Suppose there exists an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\), written as coefficients \(V_{(i_1,i_2),j}\), such that

for all \(i_1,i_2\in \{ 0,\ldots ,d-1\} \). Then \(\mathcal{E}_A^2=\mathcal{E}_A\).

The transfer map of \(A\) is idempotent if and only if there exists an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\), written as coefficients \(V_{(i_1,i_2),j}\) with \(V^\dagger V=\mathbb {1}\), such that

for all \(i_1,i_2\in \{ 0,\ldots ,d-1\} \).

The transfer map of an MPS tensor is idempotent if and only if the tensor has a physical blocking isometry.

A tensor appears as a limit of the two-site renormalization flow if and only if it has a physical blocking isometry. This is [ CPGSV16 , Theorem 3.1 ] .

CPSV asserts that a tensor \(A\) in canonical form has physical BNT zero correlation length if and only if

Both directions fail as printed.

For the reverse implication, let \(C^0=(1)\) and form the direct sum of two scalar copies with weights \(1\) and \(1/2\). The resulting tensor is

The scalar tensor \(C\) is normal, and the weights obey the source convention \(|\mu _k|\leq 1\), with at least one weight of modulus one. The singleton BNT family \((C)\) is locally orthogonal and gives \(A\) physical BNT zero correlation length, whereas

Thus zero correlation length does not imply transfer idempotence under the source’s raw weight normalization.

Independently, the Bell-pair chain of Theorem 24.2.10 is a single normal block. Taking its weight to be \(1\) and the ambient coisometry to be the identity gives a literal CPSV canonical-form tensor. Its transfer map is idempotent, but its adjacent two-region expectation is \(1\), while the expectation after an allowed one-site shift is \(0\). Hence transfer idempotence does not imply physical zero correlation length for the unrestricted quantification over disjoint regions.

Source gap (adjacent regions). The printed forward argument uses \(\mathcal{E}_A^n=\mathcal{E}_A\), which follows from idempotence only for \(n\geq 1\). Adjacent regions insert \(\mathcal{E}_A^0=\mathbb {1}\) and are included in the source definition. This gap is recorded in docs/paper-gaps/cpsv16_pure_zcl_adjacent_gap_cid_scope.tex. Theorem 24.5.10 proves the repaired forward implication when both complementary gaps are positive. It does not prove the unrestricted biconditional in [ CPGSV16 , Theorem 3.8 ] . The converse and equivalence results below require their stated unit-weight, multiplicity-one, or spectral hypotheses. The printed biconditional is recorded here as refuted, not as a theorem awaiting proof.

CPSV asserts that, for a tensor \(A\) in canonical form, the following are equivalent: \(A\) is a renormalization fixed point; \(A\) has physical zero correlation length; and, for every \(N{\gt}2\), the vector \(|V^{(N)}(A)\rangle \) belongs to the ground space of a nearest-neighbor commuting parent Hamiltonian [ CPGSV16 , Theorem 3.10 ] .

This three-way equivalence is false with the unrestricted physical zero-correlation-length definition. The Bell-pair chain is a single normal block and becomes a literal CPSV canonical-form tensor by taking unit weight and the identity ambient coisometry. Its transfer map is idempotent, so it is a renormalization fixed point, but its adjacent-region correlations are not independent of distance. Thus the implication from renormalization fixed points to zero correlation length fails, regardless of the commuting-parent condition. The raw-weight example of Theorem 24.2.9 separately refutes the reverse implication from zero correlation length to renormalization fixed points.

Source gap (adjacent regions). The obstruction and its scope are recorded in docs/paper-gaps/cpsv16_pure_zcl_adjacent_gap_cid_scope.tex. Theorem 24.5.10 repairs only the forward correlation statement when both complementary gaps are positive; it does not establish the printed unrestricted three-way equivalence. That equivalence is recorded here as refuted, not as a theorem awaiting proof.

Let \(P\) be the auxiliary BNT sector decomposition with one scalar basis tensor \(C^0=(1)\), two one-dimensional copies, and weights \(1\) and \(1/2\). Let \(A=\mathcal A(P)\), so

Then \(P\) satisfies the auxiliary BNT sector hypotheses, \(A=\mathcal A(P)\), and \(A\) and \(\mathcal A(P)\) generate the same matrix-product vectors. Moreover, the singleton family \((C)\) is a physical BNT zero-correlation-length family for \(A\), while

Its length-\(N\) matrix-product-vector coefficient is \(V^{(N)}(A)=1+2^{-N}\).

Let \(A\) be the tensor with physical index pairs \((\alpha ,\beta )\in \{ 0,1\} ^{2}\) and letters

the canonical normal-tensor renormalization fixed point with maximally mixed spectrum \(\Lambda =(1/2,1/2)\): each node carries two spins, and the pair \((|00\rangle +|11\rangle )/\sqrt2\) is shared between the second spin of one node and the first spin of the next. The transfer map is

which is idempotent since \(\operatorname{tr}\operatorname {diag}(\Lambda )=1\), and \(A\) is a single normal tensor, so no copy weights are involved. With the Pauli matrix \(Z\) on the second spin of one node and on the first spin of the next node, the two regions are adjacent, and the two-region expectation is

while after a shift by one site, which leaves one free site on each complementary arc, it is

Thus \(A\) is a renormalization fixed point whose correlations are not independent of distance in the unrestricted sense of Definition 24.5.7. The block \(\mathcal{E}_A^{0}=1\) between adjacent observables is not governed by idempotence.

Let \((A_j)_j\) be nonzero-dimensional, irreducible, left-canonical, pairwise gauge-phase-distinct BNT components. If the direct-sum tensor \(A=\bigoplus _j A_j\) satisfies \(\mathcal{E}_A^2=\mathcal{E}_A\), then \(A\) has positive-gap physical CID and \(\mathcal{E}_{j,j'}=0\) for all \(j\ne j'\). Thus this explicit direct-sum representative has positive-gap BNT zero correlation length. This does not include the adjacent-region cases in [ CPGSV16 , Theorem 3.8 ] .

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\). The multiplicity-one unit-weight tensor \(A=\bigoplus _j A_j\) is in literal CPSV canonical form, with retained weights all equal to \(1\), retained blocks \((A_j)_j\), and the identity ambient coisometry. This statement does not reconstruct the raw weighted repeated-copy tensor represented by \(P\).

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\), and let \(A=\bigoplus _j A_j\) be their direct sum with one unit-weight copy of each sector. Then \((A_j)_j\) is a basis of normal tensors for \(A\): every \(A_j\) is a normal tensor, at every positive system length \(N\)

and the states \(\{ |V^{(N)}(A_j)\rangle \} _j\) are linearly independent for all sufficiently large \(N\).

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\), and let \(A=\bigoplus _j A_j\) be their direct sum with one unit-weight copy of each sector. If \(A\) is a renormalization fixed point,

then the following restricted conclusion holds: its physical correlations are independent of the separation whenever both complementary gaps are positive, and for all distinct BNT components \(j\ne j'\)

This is a multiplicity-one, unit-weight, positive-gap statement. It does not assert the physical zero-correlation-length conclusion of [ CPGSV16 , Theorem 3.8 ] .

Suppose the simultaneous length-\(L\) word evaluations of \((A_j)_j\) span \(\bigoplus _jM_{D_j}(\mathbb {C})\). There are coefficients \(C_{(j,a,b),w}\) such that, for every sector \(k\),

Let \(A=\bigoplus _k A_k\), and suppose the simultaneous length-\(L\) word evaluations span \(\bigoplus _kM_{D_k}(\mathbb {C})\). For every sector \(j\) and matrices \(R,l\in M_{D_j}(\mathbb {C})\), there is an observable \(O\) on \(L\) sites whose inserted transfer map vanishes on every sector pair except \((j,j)\) and satisfies

In particular, arbitrary virtual matrix-unit maps supported on one sector are realized by physical observables.

Let \(P\) be an auxiliary BNT sector decomposition of total bond dimension \(D\), and form the direct sum \(A=\bigoplus _j A_j\) with one unit-weight copy of each BNT basis tensor. There is a positive length \(L\leq 3D^5\) such that, for every basis sector \(j\) and every \(R,l\in M_{D_j}(\mathbb {C})\), a physical observable on \(L\) sites realizes the sector-supported insertion (??). This is the multiplicity-one specialization of the block-injective physical-observable assertion used in [ CPGSV16 , proof of Theorem 3.8 ] , equations (1252) and (1256) in the local source. It does not realize the weighted copy-pair insertions for the full weighted tensor represented by \(P\).

Let \(P\) be an auxiliary BNT sector decomposition, let \(A=\bigoplus _k A_k\), and fix a sector \(j\). Suppose there are matrices \(r,l\in M_{D_j}(\mathbb {C})\) and a scalar \(\lambda \ne 0\) with \(|\lambda |{\lt}1\) such that

Then \(A\) does not have positive-gap physical CID.

Let \(P\) be an auxiliary BNT sector decomposition and \(A=\bigoplus _j A_j\). Assume that every non-idempotent block \(\mathcal{E}_j^2\ne \mathcal{E}_j\) has matrices \(r_j,l_j\) and a scalar \(\lambda _j\) satisfying the five spectral conditions in Theorem 24.2.20. Then

The spectral-pair hypothesis is assumed explicitly.

Let \(P\) and \(A=\bigoplus _j A_j\) satisfy the multiplicity-one unit-weight hypotheses of Theorem 24.2.21, including its normalized nonzero subleading spectral pair for every non-idempotent block. If \(A\) has positive-gap BNT zero correlation length, then for every \(N{\gt}2\) its translated two-site parent interactions satisfy all three ground-space conditions:

This is only the conditional corrected implication from (ii) to (iii) of [ CPGSV16 , Theorem 3.10 ] , not a three-way equivalence.

Under the spectral-pair hypotheses of Theorem 24.2.21, physical BNT zero correlation length for \(A=\bigoplus _j A_j\) implies \(\mathcal{E}_A^2=\mathcal{E}_A\).

Let \(P\) be an auxiliary BNT sector decomposition and \(A=\bigoplus _j A_j\). If \(A\) has positive-gap BNT zero correlation length, then \(\mathcal{E}_A^2=\mathcal{E}_A\).

Let \((\mu _k,A_k)_{k=1}^r\) be nonzero weights and injective left-canonical blocks with positive bond dimensions, non-increasing moduli \(|\mu _1|\ge \cdots \ge |\mu _r|{\gt}0\), and \(\sigma _\partial (\mathcal{E}_{A_k})=\{ 1\} \). Then \((\mu _k,A_k)_{k=1}^r\) satisfies the auxiliary hypotheses of Definition 24.3.1.1, and \(O_{A_kA_k}(N)\to 1\) for every \(k\).

If \(A\) is a normal RFP tensor, then \(A\) is injective.

Assume \(D {\gt} 0\). Let \(A\) be a normal renormalization fixed point such that \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then \(A\) is injective. This is the left-canonical injectivity step in [ CPGSV16 , Appendix B ] .

Assume \(D {\gt} 0\). Let \(A\) be a normal renormalization fixed point such that \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then there exist a unitary \(U\) and a diagonal positive definite matrix \(\Lambda \) such that, for \(B^i := U^\dagger A^i U\), \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\) and \(\mathcal{E}_B(\Lambda ) = \Lambda \). This is the diagonal fixed-point reduction in [ CPGSV16 , Appendix B ] .

Let \(A\) be an injective left-canonical RFP tensor with positive-definite fixed point \(\rho \) of the transfer map \(\mathcal{E}_A\). Then \(\mathcal{E}_A = P_\rho \), where \(P_\rho (X) = \frac{\operatorname{tr}(X)}{\operatorname{tr}(\rho )} \rho \) is the rank-one fixed-point projection.

A normal left-canonical renormalization fixed-point tensor \(A\) admits a decomposition \(A^i=X\Lambda U^iX^{-1}\), where \(\Lambda \) is diagonal positive, \(\sum _i (U^i)^\dagger U^i=I\), and

The decomposition in Theorem 24.3.1.7 additionally records that the diagonal weights satisfy \(\sum _\alpha \Lambda _\alpha ^2 = D\). This square-sum identity is the trace-normalization seed: the explicit weights are \(\Lambda _\alpha = \sqrt{D\, \rho _{\alpha ,\alpha }/\operatorname{tr}\rho }\), and \(\sum _\alpha \rho _{\alpha ,\alpha } = \operatorname{tr}\rho \) for the diagonal fixed point \(\rho \).

A normal left-canonical renormalization fixed-point tensor \(A\) admits a decomposition \(A^i=X\Lambda U^iX^{-1}\), where \(\Lambda \) is diagonal positive and

This is the unit pair-index convention of [ CPGSV16 , Section 3 ] . The statement still does not impose the trace-normalization \(\operatorname{tr}(\Lambda )=1\).

When each block \(A_k\) of a multi-block tensor is a normal, left-canonical renormalization fixed point, that block admits the isometry decomposition \(A_k^i = X_k \Lambda _k U_k^i X_k^{-1}\), with \(X_k\) invertible, \(\Lambda _k\) diagonal positive, and \(U_k = (U_k^i)\) satisfies \(\sum _i (U_k^i)^\dagger U_k^i = I\) and the pair-index orthonormality condition

This is the blockwise form of the diagonal Kraus decomposition (Theorem 24.3.1.7); see [ CPGSV16 , Section 3 ] . The source additionally imposes the normalization \(\operatorname{tr}(\Lambda _k) = 1\); the statement here gives positive \(\Lambda _k\) without it. That normalization is genuine rather than a conjugation gauge, since rescaling \(\Lambda _k \mapsto \Lambda _k/\operatorname{tr}(\Lambda _k)\) factors out as an overall scalar on \(A_k^i\) (conjugation by \(X_k\) preserves the scale); the statement is thus the unnormalized diagonal Kraus form. Scope restriction (source isometry). Corollary 3.12 also invokes the joint isometry condition

The theorem above records the contracted per-block condition \(\sum _i (U_k^i)^\dagger U_k^i=I\) and the diagonal pair-index equation with right-hand side \(D_k^{-1}\delta _{\alpha ,\alpha '}\delta _{\beta ,\beta '}\). It does not impose the trace-normalization \(\operatorname{tr}(\Lambda _k)=1\), and it does not include the cross-block equations for \(k\ne \ell \).

Under the hypotheses of Theorem 24.3.1.10, each block admits a decomposition \(A_k^i = X_k \Lambda _k U_k^i X_k^{-1}\) with \(\Lambda _k\) diagonal positive and

This is a blockwise unit pair-index decomposition. It still does not impose the source trace-normalization of \(\Lambda _k\) or the cross-block orthogonality equations for distinct blocks.

A normal left-canonical renormalization fixed-point tensor has the square-root fixed-point form. With \(\lambda _\alpha =\rho _{\alpha ,\alpha }/\operatorname{tr}\rho \) and \(\rho _0=\operatorname{diag}(\lambda _\alpha )\), it admits \(A^i=X\sqrt{\rho _0}\, U^iX^{-1}\), where \(\rho _0\) is positive and trace-normalized and \(U\) is a unit pair-index isometry. Trace normalization is the identity \(\sum _\alpha \rho _{\alpha ,\alpha }=\operatorname{tr}\rho \). This is the normal-tensor statement corresponding to lines 1278 and 1281–1283, with the square-root correction dictated by the reference tensor at line 1300 [ CPGSV16 , Appendix B ] .

A tensor in the square-root fixed-point form is a renormalization fixed point. This is the backward direction of the structural characterization of pure-state renormalization fixed points in [ CPGSV16 , Section 3 ] , which the source states as immediate.

Each block of a multi-block tensor whose blocks are normal, left-canonical renormalization fixed points has the square-root fixed-point form, with a trace-normalized diagonal weight and a unit pair-index isometry. This is the single-block trace-normalized form applied to each block; it omits the cross-block orthogonality between distinct normal-tensor blocks.

Suppose every bond dimension is positive. The transfer map of \(\bigoplus _k B_k\) is idempotent if and only if, for every pair \(j,j'\), the mixed transfer operator is idempotent:

Let \(A\) have positive bond dimension and a nonzero idempotent transfer map. Let \(\mu _q\) have unit modulus. If the literal bond direct sum \(\bigoplus _q \mu _qA\) has an idempotent transfer map, then \(\mu _q=\mu _{q'}\) for every pair \(q,q'\).

Let \((A_j)_j\) be a family of distinct irreducible left-canonical blocks, no two of which are gauge-phase equivalent, and suppose the direct sum \(\bigoplus _j A_j\) is a renormalization fixed point. Then for every pair of distinct components the mixed transfer operator vanishes, \(\mathcal{E}_{j,j'}=\sum _iA_j^i\otimes \overline{A_{j'}^i}=0\). This is the cross-block (\(\delta _{j,j'}\)) content of the isometry condition of [ CPGSV16 , Theorem 3.11 and Corollary 3.12 ] at the level of the normal-tensor blocks; converting it to the isometries \(U_j\) is a further step left open.

Let \(\mathcal V=\bigsqcup _j\{ j\} \times [D_j]\times [D_j]\) be the disjoint union of the within-sector virtual pairs. For \(x=(j,\alpha ,\beta )\) in \(\mathcal V\), set \(u_x(i)=(U_j^i)_{\alpha ,\beta }\). Then \(\sum _i u_x(i)\overline{u_y(i)}=\delta _{x,y}\).

Let \((A_j)_j\) be a family of normal, irreducible, left-canonical blocks, no two of which are gauge-phase equivalent, with \(\dim _j\ge 1\) for all \(j\), and suppose the direct sum \(\bigoplus _j A_j\) is a renormalization fixed point. Then there are invertible matrices \(X_j\), trace-one positive diagonal matrices \(\rho _j=\operatorname{diag}(\lambda _{j,\alpha })\), and tensors \(U_j\) with

such that \((U_j)_j\) satisfies the joint isometry condition. These are the isometry equations of [ CPGSV16 , Corollary 3.12 ] for the direct sum of distinct normal-tensor blocks.

Let \((B_k)_k\) be a family of tensors, each in the square-root fixed-point form, whose cross-block mixed transfer operators vanish, \(\mathcal{E}_{j,j'}=0\) for \(j\ne j'\). Then the direct sum \(\bigoplus _k B_k\) is a renormalization fixed point. This is the backward direction of the structural characterization of pure-state renormalization fixed points [ CPGSV16 , Section 3 ] , in the distinct-blocks case where each normal tensor appears once; the cross-block vanishing is the \(j\ne j'\) part of the isometry condition of that characterization, so it is part of the source joint isometry condition.

Let \((B_k)_k\) be a family of normal, irreducible, left-canonical blocks with \(\dim _k\ge 1\) for all \(k\), no two of which are gauge-phase equivalent. Then the direct sum \(\bigoplus _k B_k\) is a renormalization fixed point if and only if each block is in the square-root fixed-point form and the mixed transfer operators between distinct blocks vanish:

This is the distinct-blocks (multiplicity-one, phase-one) case of the structural characterization of pure-state renormalization fixed points [ CPGSV16 , Theorem 3.11 ] , combining its forward and backward directions.

CPSV asserts that a tensor \(A\) in canonical form is a renormalization fixed point if and only if it can be written as

where \(|\mu _{j,q}|=1\), each \(\rho _j\) is positive diagonal with \(\operatorname{tr}(\rho _j)=1\), and

The source specifies that \(\bigoplus _{j,q}\) is a simultaneous direct sum in the physical and virtual spaces, not an ordinary virtual block diagonal at each fixed physical letter \(i\) [ CPGSV16 , Theorem 3.11, lines 561–562 ] . It does not provide a coordinate map for the physical routing of \(q\), a corresponding dimension hypothesis, or a \(q\)-indexed isometry equation. Consequently the display does not yet determine a source-faithful coordinate-level predicate. Under the naive fixed-\(i\) virtual-block interpretation, the claimed converse is false by Theorem 24.4.19. The routed characterization therefore remains not ready pending a source clarification or erratum.

Local fix (square-root diagonal). CPSV prints a bare \(\Lambda _j\) in Theorem 3.11, whereas the reference tensor at line 1300 has coefficients \(\sqrt{(\rho _j)_{\alpha ,\alpha }}\), which forces the factor \(\sqrt{\rho _j}\) used above. This repair is documented in docs/paper-gaps/cpsv16_rfp_isometry_scope.tex.

Let \(B\) be the one-letter, one-dimensional tensor \(B^0=(1)\), and form two block-diagonal copies with phases \(1\) and \(-1\):

The representative \(B\) is in the square-root fixed-point form, and both copy coefficients have unit modulus. Thus \(A\) satisfies the literal virtual block-diagonal reading of the displayed repeated-copy formula in [ CPGSV16 , Section 3 ] . It does not, however, satisfy the accompanying interpretation in which the copy index also belongs to the physical direct sum: there is no scalar \(v\) such that \((A^0)^2=vA^0\). In particular, \(A\) is not a renormalization fixed point in the sense of Definition 24.1.2.

Every block satisfying the auxiliary hypotheses of Definition 24.3.1.1 is injective. This is an immediate projection of that auxiliary predicate, not a theorem about literal CPSV canonical form.

Fix one block \(A_k\) in a family satisfying Definition 24.3.1.1. Then the primitive transfer map of that block satisfies: \(\mathcal{E}_{A_k}^{2^n}\) converges pointwise to an idempotent linear map \(\mathcal{E}_{k,\infty }\) as \(n\to \infty \). Thus there exists \(\mathcal{E}_{k,\infty }\) with \(\mathcal{E}_{k,\infty }^2=\mathcal{E}_{k,\infty }\) such that \(\mathcal{E}_{A_k}^{2^n}(\rho )\to \mathcal{E}_{k,\infty }(\rho )\) for every bond matrix \(\rho \). This is the per-block primitive convergence used in the Appendix B discussion, not convergence of the full weighted canonical-family transfer matrix [ CPGSV16 , Appendix B, lines 1211–1244 ] .

If \(\mathcal{E}_A^2=\mathcal{E}_A\), then physical correlations are independent of distance whenever both complementary gaps are positive.

Suppose \(\rho ^R{\gt}0\) is a right fixed point of the transfer map. If the virtual-insertion condition holds for every positive-definite right fixed point, then the transfer map is idempotent.

The auxiliary conjunction \((\mathcal{E}_A^2=\mathcal{E}_A\ \text{and virtual-insertion distance independence})\) is equivalent to \(\mathcal{E}_A^2=\mathcal{E}_A\). This is a logical simplification of the auxiliary convention, not the physical ZCL theorem of [ CPGSV16 , Theorem 3.8 ] .

Let \((A_k)_k\) satisfy the auxiliary block-family hypotheses of Definition 24.3.1.1. For each \(k\), the transfer map of \(A_k\) is idempotent if and only if \(A_k\) satisfies the one-block auxiliary conjunction of Definition 24.5.11. The block-family hypothesis is unused; this is Theorem 24.5.13 applied to one block. It should not be read as the BNT-family ZCL assertion in [ CPGSV16 , Theorem 3.10 ] .