Rank-revealing QR sweeps give exact mixed canonical gauge

Let $M\in {\mathbb{F}}^{m\times n}$ be any core matricization. Scan its columns in order, starting with an empty list and retaining a column exactly when it is not in the span of those already retained. At termination the retained columns are independent, and every skipped column lies in their final span; hence they span the column image of $M$ and their count is $q=\mathrm{rank}(M)$ by [F2]. If $q>0$, apply [F4] to obtain a matrix $Q$ with orthonormal columns spanning those retained columns; their Gram–Schmidt coordinates form an upper triangular $R_{11}$ with nonzero diagonal. If $q=0$, then $M=0$ and use the empty factorization.

Move the retained column indices to the front by a permutation $\mathrm{\Pi }$. Every remaining column lies in the span of $Q$, so it equals $Q$ times its coordinate vector $Q^{\ast }$ applied to that column. Thus $M\mathrm{\Pi }=Q[R_{11}{R}_{12}]$ for some $R_{12}$, with $Q^{\ast }Q=I_q$; equivalently $M=QB$ where $B=[R_{11}{R}_{12}]{\mathrm{\Pi }}^{\ast }$. This is an exact rank-revealing thin QR factorization for every rectangular shape.

Starting at the left boundary, apply step 2.1 to each left matricization of cores $1,\dots ,s-1$. Replace the current core matricization $M$ by $Q$ and absorb $B$ into the next core. Since $M=QB$, the shared-index contraction in [F1] is unchanged; $Q^{\ast }Q=I_q$ makes the replaced core left-isometric by [F3]. Induction over the swept cores gives the required left-isometric prefix.

Starting at the right boundary, apply step 2.1 to the adjoint $M^{\ast }$ of each right matricization of cores $d,\dots ,t+1$. If $M^{\ast }\mathrm{\Pi }=QB$, then $M=\mathrm{\Pi }{B}^{\ast }{Q}^{\ast }$. Replace the current right matricization by $Q^{\ast }$ and absorb $\mathrm{\Pi }{B}^{\ast }$ into the preceding core. The shared-index contraction in [F1] is unchanged, and $Q^{\ast }Q=I_q$ makes the replaced matricization right-isometric by [F3]. Induction gives the right-isometric suffix and preserves the represented array, while leaving cores $k<s$ untouched. If $t<d$, the final absorption into core $t$ can change the contracted center coefficients; if $t=d$, there is no right sweep.

The scan drops only columns already in the span of retained columns, so each bond dimension changes only from its former size to the exact matrix rank. If any swept matricization had rank zero, that core would be zero and the full TT contraction would be zero, contrary to the nonzero-state hypothesis; hence every reduced swept bond has positive dimension. Steps 3.1 and 3.2 establish the exact mixed canonical gauge.