11 Proof of the Fundamental Theorem

For BNT canonical forms $P$ and $Q$, write the weighted direct sums over normal-tensor sectors as $P^i=\bigoplus _{j,q}\mu _{j,q}P_j^i$ and $Q^i=\bigoplus _{k,q}\nu _{k,q}Q_k^i$. The proportional theorem matches the normal sectors by a bijection $\beta $, unit phases $\zeta _k$, and invertible matrices $X_k$ satisfying

\begin{align} Q_k^i & =\zeta _kX_kP_{\beta (k)}^iX_k^{-1}. \notag \end{align}

It does not determine the copy weights. In the equal-MPV case, eventual linear independence gives

\begin{align} c_N^{(\beta (k))}(P) & =\zeta _k^N c_N^{(k)}(Q). \notag \end{align}

Finite power-sum comparison then gives $\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}$. Thus the sector phases do not become equal: each phase cancels against the inverse phase in its matched copy weight. The remaining block conjugacies assemble into $X=\bigoplus _k(\mathbf1_{r_k}\otimes X_k)$, and a sector-coordinate permutation yields the literal global gauge.

Throughout, “eventually” means “for all sufficiently large $N$”. At a fixed length in that range, BNT linear independence detects exact coefficient equalities even when a weight of modulus less than $1$ makes its powers decay.

11.1 Coefficient identities and copy weights

Let $P$ and $Q$ be BNT canonical forms. Their basis sectors are normalized, left-canonical, and aperiodic, and their raw copy weights satisfy the global normalization of Definition 10.6.1.11. This is the normalized surface used for the proof. The literal CPSV canonical form is brought to this surface by the reduction and normalization of Chapters 9 and 10.

The proportional statement below allows nonzero projective proportionality for all sufficiently large lengths. The source assumption of proportionality at every positive length is a special case. No copy-weight comparison is needed for its sector-level conclusion.

Theorem 11.1.1 Proportional sector matching with unit phases

✓

Let $P$ and $Q$ be BNT canonical forms whose total tensors generate eventually nonzero proportional MPV families. Then there are a bijection $\beta :J(Q)\simeq J(P)$, bond-dimension equalities $\dim P_{\beta (k)}=\dim Q_k$, scalars $\zeta _k\in \mathbb {C}$, and block gauges $X_k$ such that $|\zeta _k|=1$ and, for every sector $k$ and physical index $i$,

\begin{align} Q_k^i & = \zeta _k X_k P_{\beta (k)}^i X_k^{-1}. \label{eq:ft_gauge_relation} \end{align}

Diagrammatically, the gauge relation (??) reads

$\begin{tenkz}[physical=up] \tn[up=$i$]{Q_k} \end{tenkz}$ $ = \zeta _k$ $\begin{tenkz}[physical=up] \tnX{X_k} & \tn[up=$i$]{P_{\beta(k)}} & \tnX{X_k^{-1}} \end{tenkz}$

with every open virtual end a matrix index. Equivalently, for every positive length $N$ and word $\sigma $,

\begin{align} V^{(N)}(Q_k)_\sigma & = \zeta _k^N V^{(N)}(P_{\beta (k)})_\sigma . \notag \end{align}

This is [ CPGSV16 , Appendix MPV theorem statement, lines 1167–1170, and proof line 1182 ] .

Proof ▶

Theorem G.1.1 gives a bijection $\beta :J(Q)\simeq J(P)$ together with equal bond dimensions, gauge-phase equivalence, and

\begin{align} O_{P_{\beta (k)}Q_k}(N) & \not\longrightarrow 0 \notag \end{align}

for every matched pair. Its proof defines $T=\{ j\in J(P):O_{P_jQ_k}(N)\to 0\text{ for every }k\} $, substitutes exact $Q$-sector relations for $j\notin T$, applies linear independence to $\{ P_j:j\in T\} \cup \{ Q_k:k\in J(Q)\} $, and reverses direction; basis separation and finiteness turn the resulting matches into $\beta $.

Unpack each matched gauge-phase equivalence to choose an invertible matrix $X_k$ and a nonzero scalar $\zeta _k$. This gives $Q_k^i=\zeta _kX_kP_{\beta (k)}^iX_k^{-1}$ and, by taking traces around a positive-length word,

\begin{align} V^{(N)}(Q_k)_\sigma & =\zeta _k^N V^{(N)}(P_{\beta (k)})_\sigma . \notag \end{align}

Lemma G.1.2 applies to this MPV identity. Equivalently, the normalized self-overlaps satisfy

\begin{align} O_{Q_kQ_k}(N) & =|\zeta _k|^{2N}O_{P_{\beta (k)}P_{\beta (k)}}(N), \notag \end{align}

and both outer sequences tend to $1$, which forces $|\zeta _k|=1$.

Lemma 11.1.2 Full-basis coefficient identity from matched phases

✓

Let $P$ be a BNT canonical form, let $Q$ be a sector decomposition, and assume $P_{\mathrm{tot}}$ and $Q_{\mathrm{tot}}$ have the same MPV family at every positive length. Suppose that there is a bijection $\beta :J(Q)\simeq J(P)$ and scalars $\zeta _k \in \mathbb {C}$ such that, for every $k \in J(Q)$, every positive length $N$, and every spin configuration $\sigma $ of length $N$,

\begin{align} V^{(N)}(Q_k)_\sigma & = \zeta _k^N\, V^{(N)}(P_{\beta (k)})_\sigma . \label{eq:ft_phase_relation_hyp} \end{align}

Then for every $k \in J(Q)$ there is $N_0$ such that, for all $N{\gt}N_0$, $c_N^{(\beta (k))}(P)=\zeta _k^N\, c_N^{(k)}(Q)$. This is the coefficient-comparison step of [ CPGSV16 , Appendix MPV proof, lines 1187–1188 ] in full-basis form.

Proof ▶

For every $N$, expand the equality of total MPVs as

\begin{align} \sum _{j\in J(P)} c_N^{(j)}(P)|V^{(N)}(P_j)\rangle & = \sum _{k\in J(Q)} c_N^{(k)}(Q)|V^{(N)}(Q_k)\rangle . \notag \end{align}

The phase relation (??) gives $|V^{(N)}(Q_k)\rangle =\zeta _k^N|V^{(N)}(P_{\beta (k)})\rangle $, so reindexing the $Q$-sum by $\beta $ onto the $P$-basis yields

\begin{align} \sum _{j\in J(P)} c_N^{(j)}(P)|V^{(N)}(P_j)\rangle & = \sum _{j\in J(P)} \zeta _{\beta ^{-1}(j)}^N c_N^{(\beta ^{-1}(j))}(Q)|V^{(N)}(P_j)\rangle . \notag \end{align}

For all sufficiently large $N$ the $P$-basis MPV family $\{ |V^{(N)}(P_j)\rangle \} _{j\in J(P)}$ is linearly independent, by the BNT eventual linear independence of $P$. Lemma F.1.11 then extracts the coefficient equalities $c_N^{(\beta (k))}(P)=\zeta _k^Nc_N^{(k)}(Q)$ from this eventual linear independence, for all sufficiently large $N$.

Definition 11.1.3 Copy-weight matching for matched BNT sectors

✓

Let the sectors of two decompositions $P$ and $Q$ be matched by $\beta :J(Q)\simeq J(P)$, with sector phases $\zeta _k$. A copy-weight matching consists of copy permutations $\tau _k$ such that, for all sectors $k$ and copies $q$, $\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}$. This is the copy-level conclusion of [ CPGSV16 , Appendix MPV proof, line 1188 ] .

Definition 11.1.4 Copy-weight matching from matched-sector coefficient identities

✓

The construction takes as input a sector matching $\beta :J(Q)\simeq J(P)$, nonzero sector phases $\zeta _k$, and the eventual matched-sector coefficient identities

\begin{align} c_N^{(\beta (k))}(P) & = \zeta _k^N\, c_N^{(k)}(Q). \label{eq:ft_sector_coeff_identity_hyp} \end{align}

It returns the copy-weight matching obtained by choosing, in every sector, a copy permutation $\tau _k$ satisfying $\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}$. This is the sectorwise construction in [ CPGSV16 , Appendix MPV proof, line 1188 ] .

Proof ▶

Eventually the coefficient identity (??) in sector $k$ reads

\begin{align} \sum _r \mu _{\beta (k),r}^{\, N} & = \zeta _k^N\sum _q\nu _{k,q}^{\, N} = \sum _q(\zeta _k\nu _{k,q})^{N}. \notag \end{align}

Newton–Girard finite power-sum comparison (Lemma 10.7.1.2) applies sector by sector, producing a copy permutation $\tau _k$ with $\mu _{\beta (k),\tau _k(q)}=\zeta _k\nu _{k,q}$; since $\zeta _k\ne 0$ this is $\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}$.

Theorem 11.1.5 Equal-case sector and copy matching with differing bond dimensions

✓

Let $P$ and $Q$ be BNT canonical forms, possibly with different sector counts and different bond dimensions before matching. If their total tensors generate the same MPV family at every positive length, then there is a bijection $\beta :J(Q)\simeq J(P)$ such that each matched basis pair has equal bond dimension and is gauge-phase equivalent. The matched copy multiplicities agree, and there are unit phases $\zeta _k$ and copy bijections $\tau _k$ satisfying

\begin{align} Q_k^i & =\zeta _kX_kP_{\beta (k)}^iX_k^{-1}, \notag \\ \nu _{k,q} & =\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}. \notag \end{align}

Only the global weight normalization in Definition 10.6.1.11 is assumed; no unit-modulus copy is required in each sector. This is the sector-data form of [ CPGSV16 , Corollary 2.11 and Appendix A, lines 1182–1188 ] .

Proof ▶

First, Theorem 10.8.1.1 gives the sector bijection, the bond-dimension equalities, and the gauge-phase relations. Choose their phases $\zeta _k$ and use the normalized self-overlaps to obtain $|\zeta _k|=1$. Second, Lemma 11.1.2 expands the equality of the two total MPVs in the matched $P$-basis. Eventual linear independence gives, for every $k$ and all sufficiently large $N$,

\begin{align} c_N^{(\beta (k))}(P) & =\zeta _k^N c_N^{(k)}(Q). \notag \end{align}

Third, expand both coefficients as finite power sums. Their eventual equality determines the two multisets of nonzero copy weights, so the multiplicities agree and a copy bijection $\tau _k$ realizes the identity $\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}$.

11.2 The equal-MPV global gauge

For equal MPV families, the sector phases occur twice: in the matched tensor relation and with the inverse power in the matched copy weights. They therefore cancel before the block gauges are assembled.

Definition 11.2.1 Flattened block-diagonal gauge

✓

Given per-block gauges $X_k \in \mathrm{GL}_{D_k}(\mathbb {C})$, form the block-diagonal element $\bigoplus _k X_k$ and then reindex the dependent block-coordinate space with the standard bond index $\{ 0,\ldots ,\sum _k D_k-1\} $ of the assembled tensor. The result is an element of $\mathrm{GL}_{\sum _k D_k}(\mathbb {C})$.

Theorem 11.2.2 Global gauge from matched sectors and copy weights

✓

Suppose that the sectors of two sector decompositions $P$ and $Q$ are matched by a bijection $\beta :J(Q)\simeq J(P)$, with bond-dimension equalities, copy permutations $\tau _k$, nonzero phases $\zeta _k$, and block gauges $X_k$. If, for every sector $k$, copy $q$, and physical index $i$,

\begin{align} Q_k^i & = \zeta _k\, X_k P_{\beta (k)}^i X_k^{-1}, \notag \\ \nu _{k,q} & = \zeta _k^{-1} \mu _{\beta (k),\tau _k(q)}, \label{eq:ft_matched_weights_hypotheses} \end{align}

then, in the flattened $Q$-coordinates,

\begin{align} Q_{\mathrm{flat}}^i & = X\, P_{\mathrm{matched}}^i\, X^{-1}, \notag \\ X & = \bigoplus _k(\mathbf{1}_{r_k}\otimes X_k). \notag \end{align}

This is the direct-sum gauge construction of [ CPGSV16 , Appendix MPV proof, lines 1189–1192 ] , separated from the coefficient-comparison step that supplies the weight identities.

Proof ▶

For each matched copy,

\begin{align} \nu _{k,q}Q_k^i & = (\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}) (\zeta _kX_kP_{\beta (k)}^iX_k^{-1}) = \mu _{\beta (k),\tau _k(q)}X_kP_{\beta (k)}^iX_k^{-1}. \notag \end{align}

Thus every flattened weighted block is conjugated by its sector gauge $X_k$. Lemma G.4.3 combines these block conjugacies into the direct-sum gauge $X=\bigoplus _k(\mathbf{1}_{r_k}\otimes X_k)$.

Lemma 11.2.3 Equal-MPV matched copy-weight witnesses

✓

Let $P$ and $Q$ be BNT canonical forms whose total tensors have the same MPV family at every positive length. Then there are a bijection $\beta :J(Q)\simeq J(P)$, bond-dimension equalities, unit-modulus phases $\zeta _k$, and block gauges $X_k$ such that $Q_k^i=\zeta _k X_kP_{\beta (k)}^iX_k^{-1}$. For the same phases there is a copy-weight matching: after a permutation of the copies inside each matched sector, $\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}$.

Proof ▶

The full-basis matching theorem gives the bijection and block gauges. The self-overlap comparison

\begin{align} \langle V_N(Q_k),V_N(Q_k)\rangle & = |\zeta _k|^{2N} \langle V_N(P_{\beta (k)}),V_N(P_{\beta (k)})\rangle \notag \end{align}

together with the BNT normalization limits on both sides forces $|\zeta _k|=1$. The matched MPV phase identities also give the eventual coefficient identities $c_N^{(\beta (k))}(P)=\zeta _k^N c_N^{(k)}(Q)$. Definition 11.1.4 applies the finite power-sum comparison in each sector and gives the copy-weight identities.

Theorem 11.2.4 Full-basis global gauge in matched coordinates

✓

Let $P$ and $Q$ be BNT canonical forms whose total tensors have the same MPV family at every positive length. Then there exist:

a bijection $\beta :J(Q)\simeq J(P)$,
bond-dimension equalities $\dim P_{\beta (k)}=\dim Q_k$,
copy-number equalities $r_{\beta (k)}(P)=r_k(Q)$ and copy permutations $\tau _k$,
unit-modulus phases $\zeta _k$ with $|\zeta _k|=1$ and block gauges $X_k$,

such that, for every sector $k$, copy $q$, and physical index $i$,

\begin{align} Q_k^i & = \zeta _k\, X_k P_{\beta (k)}^i X_k^{-1}, \notag \\ \nu _{k,q} & = \zeta _k^{-1} \mu _{\beta (k),\tau _k(q)}. \notag \end{align}

Moreover, if $X=\bigoplus _k (\mathbf{1}_{r_k}\otimes X_k)$ is the flattened block-diagonal gauge, then, for every physical index $i$, the flattened $Q$-coordinates satisfy

\begin{align} Q_{\mathrm{flat}}^i & = X\, P_{\mathrm{matched}}^i\, X^{-1}. \notag \end{align}

This is the explicit direct-sum gauge construction of [ CPGSV16 , Appendix MPV proof, lines 1189–1192 ] .

Proof ▶

Lemma 11.2.3 gives $\beta $, the bond-dimension identifications, the phases $\zeta _k$, the block gauges $X_k$, and copy permutations $\tau _k$ satisfying $\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}$. Theorem G.4.4 combines the block conjugacies into the flattened direct-sum conjugation by $X=\bigoplus _k(\mathbf{1}_{r_k}\otimes X_k)$.

Theorem 11.2.5 Literal equal-MPV gauge after sector-coordinate permutation

✓

Let $P$ and $Q$ be BNT canonical forms whose total tensors have the same MPV family at every positive length. Then the total bond dimensions agree; write their common value as $D$. There is

\begin{align} Y & \in \mathrm{GL}_D(\mathbb {C}) \label{eq:ft_literal_gauge_Y} \end{align}

such that, for every physical index $i$,

\begin{align} Q_{\mathrm{tot}}^i & = Y\, P_{\mathrm{tot}}^i\, Y^{-1}, \notag \end{align}

where $P_{\mathrm{tot}}^i$ and $Q_{\mathrm{tot}}^i$ are both regarded as matrices in $M_{D}(\mathbb {C})$ using the total bond-dimension equality. This is the BNT form of [ CPGSV16 , Corollary II.2 and Appendix MPV proof, lines 1189–1192 ] .

Proof ▶

Apply Theorem 11.2.4 to obtain the flattened gauge after the sectors of $P$ have been reindexed by the bijection between basis blocks of $P$ and $Q$. The sector-coordinate permutation $\rho $, induced by this bijection, the copy permutations, and the bond-dimension equalities, identifies the reindexed $P$-side direct sum with the total tensor $P_{\mathrm{tot}}$, transported along the total bond-dimension equality. Composing the permutation gauge for $\rho $ with the sector-permuted gauge gives the single matrix $Y$ of (??).

Remark 11.2.6 Proportional sector matching does not determine copy weights

The proportional theorem determines the normal sectors, but it does not determine the copy weights or imply conjugacy of the total tensors. Let $C$ be a normalized normal tensor and consider the one-sector BNT forms

\begin{align} P & =C\oplus \tfrac 12 C, & Q & =C\oplus \tfrac 13 C. \notag \end{align}

Their positive-length MPV families are

\begin{align} V^{(N)}(P) & =(1+2^{-N})V^{(N)}(C), & V^{(N)}(Q) & =(1+3^{-N})V^{(N)}(C). \notag \end{align}

They are nonzero and proportional with

\begin{align} c_N & =\frac{1+2^{-N}}{1+3^{-N}}, \notag \end{align}

but $(c_N)_N$ is not a geometric sequence $\xi ^N$. For a one-dimensional choice of $C$, the total matrices have eigenvalue multisets $\{ 1,\tfrac 12\} $ and $\{ 1,\tfrac 13\} $, so they are not conjugate, even after multiplication by a common phase.

Thus the scalar-free coefficient identity used in copy-weight recovery is an additional hypothesis in the proportional setting, not a consequence awaiting an analytic estimate. CPSV16 Theorem 2.10 asserts the sector matching proved in Theorem 11.3.1. The copy-weight recovery and global block gauge belong to the equal-MPV Corollary 2.11, where the proportionality scalar is identically $1$.

11.3 The proportional and equal source theorems

Theorem 11.3.1 Multi-block fundamental theorem for normal tensors

✓

Let $P$ and $Q$ be BNT canonical forms (Definition 10.6.1.11), with bases of normal tensors $(P_j)_{j=1}^{g_a}$ and $(Q_k)_{k=1}^{g_b}$. If the total tensors $P_{\mathrm{tot}}$, $Q_{\mathrm{tot}}$ generate eventually nonzero proportional MPV families — for all sufficiently large $N$ there is a nonzero scalar $c_N$ with $V^{(N)}(P_{\mathrm{tot}})=c_NV^{(N)}(Q_{\mathrm{tot}})$ — then $g_a=g_b$. After relabelling the basis of normal tensors of $P$ by a bijection $\beta :J(Q)\simeq J(P)$, there are scalars $\zeta _k\in \mathbb {C}$ with $|\zeta _k|=1$ and invertible matrices $Y_k$ such that, for every $k$ and every physical index $i$,

\begin{align} Q_k^i & =\zeta _k\, Y_k\, P_{\beta (k)}^i\, Y_k^{-1}. \notag \end{align}

Thus the paired normal tensors have the same bond dimension ($\dim P_{\beta (k)}=\dim Q_k$) and the same gauge-phase class.

Proof ▶

Apply Theorem 11.1.1. Its partition-and-substitution argument gives a bijection $\beta :J(Q)\simeq J(P)$, equal bond dimensions, invertible block gauges $X_k$, and phases $\zeta _k$ of unit modulus. Taking $Y_k=X_k$ gives exactly the asserted relations.

This is [ CPGSV16 , Theorem 2.10 ] ; see also [ CPGSV21 , Theorem IV.4 and Corollary IV.5 ] . The theorem proved here uses the normalized, left-canonical, aperiodic BNT surface of Definition 10.6.1.11. The source canonical form is brought to this surface by the reductions and normalizations of Chapters 9 and 10. Our hypothesis only requires eventual nonzero projective proportionality, so the source assumption at every positive length is a special case. The reduction of an arbitrary proportional-MPV pair to a common primitive block decomposition is Theorem 9.3.1.4 of Chapter 9.

11.3.1 Unitary gauges on normalized sectors

Lemma 11.3.1.1 Phase normalization for an irreducible gauge

✓

Let $A$ and $B$ be left-canonical tensors of common positive bond dimension, with $B$ irreducible. If $X$ is invertible, $\zeta \ne 0$, and $B^i=\zeta XA^iX^{-1}$ for every $i$, then $|\zeta |=1$.

Proof ▶

Let $\rho $ be a nonzero positive fixed point of $\mathcal{E}_A$ and set $\sigma =X\rho X^\dagger $. Then $\sigma $ is nonzero and positive, and $\mathcal{E}_B(\sigma )=|\zeta |^2\sigma $. The map $\mathcal{E}_B$ also has a nonzero positive fixed point. Its irreducibility therefore forces $|\zeta |^2=1$.

Lemma 11.3.1.2 Unitary replacement with the same phase

✓

Let $A$ and $B$ be left-canonical irreducible tensors of common positive bond dimension satisfying $B^i=\zeta XA^iX^{-1}$ for an invertible $X$ and a nonzero scalar $\zeta $. There is a unitary matrix $U$, with the same scalar $\zeta $, such that $|\zeta |=1$ and $B^i=\zeta UA^iU^\dagger $ for every $i$.

Proof ▶

Lemma 11.3.1.1 gives $|\zeta |=1$. Substitution into the left-canonical identities gives

\begin{align} \sum _i A^{i\dagger }(X^\dagger X)A^i & =X^\dagger X. \notag \end{align}

Thus $X^\dagger X$ is a positive-semidefinite fixed point of the adjoint transfer map of $A$. Irreducibility gives $X^\dagger X=c\mathbb {1}$ for some $c{\gt}0$, so $X^{-1}=c^{-1}X^\dagger $ and $U=c^{-1/2}X$ is unitary. Moreover,

\begin{align} UA^iU^\dagger & =c^{-1}XA^iX^\dagger =XA^iX^{-1}. \notag \end{align}

Corollary 11.3.1.3 Canonical Form II unitary refinement of proportional matching

✓

Under the hypotheses of Theorem 11.3.1, there are a bijection $\beta :J(Q)\simeq J(P)$, unitary matrices $U_k\in \mathrm{U}(\dim Q_k)$, and phases $\zeta _k\in \mathbb {C}$ with $|\zeta _k|=1$, such that $\dim P_{\beta (k)}=\dim Q_k$ and, for every $k$ and every physical index $i$,

\begin{align} Q_k^i & =\zeta _k\, U_k\, P_{\beta (k)}^i\, U_k^\dagger . \notag \end{align}

This is the proportional part of [ CPGSV16 , Corollary A.6, lines 1197–1199 ] .

Proof ▶

The proportional sector-matching theorem gives a bijection $\beta $, invertible matrices $X_k$, and unit-modulus phases $\zeta _k$ with $Q_k^i=\zeta _kX_kP_{\beta (k)}^iX_k^{-1}$. The matched tensors are left-canonical and irreducible. Hence Lemma G.8.1 gives unitary matrices $U_k$ satisfying $X_kP_{\beta (k)}^iX_k^{-1}=U_kP_{\beta (k)}^iU_k^\dagger $, and therefore $Q_k^i=\zeta _kU_kP_{\beta (k)}^iU_k^\dagger $.

11.3.2 The equal case

Theorem 11.3.2.1 Equal BNT canonical forms are globally conjugate

✓

Let $P$ and $Q$ be BNT canonical forms (Definition 10.6.1.11) whose total tensors $P_{\mathrm{tot}}$, $Q_{\mathrm{tot}}$ generate the same MPV family for every positive system size $N$. Then their total bond dimensions coincide; writing the common value as $D$, there is one invertible matrix $X\in \mathrm{GL}_D(\mathbb {C})$ such that, for every physical index $i$,

\begin{align} Q_{\mathrm{tot}}^i & =X\, P_{\mathrm{tot}}^i\, X^{-1}. \notag \end{align}

This is [ CPGSV16 , Corollary 2.11 ] on the normalized BNT surface; the literal source surface is reached by the reduction and normalization of Chapters 9 and 10. Moreover, the canonical form of [ CPGSV16 , Section 2.3 ] , a direct sum of normal tensors with scalar weights, is a BNT canonical form. The reduction of an arbitrary pair of same-MPV tensors to a common BNT canonical form is Chapter 9.

Proof ▶

This is Theorem 11.2.5. The BNT comparison of Chapter 10 provides the basis relabelling, the per-block gauge-phase equivalences, and the copy-weight multiset matching, which combine into the block-diagonal global gauge. The exact linear-independence matching needs no per-sector unit-modulus hypothesis, and the BNT canonical form needs no one-site injectivity.

This is also [ CPGSV21 , Corollary IV.5 ] .

Corollary 11.3.2.2 Canonical Form II unitary refinement in the equal case

✓

Let $P$ and $Q$ be BNT canonical forms whose total tensors generate the same MPV family at every positive length. Their total bond dimensions agree; writing the common value as $D$, there is a unitary matrix $U\in \mathrm{U}(D)$ such that, for every physical index $i$,

\begin{align} Q_{\mathrm{tot}}^i & = U P_{\mathrm{tot}}^i U^\dagger . \notag \end{align}

This is the equal case of [ CPGSV16 , Corollary A.6, lines 1197–1199 ] .

Proof ▶

For each matched sector, retain the phase in both identities

\begin{align} Q_k^i & = \zeta _k U_kP_{\beta (k)}^iU_k^\dagger , \notag \\ \nu _{k,q} & = \zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}. \notag \end{align}

For every matched copy $(k,q)$,

\begin{align} \nu _{k,q}\zeta _k & = \zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}\zeta _k = \mu _{\beta (k),\tau _k(q)}. \notag \end{align}

Hence $Q_{\mathrm{flat}}^i=XP_{\mathrm{match}}^iX^{-1}$. Repeating $U_k$ on its matched copies gives

\begin{align} X & = \bigoplus _{k,q}U_k, \notag \\ X^\dagger X & = \mathbb 1. \notag \end{align}

If $\Pi $ is the permutation matrix restoring the original sector and copy order, then $\Pi ^\dagger \Pi =\mathbb 1$. Hence

\begin{align} U & = X\Pi , \notag \\ U^\dagger U & = \mathbb 1, \notag \end{align}

and the permutation of matched coordinates gives

\begin{align} Q_{\mathrm{tot}}^i & = UP_{\mathrm{tot}}^iU^\dagger . \notag \end{align}