= Solution
For the injective translationally invariant case, the <fundamental theorem of matrix product states> states that two tensors $A=\{A^i\}$ and $B=\{B^i\}$ of the same minimal bond dimension generate the same periodic MPS for every sufficiently large length if and only if
$$
A^i=e^{i\theta}XB^iX^{-1}
$$
for every physical index $i$, with one invertible matrix $X$. Equality as normalized rays permits the phase $e^{i\theta}$; equality as vectors for all lengths restricts the resulting factor $e^{iN\theta}$ accordingly. The converse is immediate from cyclicity of the <matrix trace>:
$$
\operatorname{Tr}(A^{i_1}\cdots A^{i_N})
=e^{iN\theta}\operatorname{Tr}(B^{i_1}\cdots B^{i_N}).
$$
For the nontrivial direction, <blocking a matrix product state>[block] enough sites that both tensors are injective and define
$$
\Gamma_A^L(X)
=\sum_{i_1,\ldots,i_L}
\operatorname{Tr}(XA^{i_1}\cdots A^{i_L})
|i_1\cdots i_L\rangle,
$$
with $\Gamma_B^L$ defined similarly. Injectivity means that $\Gamma_A^L$ and $\Gamma_B^L$ have trivial <kernel of a linear map>[kernels]. Equality of all sufficiently long periodic states implies equality of the local support spaces $\operatorname{im}\Gamma_A^L=\operatorname{im}\Gamma_B^L$, so there is an invertible linear map $F$ on the virtual matrix algebra satisfying
$$
\Gamma_A^L=\Gamma_B^L\circ F.
$$
Compare two adjacent blocks and contract arbitrary environments on their left and right. Because both block maps are injective, equality of the physical contractions forces
$$
F(XY)=F(X)F(Y),
\qquad
F(I)=I.
$$
Thus $F$ is a unital algebra automorphism of $M_\chi(\mathbb C)$. By <every automorphism of a full matrix algebra is inner>, $F(X)=X_0XX_0^{-1}$ for an invertible $X_0$. Applying this relation to a block with one physical site exposed gives
$$
A^i=e^{i\theta}X_0B^iX_0^{-1}.
$$
This proves the theorem and identifies the freedom as the <gauge equivalence of injective matrix product state tensors>. For noninjective tensors, their canonical forms first split into injective blocks; equality then permits a permutation of equivalent blocks together with a similarity transformation and phase on each block.
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