G Proof of the Fundamental Theorem: Supporting Results

This appendix supports Chapter 11. It proves phase normalization, coefficient extraction from matched gauges, direct-sum conjugation, the exact consequences of length-dependent proportionality, coordinate reindexing, and unitary refinements. The first part concerns coefficients and block gauges; the second identifies matched coordinates with the literal total bond space.

G.1 Matched phases and coefficient extraction

This section isolates two elementary steps used by the main proof. First, the sector partners obtained in both directions determine a finite bijection. Second, normalization of the matched normal tensors forces every gauge scalar to be a phase.

Theorem G.1.1 Bijective matching under eventual proportionality

✓

Let \(P\) and \(Q\) be BNT canonical forms whose total MPV families are eventually nonzero and proportional. There is a bijection \(\beta :J(Q)\simeq J(P)\) such that every \(Q_k\) has an equal-dimensional, gauge-phase-equivalent partner \(P_{\beta (k)}\) satisfying

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

Proof ▶

Apply the full-basis matching theorem first from \(Q\) to \(P\) and then to the reverse proportionality from \(P\) to \(Q\). If \(P_j\) and \(P_{j'}\) had the same partner \(Q_k\), then symmetry and transitivity would give

\begin{align} P_j\sim _{\mathrm{gp}}Q_k\sim _{\mathrm{gp}}P_{j'} & \Longrightarrow P_j\sim _{\mathrm{gp}}P_{j'}, \notag \end{align}

contradicting BNT basis separation. Both partner maps are therefore injective. The two finite sector sets have equal cardinality, so the forward map is bijective and retains the displayed overlap limit together with the dimension and gauge data.

Lemma G.1.2 Unit phase from matched normalized BNT blocks

✓

Let \(P\) and \(Q\) be BNT canonical forms. If, for every positive length \(N\) and word \(\sigma \), a sector \(P_j\) and a sector \(Q_k\) have MPV families related by

\begin{align} V^{(N)}(Q_k)_\sigma & = \zeta ^N V^{(N)}(P_j)_\sigma , \label{eq:ft_matched_mpv_phase_norm} \end{align}

then \(|\zeta |=1\).

Proof ▶

The BNT normalization gives self-overlap limits of modulus \(1\) for both sectors. The MPV identity (??) gives, for every positive \(N\),

\begin{align} O_{Q_kQ_k}(N) & = |\zeta |^{2N}\, O_{P_jP_j}(N). \notag \end{align}

Since \(O_{Q_kQ_k}(N)\to 1\) and \(O_{P_jP_j}(N)\to 1\), taking limits forces \(|\zeta |^{2N}\to 1\) and hence \(|\zeta |=1\).

G.2 Coefficient identities from gauge-phase equalities

The main coefficient theorem takes the matched phases as input. The next result shows how those phases are selected from gauge-phase equivalences before the linear-independence argument is applied.

Lemma G.2.1 Full-basis coefficient identity from matched sectors

✓

Let \(P\) and \(Q\) be BNT canonical forms with the same MPV family at every positive length, matched by a bijection \(\beta :J(Q)\simeq J(P)\) with bond-dimension equalities and a gauge-phase equivalence between \(Q_k\) and \(P_{\beta (k)}\) for every \(k\). Then each sector carries a unit-modulus phase \(\zeta _k\), and eventually

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

This is the equal-MPV coefficient comparison of [ CPGSV16 , Appendix MPV proof, lines 1187–1188 ] .

Proof ▶

For each matched sector, the gauge-phase equivalence gives \(V^{(N)}(Q_k)_\sigma =\zeta _k^NV^{(N)}(P_{\beta (k)})_\sigma \). Lemma G.1.2 gives \(|\zeta _k|=1\). Applying Lemma 11.1.2 to the same phases gives \(c_N^{(\beta (k))}(P)=\zeta _k^Nc_N^{(k)}(Q)\) for all sufficiently large \(N\).

G.3 The scalar-threaded proportional identity

For proportional total MPVs, coefficient extraction retains the one scalar that relates the two total vectors at each length. This identity is exact, but it does not imply the scalar-free identity needed to recover copy weights; the counterexample in Remark 11.2.6 shows that no such implication is possible.

Lemma G.3.1 Proportional matched-phase coefficient identity, scalar-threaded

✓

Let \(P\) be a BNT canonical form and \(Q\) a sector decomposition whose total tensors generate eventually nonzero proportional MPV families. Suppose a bijection \(\beta :J(Q)\simeq J(P)\) and scalars \(\zeta _k\) satisfy the matched MPV phase identities (??) for every sector \(k\). Then there is a single eventually-nonzero scalar sequence \(c_N\) and a threshold \(N_0\) such that, for all \(N{\gt}N_0\) and every \(k\in J(Q)\),

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

The scalar \(c_N\) is the same for every sector \(k\): it is the single global proportionality scalar of [ CPGSV16 , Theorem thm1, lines 1167–1170 ] at length \(N\).

Proof ▶

For each \(N\) on the proportionality tail, expand both total MPVs in their sector bases and use the global proportionality scalar \(c_N\):

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

The matched phase identities (??) give \(|V^{(N)}(Q_k)\rangle =\zeta _k^N|V^{(N)}(P_{\beta (k)})\rangle \); because every sector is matched, reindexing the \(Q\)-sum by \(\beta \) rewrites the right-hand side entirely in the \(P\)-basis. The BNT eventual linear independence of the \(P\)-basis and Lemma F.1.11 then extract the coefficient identities (??) for all sufficiently large \(N\).

G.4 Direct-sum conjugation

We now prove the general block-matrix identities used by the global gauge in Section 11.2. These statements do not choose or match sectors; they only combine conjugacies that have already been obtained.

Definition G.4.1 Invertible matrix associated to a unitary

✓

A unitary matrix \(U\) determines an invertible matrix whose inverse is \(U^\dagger \).

Lemma G.4.2 Unitary block-diagonal gauges

✓

If every block gauge \(U_k\) is unitary, then the flattened block-diagonal gauge \(U=\bigoplus _k U_k\) is unitary.

Proof ▶

The products are computed blockwise:

\begin{align} U U^\dagger & = \bigoplus _k U_kU_k^\dagger = \bigoplus _k\mathbb 1 = \mathbb 1. \notag \end{align}

Lemma G.4.3 Direct-sum conjugation by the flattened gauge

✓

If, for every block \(k\) and physical index \(i\), \(B_k^i=X_kA_k^iX_k^{-1}\), then the weighted direct sums satisfy

\begin{align} \bigoplus _k \mu _k B_k^i & = X\left(\bigoplus _k \mu _k A_k^i\right)X^{-1}, \notag \end{align}

where \(X\) is the flattened block-diagonal gauge from Definition 11.2.1. Hence the two assembled tensors are gauge equivalent.

Proof ▶

Multiplication of block-diagonal matrices is computed blockwise. The scalar weight \(\mu _k\) commutes with the conjugation on the \(k\)th block, and the reindexing from dependent block coordinates to the flattened bond index preserves products and inverses.

Theorem G.4.4 Global gauge from a copy-weight matching

✓

Suppose that the sectors of \(P\) and \(Q\) are matched, with bond-dimension equalities, nonzero phases, and block gauges satisfying \(Q_k^i=\zeta _k X_kP_{\beta (k)}^iX_k^{-1}\). If the same phases also give a copy-weight matching, then the flattened \(Q\)-tensor is obtained from the matched-coordinate \(P\)-tensor by the direct-sum gauge \(X=\bigoplus _k(\mathbf{1}_{r_k}\otimes X_k)\). This is only a reformulation of Theorem 11.2.2.

Proof ▶

The copy-weight matching supplies \(\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}\). Together with \(Q_k^i=\zeta _kX_kP_{\beta (k)}^iX_k^{-1}\), these are exactly the two hypotheses (??) of Theorem 11.2.2.

G.5 Additional proportional consequences

The following implications are useful when copy weights satisfy extra relations. They are not conclusions of CPSV16 Theorem 2.10. In particular, their additional hypotheses fail for the proportional BNT forms in Remark 11.2.6.

Theorem G.5.1 Additional proportional global gauge under copy-weight matching

✓

Let \(P\) and \(Q\) be BNT canonical forms whose total tensors generate eventually nonzero proportional MPV families. Then the proportional sector-matching theorem gives \(\beta \), the bond-dimension equalities, the unit phases \(\zeta _k\), and the block gauges \(X_k\). If one additionally assumes a copy-weight matching for these same phases, then the flattened \(Q\)-tensor is obtained from the matched-coordinate \(P\)-tensor by the direct-sum gauge \(X=\bigoplus _k(\mathbf{1}_{r_k}\otimes X_k)\).

Proof ▶

The proportional sector-matching theorem supplies \(\beta \), \(X_k\), and unit phases \(\zeta _k\) satisfying \(Q_k^i=\zeta _kX_kP_{\beta (k)}^iX_k^{-1}\). Since \(|\zeta _k|=1\), each \(\zeta _k\) is nonzero. A copy-weight matching gives \(\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}\), so Theorem G.4.4 gives the direct-sum gauge.

Theorem G.5.2 Additional proportional global gauge under coefficient identities

✓

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, 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 every sector \(k\) and physical index \(i\). If one additionally assumes that these same phases satisfy the matched-sector coefficient identities \(c_N^{(\beta (k))}(P)=\zeta _k^N\, c_N^{(k)}(Q)\) for all sufficiently large \(N\), then they determine a copy-weight matching and hence the flattened \(Q\)-tensor is obtained from the matched-coordinate \(P\)-tensor by the direct-sum gauge \(X=\bigoplus _k(\mathbf{1}_{r_k}\otimes X_k)\).

Proof ▶

The sector-matching theorem gives \(\beta \), \(X_k\), and \(|\zeta _k|=1\) with \(Q_k^i=\zeta _kX_kP_{\beta (k)}^iX_k^{-1}\). Since \(\zeta _k\ne 0\), the coefficient identities \(c_N^{(\beta (k))}(P)=\zeta _k^Nc_N^{(k)}(Q)\) produce \(\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}\) by Definition 11.1.4. Theorem G.5.1 then applies to this copy-weight matching.

G.6 Matched flattened coordinates

Let \(\beta :J(Q)\simeq J(P)\) match the basis sectors and let \(\tau _k\) match the copies in sector \(k\). Flatten the pair \((k,q)\) to the copy index of \(Q\). The matched \(P\)-data on that index are the weight and tensor

\begin{align} \widetilde\mu _{k,q} & =\mu _{\beta (k),\tau _k(q)}, & \widetilde P_{k,q} & =P_{\beta (k)}, \notag \end{align}

where the second expression is read after the matched bond-dimension identification. The block gauge repeats \(X_k\) on every copy \(q\).

Definition G.6.1 Matched weights, tensors, and repeated gauges

✓

The matched flattened data are the functions \(\widetilde\mu \), \(\widetilde P\), and \((k,q)\mapsto X_k\) defined above on the flattened copy coordinates of \(Q\).

Lemma G.6.2 Equality of the matched total bond dimensions

✓

If matched sectors have equal bond dimensions and matched copy sets are bijective, then \(P\) and \(Q\) have the same total bond dimension.

Proof ▶

Reindex first by the sector bijection and then by each copy bijection. Every summand \(\dim P_{\beta (k)}\) becomes \(\dim Q_k\), so

\begin{align} \sum _{j\in J(P)}r_j(P)\dim P_j & =\sum _{k\in J(Q)}r_k(Q)\dim Q_k. \notag \end{align}

Definition G.6.3 Sector and bond-coordinate equivalences

✓

The sector bijection, copy bijections, and bond-dimension equalities induce an equivalence of the dependent triples \((j,q,a)\) and \((k,q',a')\). After flattening the dependent bond coordinate, this gives an equivalence of the standard total bond-index sets.

Proof ▶

Send \((k,q,a)\) to \((\beta (k),\tau _k(q),a)\), using the bond-dimension equality to identify the last coordinate. The inverse uses \(\beta ^{-1}\) and \(\tau _k^{-1}\). Both composites are the identity componentwise. Conjugating by the standard equivalence between dependent triples and a finite interval gives the flattened bond-coordinate equivalence.

Theorem G.6.4 Equal-MPV gauge with all matched witnesses

✓

Under the hypotheses of the equal-MPV theorem, one may choose simultaneously the sector bijection, bond-dimension equalities, copy bijections, phases, block gauges, total-dimension equality, and the flattened global gauge. They satisfy the matched block and weight identities of Theorem 11.2.4, together with the conjugation of \(Q\) by the matched-coordinate \(P\)-tensor.

Proof ▶

Theorem 11.2.4 supplies all witnesses except the displayed equality of total dimensions. Apply Lemma G.6.2 to its sector and copy bijections. The resulting equality identifies the total bond spaces, while the global conjugation equation is unchanged.

G.7 Permutation gauges and literal coordinates

The matched \(P\)-tensor has the correct blocks but lists them in \(Q\)-order. A permutation matrix restores the literal order of \(P\). We record its invertible and unitary forms before proving the coordinate conversion used in Theorem 11.2.5.

Definition G.7.1 Invertible permutation gauge

✓

For a permutation \(\rho \) of a finite bond-index set, let \(\Pi _\rho \) be its permutation matrix. The corresponding element of the general linear group is denoted \(G_\rho \).

Lemma G.7.2 Matrix and inverse identities for a permutation gauge

✓

The matrix of \(G_\rho \) is \(\Pi _\rho \), and the matrix of its inverse is \(\Pi _{\rho ^{-1}}\).

Proof ▶

The first identity is the definition of the permutation gauge. Since the inverse permutation reverses the coordinate relabelling, \(G_\rho ^{-1}=G_{\rho ^{-1}}\), which gives the second identity.

Every row and every column of \(\Pi _\rho \) contains exactly one nonzero entry, equal to \(1\). Hence

\begin{align} \Pi _\rho \Pi _{\rho ^{-1}} & =\mathbb {1}, & \Pi _\rho ^\dagger & =\Pi _{\rho ^{-1}}. \notag \end{align}

Together with Lemma G.7.2, these identities show that \(G_\rho ^\dagger =G_\rho ^{-1}\), so the permutation gauge is unitary.

The coordinate equivalence of Definition G.6.3 induces a permutation \(\rho \) after the total-dimension identification. With the permutation-matrix convention used here, conjugation reindexes both matrix coordinates by \(\rho \):

\begin{align} \bigl(G_\rho M G_\rho ^{-1}\bigr)_{ij} & =M_{\rho (i),\rho (j)}. \label{eq:app_ft_perm_conjugation} \end{align}

Equivalently, the right-hand side is the submatrix of \(M\) along \(\rho \times \rho \). A convention with inverse indices merely replaces the chosen permutation by \(\rho ^{-1}\); we use the convention in (??) to match the formal coordinate identity. Applying that identity to \(P_{\mathrm{tot}}^i\) gives the matched-coordinate tensor. Composing \(G_\rho \) with the block-diagonal gauge therefore gives the literal gauge of Theorem 11.2.5. This also explains why the total-dimension cast and the sector permutation must occur in that order.

G.8 Unitary witness refinements

The main text proves that a specified gauge between left-canonical irreducible sectors can be replaced by a unitary without changing its phase. We record the existential form and then repeat the construction over every matched copy.

Lemma G.8.1 Unitary gauge for left-canonical irreducible tensors

✓

Let \(A\) and \(B\) be left-canonical irreducible tensors of common positive bond dimension that are gauge-phase equivalent. Then the gauge can be taken unitary: there are a unitary matrix \(U\) and a scalar \(\zeta \) with \(|\zeta |=1\) such that, for every physical index \(i\),

\begin{align} B^i & =\zeta \, UA^iU^\dagger . \notag \end{align}

Proof ▶

Unpack the gauge-phase equivalence as \(B^i=\zeta XA^iX^{-1}\) with \(X\) invertible and \(\zeta \ne 0\). Lemma 11.3.1.2 replaces \(X\) by a unitary \(U\) while retaining this same phase and proves \(|\zeta |=1\).

Lemma G.8.2 Unitary gauge in matched sector coordinates

✓

Let \(P\) and \(Q\) be BNT canonical forms whose total tensors generate the same MPV family at every positive length. There are a sector bijection \(\beta \), copy bijections \(\tau _k\), phases \(\zeta _k\), sector unitaries \(U_k\), and a unitary block-diagonal matrix \(X\) such that

\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 \\ |\zeta _k| & = 1. \notag \end{align}

The matched sectors and copies have equal bond dimensions and multiplicities. If \(P_{\mathrm{match}}\) denotes the corresponding reordered block assembly, then \(Q_{\mathrm{flat}}^i=XP_{\mathrm{match}}^iX^{-1}\).

Proof ▶

Apply Lemma 11.3.1.2 to each matched sector, retaining the phase in the copy-weight identity. Repeating \(U_k\) on every copy gives \(X=\bigoplus _{k,q}U_k\), which is unitary by Lemma G.4.2. For every matched copy \((k,q)\),

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

These are precisely the blocks of \(Q_{\mathrm{flat}}^i=XP_{\mathrm{match}}^iX^{-1}\).