {im}F\). The product-range criterion gives the full \(E\)-fold range.

Partitioning the defining product for \(M(F)\) into exterior, crossing, and internal edges yields \(c_R\) on the exterior edges, while fixing the outside endpoint of each crossing edge gives (75). The internal factors are unchanged. The coordinate bijections identify their product with (76).

Conjugation distributes over each coefficient product. For composition, the sum over the intermediate site configuration is, under the bond-coordinate bijection, a sum over independent intermediate edge labels. It factors into the products \(F_eL_e\). The product of edgewise Kronecker deltas is the delta of the complete configuration, proving the identity assertion.

The coefficient of \(T(Q)\) is \(Q_{a,c}\overline{Q_{b,d}}\). Exchanging both endpoint pairs gives \(\overline{Q_{a,c}}Q_{b,d}\), which is \(T(\overline Q)_{(a,b),(c,d)}\). Summing this coefficient against \(M_{d,c}\) gives \((QMQ^\dagger )_{b,a}\).

Taking entrywise conjugates of the finite sums defining \(AB\) gives \(\overline A\, \overline B=1\). Applying this identity to each \(Q_e\), its conjugate on a retained tail, and the two factors of every internal pair, and multiplying the independent boundary and internal tensor factors proves the regional assertion.

At a native head, \(L_gQ=QU_g\), and at a native tail, \(Q^\dagger L_g=U_gQ^\dagger \), with conjugates as specified by \(B_{e,R}(Q)\), which gives (79). Internal factors follow by conjugating the relative representation matrix, using (77) on reversed edges. Applying these identities inside (65) and factoring the independent boundary sums yields (80). Linearity gives (81) and the forward range inclusion. At \(g=1\), the representation identity gives \(1=L_1=QQ^\dagger \). The reverse coordinate change \(Q^\dagger \) therefore satisfies the same hypotheses, and cancellation proves (82).

A crossing coefficient of \(Q_e\otimes \overline{Q_e}\) factors into its fixed exterior entry and its retained endpoint entry, and taking their product gives (85). By (76), each global block is \(c_Rb_RJ_{Y,R}^{-1}T_R^o(Q)J_{X,R}\). Equations (68) and (82) show that this block preserves the required regional range. Each output slice of a global vector is a sum of such blocks applied to its input slices, proving (86).

Each edge factor and its inverse cancel, as do the conjugated factors on reversed edges, so their global product cancels by Theorem 38.3.2.4. Applying the same identity to \(Q^\dagger \) yields a left inverse under \(Q^\dagger Q=1\) and hence injectivity.

Evaluating the intertwining identity at \(1\) gives \(QQ^\dagger =1\). Applying the block inclusion in both directions and cancelling the global coordinate matrices shows that the restriction of \(M_o(Q)\) is bijective, which gives equal dimension. Positivity identifies each Hamiltonian kernel with the intersection of its regional parent conditions, giving the kernel assertion.

Within each diagonal block, the scalar weight \(m_i^{1/4}\) commutes with \(D_i(g)\) and satisfies \((m_i^{1/4})^2=\sqrt{m_i}\).

The internal factor is \(W_m^2D(q_hq_t^{-1})\). Equation (89) gives \(\sqrt{m_i}\) in each block, canceled by the normalized copy map. Its adjoint sums \(m_i\) equal-copy terms with coefficient \(m_i^{-1/2}\), recovering the same factor. At a crossing, the bond factor is \(W_mD(q_t^{-1})\) when the tail lies in \(R\) and \(D(q_h)W_m\) when the head lies in \(R\). The sparse coefficient of \(B_z\) fixes the base coordinate and copy label; moving the remaining \(w_y\) to the virtual boundary coefficient gives (91). Conjugating the same sparse coefficient and dividing by the nonzero \(w_y\) gives (92).

Expanding each open contraction as \(|G|^{-|R|}\) times a sum over \(q:R\to G\) of the product of its crossing and internal factors and applying Theorem 38.5.1.4 to each factor, distributing the crossing sums gives (93) while collecting their scalar coefficients gives (94). The transformation never restricts the original virtual boundary.

The vectors \(a_R(\theta )\) and \(b_R(\eta )\) span the respective full regional ranges. Equations (93) and (94) send every generator into the required span. The coordinate permutations identify these spans with the original numbered contraction ranges.

In the case of Theorem 38.8.1.4 with no edge reversed (\(o_e=0\) for every \(e\)), the oriented tensors \(A^{D,W_m,o}\) and \(A^{D^{[m]},1,o}\) are the ordered tensors \(A_m\) and \(B_m\), and the oriented product of the bond maps \(F_m\) is \(M_m\). Taking \(m=d\) gives the dimension-multiplicity inclusions.

Setting \(o_e=0\) for every \(e\) in Theorem 38.8.1.4 identifies the oriented tensors with the ordered tensors \(A_m\) and \(B_m\), reducing (124) and (125) to (97), together with injectivity on the source parent space. Taking \(m=d\) gives the dimension-multiplicity identities.

Taking every \(o_e=0\) in Theorem 38.8.1.5 gives \(M_m\mathcal P_{\mathcal R}(A_m)=\mathcal P_{\mathcal R}(B_m)\). By Theorem 38.5.2.2, \(M_m\) is injective on the source parent space and its restriction has inverse the restriction of \(M_m^\dagger \). The dimension assertion follows. Positivity identifies the Hamiltonian kernels with the common regional parent spaces, so the same map identifies those kernels.

On each incident leg, the block-scalar weight can be read from the physical sector label. Taking the product of these weights gives \(c_v(\sigma )\), independently of the virtual column. Positive \(m_i\) make the diagonal matrices invertible. Physical deformation commutes with every open contraction. Applying \(C\) and \(C^{-1}\) to regional slices proves (99).

Composing the isomorphism \(C\) of (99) with the isomorphism \(M_m\) of (98) maps \(\mathcal P_{\mathcal R}(A_0)\) onto \(\mathcal P_{\mathcal R}(B_m)\) and preserves dimensions; inverting the comparison for \(m\) and composing with that for \(n\) identifies the parent spaces of any two positive multiplicity families. The kernel statement again follows from positivity and the exact local-kernel hypotheses.

Evaluating the first identity at \(g=1\) with \(U(1)=L(1)=I\) gives \(QQ^\dagger =I\). Multiplying \(L(g)=QU(g)Q^\dagger \) on the left by \(Q^\dagger \) and on the right by \(Q\) and cancelling \(Q^\dagger Q\) yields the second identity.

In matrix multiplication, the sum over an intermediate product configuration factors into the product of the individual sums. Conjugation commutes with multiplication, proving the bond formula. For a tail crossing the two factors are \(\overline Q\) and \(\overline{Q^\dagger }\), with product \(\overline{QQ^\dagger }=I\); at a head they are \(Q,Q^\dagger \). The internal product is \(B(QQ^\dagger )=I\), proving the regional cancellation identity.

For an internal edge, vectorization of \(L(q_hq_t^{-1})=QU(q_hq_t^{-1})Q^\dagger \) gives its factor identity. At a head crossing \(L(g)Q=QU(g)\), while at a tail crossing \(Q^\dagger L(g)=U(g)Q^\dagger \) with \(g=q_t^{-1}\) in tail-ordered coefficients; both identities leave the oriented virtual boundary matrix \(Q_f\). Expanding the site averages and distributing the product of crossing sums yields (102). Summing against \(c(\theta )\) and interchanging finite sums proves (103).

Equation (102) maps every source generator into the target span. Applying it to \(Q^\dagger \) via Theorem 38.6.1.1 sends every target range vector back into the source range, and the identity \(Q_R(Q^\dagger )_R=I\) then gives its preimage.

This is the case of Theorem 38.4.6 in which no edge is reversed. When every \(o_e=0\), each \(Q_e\) equals \(Q\), so the oriented global coordinate matrix is \(\mathcal Q\), and the oriented averaging tensors are the ordered ones.

Setting every \(o_e=0\) in Theorems 38.4.7 and 38.4.8 makes the oriented global coordinate matrix equal to \(\mathcal Q\) and the oriented averaging tensors equal to the ordered ones, which gives the inverse identities, injectivity, and the equality of parent spaces. Restricting \(\mathcal Q\) to the source parent space is therefore injective with image the target parent space, which gives the isomorphism and equality of dimensions. Positivity identifies the Hamiltonian kernels with those spaces.

Copy restoration identifies the full weighted single-copy parent space with the unweighted \(D^{[d]}\) parent space, and the Fourier isomorphism identifies that entire space with the regular parent space. Composing the two maps preserves dimension, and positivity identifies each Hamiltonian kernel with its parent space.

In the orthonormal irreducible rows of Theorem 35.10.70, the matrices of the action on one row of each component give \(D_i\), while the common action on its \(m_i\) rows gives \(D_i\otimes I_{m_i}\) after exchanging the row and matrix indices. Changing between these orthonormal coordinates and the original basis via \(Q\) yields (106), with the character multiplicity equal to the number of rows. Theorem 38.7.1.2 then gives semi-regularity, and the direct sum of the unitary \(D_i\) is unitary.

Semi-regularity is equivalent to linear independence of the group operators. If \(\sum _gc_gD(g)=0\), each diagonal block gives \(\sum _gc_gD_i(g)=0\). Tensoring with \(I_{m_i}\) and conjugating by \(Q\) yields \(\sum _gc_gU(g)=0\). Hence every \(c_g\) is zero.

Traces add under direct sums and multiply under tensor products. Character orthogonality shows that an irreducible character absent from all the summands has zero multiplicity. Conversely, semi-regularity includes every irreducible character of the regular representation. Its character expansion is therefore \(\sum _id_i\chi _{D_i}\), with no repetitions. Evaluating at the identity gives the dimension formula.

Multiplicativity of \(D_i\) gives \(TL(g)=D^{[d]}(g)T\). If \(Tc=0\), all matrix coefficients of \(\sum _gc_gD(g)\) vanish; semi-regularity forces \(c=0\). The dimension identity in (110) makes \(T\) bijective. For unitary blocks, direct multiplication gives

Equal dimensions give the other inverse identity. The row weights commute with \(D^{[d]}\), so the intertwining identity also holds for \(\widehat T\), proving (112).

Products of independent incident indices separate into products of matrix products. At a tail, inversion and transposition each reverse the order, giving multiplicativity. Expanding the averaging site yields \(P_v\). Averaging gives \(P_v\rho _v(g)=P_v\) and fixes every invariant vector, hence is injective on the invariant subspace. Conjugation by \(J_v\) preserves invariance, and \(E_v\) is injective; therefore \(E_vP_vJ_v\) has the same two \(G\)-injectivity properties.

With \(D_i,d_i,m_i,Q\) obtained from Theorem 38.7.1.1, \(Q^\dagger \) maps canonical parents from \(U\) to \(D^{[m]}\), Theorem 38.5.3.4 changes the positive copy family \(m_i\) to \(d_i\), and the normalized Fourier matrix maps \(D^{[d]}\) to \(L\) by Theorem 38.6.2.2. Constant incidence provides the intermediate common alphabets, and edge coverage provides the supported inverse for the copy change. Composing these three isomorphisms gives the parent-space isomorphism, while positivity identifies the kernel of each parent Hamiltonian with its common regional parent space.

The canonical averaging tensor is \(G\)-injective for the same incident action. The fixed-representation comparison provides physical maps \(F_v\) that are injective on its site images and satisfy \(A_v=F_vA(U,e)_v\). Vertex coverage makes the restriction of \(\bigotimes _vF_v\) an isomorphism of the parent spaces, and its inverse gives the stated isomorphism.

In the common incident enumeration \(e^X\), the composition

combines the vertex-coverage comparison with the edge-coverage, constant-incidence, and semi-regularity comparison. Dimensions are preserved by linear isomorphisms. For positive interactions, \(\langle \psi ,H\psi \rangle =0\) holds exactly when every regional term annihilates \(\psi \); hence \(\ker H=\mathcal P_{\mathcal R}(A)\) and likewise for \(K\).

Applying the ordinary restoration and adjoint formulas to \(M_i^T\) gives both identities, since transposition commutes with block sums and scalar multiplication, and \((M_i\otimes 1_{m_i})^T=M_i^T\otimes 1_{m_i}\).

Since \(W_m^2D_g=\bigoplus _i\sqrt{m_i}D_i(g)\), the ordinary bond identities give the internal claim on unreversed edges; Theorem 38.8.1.1 gives it on reversed edges. On a crossing edge the sparse coefficient \(\alpha (y,z)\delta _{\pi y,x}\) retains exactly one factor \(w_y\). Taking its adjoint gives the factor \(w_y^{-1}\alpha (y,z)\), which proves (120) and (121). Substituting into (65) and distributing the independent boundary sums gives (122) and (123).

Equations (122) and (123) put every image of a boundary column in the required span. Extending by linearity and using the range identification (68) yields the inclusions.

Each regional block of \(M_m\) is a scalar times the numbered \(F_{m,R}(\gamma )\); the adjoint block has the conjugate scalar. The regional inclusions therefore give the two global inclusions. The source factors \(\bigoplus _i\sqrt{m_i}D_i(g)\) and their transposes lie in the matching-sector subspace. The target factors \(\bigoplus _i(D_i(g)\otimes 1_{m_i})\) and their transposes lie in \(\operatorname {im}F_m\). Theorem 38.3.2.1 derives these product supports for every parent vector from edge coverage. On the source support \(F_m^\dagger F_m\) is the identity; on the target support \(F_mF_m^\dagger \) is the identity. Taking products proves (124) and (125).

For \(\phi \in \mathcal P_{\mathcal R}(B_m)\), its preimage \(M_m^\dagger \phi \) belongs to the source parent space and satisfies \(M_mM_m^\dagger \phi =\phi \). The source identity proves injectivity. The kernel statement follows from positivity and the exact local kernel conditions.

Each \(D_i(g)\) preserves its sector. A diagonal factor \(m_i^{1/4}\) can therefore be moved to the physical index at either end of an incidence, regardless of \(o_e\). Factoring the product of these scalars out of the group average gives the coefficient identity. Positive multiplicities make every diagonal entry nonzero, and applying the invertible physical maps gives (126).

The decomposition theorem yields inequivalent irreducibles \(D_i\), dimensions \(d_i{\gt}0\), multiplicities \(m_i{\gt}0\), and coordinates \(Q\) such that \(U_g=QD^{(m)}_gQ^\dagger \), \(Q^\dagger Q=1\). Semi-regularity includes every irreducible sector, so normalized Fourier coordinates give \(L_g=FD^{(d)}_gF^\dagger \) with \(F^\dagger F=1\). Constant degree provides physical numberings of all intermediate incidence alphabets. The equivalence is the composition

Here all parent spaces use \(\mathcal R\). Dimensions are preserved by each equivalence; the positive-parent common-kernel theorem gives the Hamiltonian assertion.

The averaging tensor is \(G\)-injective by Theorem 38.3.1.6. The two local maps have the same kernel, namely the complement of the invariant subspace killed by the group average. The fixed-representation physical comparison constructs local maps \(F_v\) with \(a_v=F_vA^{U,1,o}_v\), injective on the local image. Vertex coverage makes their product injective on the parent space and identifies its image with \(\mathcal P_{\mathcal R}(a)\); inverting that equivalence gives \(\mathcal P_{\mathcal R}(a)\simeq \mathcal P_{\mathcal R}(A^{U,1,o})\).

Vertex coverage compares \(a\) with \(A^{U,1,o}\), while edge coverage and (127) compare \(A^{U,1,o}\) with \(A^{L,1,o}\). Composing these equivalences yields the dimension and positive-parent kernel assertions. The positive filters \(S_v\) need not be unitary, so this assertion concerns ground spaces and kernels, without a spectral comparison.

The horizontal and vertical edge bijection identifies the selected tails with edge endpoints. The four-leg incidence bijection gives degree four. Evaluating the tail on its four legs gives \(v\) on right and down; on top and left it gives a neighboring vertex, different from \(v\). The definition of \(o_e\) then identifies this test with the ordered-graph native-tail predicate.

The outgoing indices \(1,2\) contribute \(U_{g^{-1}}^T\); indices \(0,3\) contribute \(U_g\). Reindexing the four incidence factors gives (129). Reindexing the finite coefficient sum by \(j_v\) gives both intertwining identities. Since \(J_v\) is invertible, invariance and injectivity on invariant vectors pass from \(\mathcal A\) to \(\mathcal A_v\). Reindex once more by \(E_v\) to obtain the numbered assertion.

Let \(\Pi =|G|^{-1}\sum _g\rho _\square (g)\) and enumerate the coefficient basis of \(\mathbb {C}^{X^4}\) by a linear isomorphism \(E\) to \(\mathbb {C}^p\), with \(p=|X|^4\). Take \(b\) to be the matrix entries of \(E\Pi \). The average satisfies \(\Pi \rho _\square (g)=\Pi \) and is the identity on invariant vectors. Composing with \(E\) therefore gives the required invariance and restricted injectivity.

Split endpoint assignments into internal labels and pairs \(\zeta \) on all noninternal edges. The identity matrices on internal edges force their two endpoint labels to agree. For fixed \(\zeta \), the remaining coefficient is

Here \(\theta (\zeta )\) selects the retained endpoint on every crossing edge. Each summand is a boundary column of the original regional map, so the sum belongs to its range.

Reindex all native top/right/down/left labels by the incidence bijections \(j_v\), then regroup them into ordered endpoint pairs. The horizontal and vertical bond coefficients become, respectively,

Equations (134) and (137) give exactly these entries in both possible vertex orders. The site factors are unchanged, so the reindexed finite sums agree.

The first coordinates in \(R_v\) are \(x,x+1\), whereas \(x+2\) differs from both modulo \(m\ge 3\). The analogous statement holds for \(y,y+1,y+2\) modulo \(n\). An internal rightward edge cannot end at the horizontal seam, and an internal upward edge cannot end at the vertical seam. Their insertion matrices are therefore \(1\); any transpose required by ordered coordinates leaves \(1\) unchanged.

Local invariance and \(gh=hg\) move both closure seams to \(x+2,y+2\) without changing the vector. Theorem 38.9.2.4 makes every internal insertion on \(R_v\) the identity. After the native-to-ordered contraction identity, formula (131) expresses each physical slice as a sum of unrestricted regional boundary columns. For the four-site plaquette, collecting equal boundary labels gives the contraction below. All four sites carry \(a\).

\(\displaystyle \lvert a;(g,h)\rangle (\, \cdot \, ,\tau ) =\sum _\theta c_\tau (\theta )\)

The coefficients \(c_\tau \) include the exterior contraction and every crossing insertion. Each local term therefore annihilates every closure slice. Summing the terms gives \(H\lvert a;(g,h)\rangle =0\); linearity proves (139).

Native incidence coordinates make every graph tensor \(G\)-injective for the rightward/downward orientation. The torus has degree four. Every vertex belongs to its plaquette, and every edge has both endpoints in a native plaquette. Theorem 38.8.2.5 therefore identifies the entire parent space of \(a\) with that of the oriented regular averaging tensor. Independently construct a native \(G\)-injective site \(b\) for the regular representation using Theorem 38.9.1.5. Fixed-representation comparison identifies its parent space with the same oriented regular canonical space. The already established regular native-torus theorem gives dimension \(|\mathcal K_G|\) for \(b\). Thus

Finally, positivity and the genuine local kernel conditions give \(\ker H=\mathcal P_{\{ R_v\} }(a)\).

Let \(\mathcal V\) be the span on the right of (142). Invariance gives \(\mathcal V\subseteq \ker H\) by (139). Closures in the same simultaneous-conjugacy class coincide, and semi-regular \(G\)-injectivity makes the vectors from distinct commuting classes linearly independent. Hence \(\dim \mathcal V=|\mathcal K_G|\) by Theorem 35.1.6.10. Equation (140) gives \(\dim \ker H=|\mathcal K_G|\). Inclusion and equal finite dimension imply \(\mathcal V=\ker H\). Taking canonical positive plaquette interactions identifies the same span with the full intersection of regional range conditions.

The bond set has \(2\cdot 2\cdot 2=8\) elements. For each of the four choices of \((c,r)\), exactly two horizontal and two vertical bonds satisfy (144). Each has two endpoints. A bond anchored at \((x,y)\) is outside \(E_{x,y}\) because \(x+1\ne x\) and \(y+1\ne y\) in \(\mathbb Z_2\).

Multiplying (147) on the left by \(q_{x+1,y}\) and on the right by \(q_{x,y+1}^{-1}\) and substituting (146) cancels the two intermediate vertex factors on each side. Applying the same calculation with \(q^{-1}\) gives the converse.

Multiplying (147) on the left by \(b_{x+1,y}^{-1}\) and on the right by \(b_{x,y}^{-1}\) and reversing the resulting equality gives the rightward-upward condition, while multiplication by the inverse factors gives the converse.

The plaquette at \((-1,-1)\) gives \(gh=hg\). Conversely, substituting (152) into (147) gives either \(gh=hg\) or an equality with an identity factor.

Applying the two rooted-path identities of Theorem 35.1.4.150 to \((a,b^{-1})\) yields the claimed horizontal component directly, while inverting the vertical identity gives the downward component.

The based holonomies \(B_{0,\ell }\) and \(A_w\) commute, hence so do \(g=B_{0,\ell }^{-1}\) and \(h=A_w\). Theorem 38.9.5.10 gives \(q\cdot p=\kappa (g,h)\); applying \(q^{-1}\) gives the other form. Conversely, commuting closure labels are flat and remain flat under a vertex gauge.

Substituting \(r\cdot (q\cdot p)=(rq)\cdot p\) and reindexing the sum by the bijection \(r\mapsto rq\) of vertex assignments gives the equality.

The one-dimensional seam identity gives the changed horizontal component. Every value of \(s_{h,c}\) commutes with \(g\), so conjugation leaves the vertical component fixed. The inverse-label seam identity gives the vertical movement, while commutation fixes the horizontal component. Composing the horizontal gauge with the vertical gauge gives their pointwise product \(q_{c,r}\).

Replacing \(\kappa _{c,r}(g,h)\) via (153) and reindexing the orbit sum by multiplication with \(q_{c,r}\) gives the equality.

In each direction, being outside the cut makes the seam condition false.

Taking the gauge of Theorem 38.9.6.3 and reading its horizontal or vertical equality according to the direction of \(e\) gives the bondwise identity.

Classifying the horizontal and vertical components and evaluating the resulting pair of equalities in the direction of each bond gives the claim.

The four bonds in this equality are uncut. Substitution gives

The sets \(I_v\) partition the incidence set. Regroup first by this partition and then by the two signs of each bond. For the sum, change coordinates by \(S\) and apply finite distributivity.

Expand \(M=\sum _\eta M(\eta )\delta _\eta \) and use linearity. In the contraction with \(M_B\), the factors on \(C\) are the entries of \(B_e\), and the remaining factors are identity entries. Products of matrix-unit entries equal one exactly when all cut coordinates equal \(\eta \), proving the last assertions without factoring a general \(M\).

Finite distributivity separates the intermediate sum in a matrix product into one sum at each coordinate. Identity entries force equality of the remaining coordinates. For the fixed-vector assertion, induct on the finite set of coordinates at which \(P_v\) replaces the identity, using the one-coordinate formula and coordinatewise composition.

For the forward implication, expand a slice as a sum of columns of \(F_v\), with coefficients obtained by contracting all other coordinates. For each \(v\), choose a linear right inverse on \(\operatorname {im}F_v\) and extend it to a map \(L_v\) on the output space. Thus \(F_vL_v\) fixes every slice. The preceding theorem gives \((\bigotimes _vF_vL_v)\psi =\psi \), so \((\bigotimes _vL_v)\psi \) is a preimage.

Right multiplication by \(\rho _v(g)\) permutes the group average, so \(P_v\rho _v(g)=P_v\). On the invariant subspace \(P_v\) is the identity, hence is injective there.

Expanding each local average collects the factor \(|G|^{-n}\), and regrouping the incident matrix factors by bonds turns the independent sum over the two indices of each bond into the displayed three-matrix product by (155).

Changing the vertex summation variable from \(r\) to \(rq\) in (162), the inverse at a tail reverses the product in precisely the required order.

Expanding the product physical map, interchanging the finite physical and incidence sums, and distributing the product of the local sums leaves the factors \(M(r_C\beta )w_C(\beta )\) and \(w_B(\beta )\) unchanged.

Each \(L_v\) is given by the left-inverse characterization of \(G\)-injectivity, and averaging the invariance equations gives \(T_vP_v=T_v\), so the physical-contraction identities yield (163) with the same \(M\). If \(\psi \) lies in every original cut range, then \(\mathcal L\psi \) lies in every canonical range, and choosing one cut in the nonempty family gives \(\mathcal T\mathcal L\psi =\psi \); conversely, \(\mathcal T\) carries each canonical range into the corresponding original range.

Canonical recovery applied to the \(G\)-injective tensors \(A^0\) gives the first equality; for the second, applying \(P_v\) to the physical leg of \(I_v\) gives \(A_v^0\), and commuting these local changes through the network completes the proof.

Distributing (165) over the bond pairs evaluates the \(e\)-th sum as \(K_e(B_e)\) by expansion in matrix units. Trace duality then gives \(\ell _{e,p_e}(U_e(q_e))=\delta _{p_e,q_e}\), whose product is \(\delta _{p,q}\), and applying linearity to the entire coherent sum identifies each coefficient.

Separating the \(e\)-th factor establishes the slice identity, while regrouping (162) by \(J\) yields the first two formulas and applying the product-functional identity to each term gives the third.

In a summand of (167), an identity insertion gives \(B_e(q)=U_e(q_{h(e)}q_{t(e)}^{-1})\), so every slice is a scalar multiple of this representation matrix and sums retain membership in \(\operatorname {im}F_e\). Expanding a general boundary into fixed configurations produces matrix-unit networks that still have \(B_e=I\) when \(e\notin C\).

Choosing a cut for each bond that leaves it uncut, the preceding theorem places every slice in \(\operatorname {im}F_e\). The product-range criterion then provides a preimage under \(\bigotimes _eF_e\) whose entries are the coefficients \(a_p\), and applying \(c_p\) to the whole expansion identifies \(a_p=c_p(\psi )\) in the semi-regular case.

For any vertex assignment \(q\) in the extracted sum of (167), incompatibility yields an uncut bond with \(p_e\ne q_{h(e)}q_{t(e)}^{-1}\) whose trace-dual factor is zero, so every summand vanishes. Expanding an arbitrary boundary into fixed-coordinate columns and using their matrix-unit realization extends this vanishing, whose contrapositive is the stated implication.