doc(MPS/ParentHamiltonian): align PGVWC boundary-word terminology

TNLean/MPS/ParentHamiltonian/BNTBlockDiagonalPGVWCComparison.lean

@@ -7,19 +7,19 @@ import TNLean.MPS.ParentHamiltonian.BNTBlockDiagonalCrossingTrace
 /-!
 # Boundary-condition comparisons for block-diagonal parent spaces
 
-The word-indexed comparison from arXiv:quant-ph/0608197
+The boundary-crossing comparison from arXiv:quant-ph/0608197
 \[
   A^j_\beta C^j_{i,\rho}
   =
   \bigl((\mu_j^NX_j)A^j_\beta\bigr)A^j_\rho
 \]
-is the local boundary-crossing coordinate form of the \(C^j,D^j\) trace
+is the cut-adapted boundary-crossing form of the \(C^j,D^j\) trace
 comparison from arXiv:quant-ph/0608197, Theorem 12, proof lines 1446--1451,
 after specializing \(D^j_\beta\) to \((\mu_j^N X_j)A^j_\beta\). The normalized
 \(E^j\)-calculation then implies the periodic-boundary single-block constraints
 and the finite-range block-diagonal periodic-boundary equality used here.
 
-The source proof writes this comparison with site-indexed matrices
+The source proof writes this comparison with boundary-indexed matrices
 \(C^j_{i_1}\), \(D^j_{i_{m+1}}\), and the derived matrix \(E^j\). The words
 \(\beta\) and \(\rho\) below are cut-adapted coordinates for the same
 periodic-boundary comparison, not additional source terminology.
@@ -36,7 +36,7 @@ boundary representation to periodic single-block states.
 
 Under the normalized BNT hypotheses, every vector in the block-diagonal
 periodic-boundary ground space has block-diagonal boundary conditions \(X_j\). If those
-same boundary conditions satisfy the source comparison, in the word-indexed
+same boundary conditions satisfy the source comparison, in the cut-adapted
 boundary-crossing form
 \[
   A^j_\beta C^j_{i,\rho}
@@ -119,7 +119,7 @@ single-block states.
 Assume the vector \(\psi\in\mathcal G_{N,L}(\oplus_j\mu_jA_j)\) has already
 been written with block-diagonal boundary matrices \(X_j\). If every
 boundary-crossing interval has the simultaneous block-word spanning property,
-then the local constraint produces the word-indexed \(C^j,D^j\) comparison,
+then the local constraint produces the cut-adapted \(C^j,D^j\) comparison,
 and the normalized \(E^j\)-calculation gives
 \[
   \Gamma_N^{A_j}(\mu_j^NX_j)\in\mathcal G_{N,L}(A_j).
@@ -433,16 +433,16 @@ theorem chainGroundSpace_toTensorFromBlocks_eq_iSup_of_crossing_pgvwc_comparison
 periodic-boundary equality in the finite BNT range.
 
 This theorem assumes the source \(C^j,D^j\) comparison only up to the
-word-indexed matrix identity
+cut-adapted matrix identity
 \[
   A^j_\beta C^j_{i,\rho}
   =
   \bigl((\mu_j^NX_j)A^j_\beta\bigr)A^j_\rho
 \]
 for every boundary-crossing interval, local word \(\beta\) before the cut,
 and outside word \(\rho\), with \(D^j_\beta\) already specialized to
-\((\mu_j^NX_j)A^j_\beta\). The words \(\beta\) and \(\rho\) are formal
-word coordinates for the opened boundary comparison; they reindex the source
+\((\mu_j^NX_j)A^j_\beta\). The words \(\beta\) and \(\rho\) are
+cut-adapted coordinates for the opened boundary comparison; they reindex the source
 end-site comparison by blocked words rather than replacing the source indices.
 The normalized \(E^j\)-calculation
 and the block-injective crossing-window argument then give the
@@ -518,7 +518,7 @@ the equality conclusion
   =
   \bigvee_j\mathcal G_{N,L}(A_j),
 \]
-under the assumed word-indexed \(C^j,D^j\) comparison.
+under the assumed cut-adapted \(C^j,D^j\) comparison.
 
 **Scope restriction (length-\(L_0\) injectivity range):** Theorem 12 of
 arXiv:quant-ph/0608197 assumes \(L\ge 3(b-1)(L_0+1)+1\). This theorem is stated in the current BNT

blueprint/src/chapter/ch13_parent_hamiltonian_block_diagonal_chain_equality.tex

@@ -12,15 +12,14 @@ \subsection{Periodic-boundary ground space for a block-diagonal tensor}
 comparison in the periodic ring. The source proof
 \cite{PerezGarcia2007Matrix} writes this comparison with $C^j_{i_1}$,
 $D^j_{i_{m+1}}$, and the normalized matrix $E^j$; the symbols $\beta$ and
-$\rho$ below are formal word variables introduced after opening the periodic
-cut, not additional terminology from the paper. In these
-coordinates the source comparison becomes
+$\rho$ below are the cut-adapted words obtained after opening the periodic
+cut. In these coordinates the source comparison becomes
 \[
     A^j_\beta C^j_{i,\rho}
     =
     \bigl(\mu_j^N X_j A^j_\beta\bigr)A^j_\rho,
 \]
-which is the formal word-coordinate counterpart of
+which is the blocked-word form of
 $A^j_{i_{m+1}}C^j_{i_1}=D^j_{i_{m+1}}A^j_{i_1}$ with
 $D^j_\beta=(\mu_j^NX_j)A^j_\beta$.
 
@@ -385,7 +384,7 @@ \subsection{Periodic-boundary ground space for a block-diagonal tensor}
     \]
     and the $N$-site ground spaces $G_N(A_j)=\mathcal G_{N,N}(A_j)$
     have internal sum.
-    The present formal statement is the equality-level consequence of the
+    The present statement is the equality-level consequence of the
     boundary-condition comparison in
     \cite[Theorem~12, proof lines~1436--1451]{PerezGarcia2007Matrix} in the
     length-$L_0$ injectivity range

docs/paper-gaps/cpgsv21_block_diagonal_parent_ground_space.tex

@@ -244,7 +244,7 @@ \section{The remaining boundary condition}
 block-diagonal boundary matrices whose block components again belong to the
 blockwise periodic-boundary spaces.
 
-The present formal boundary isolates the remaining source comparison in local
+The present boundary isolates the remaining source comparison in cut-adapted
 coordinates for boundary-crossing windows.  Suppose
 \begin{align}
   \psi
@@ -255,7 +255,7 @@ \section{The remaining boundary condition}
 For a cyclic interval beginning at \(i\) and crossing the boundary cut, put
 \(a=i+L-N\).  The word \(\beta\) records the part of the interval before the
 chosen cut, while \(\rho\) records the outside word of length \(N-L\).  These
-are local coordinates for the proof, not names used in the source statement.
+are the blocked words obtained by opening the source boundary-index comparison.
 The conditional theorem assumes that, for every block \(j\) and every such
 outside word \(\rho\), there is a matrix \(E_{j,i,\rho}\) such that for every
 word \(\beta\) of length \(a\),
@@ -283,7 +283,7 @@ \section{The remaining boundary condition}
 \leanid{MPSTensor.chainGroundSpace_toTensorFromBlocks_eq_iSup_and_iSupIndep_of_complementary_identities}.}
 
 The live blueprint records that larger source statement with explicit citations
-and separately marks the local cut coordinates used by the formalization,
+and separates the cut-adapted coordinates from the paper's boundary indices,
 without claiming that the source statement is proved.\footnote{Blueprint locations:
 \path{def:parent_hamiltonian_ground_space_bnt} for the block-diagonal
 periodic-boundary ground space,
@@ -495,7 +495,7 @@ \section{Conclusion}
 and
 \leanid{MPSTensor.chainGroundSpace_toTensorFromBlocks_eq_iSup_and_iSupIndep_of_trace_decomposition_bnt_c1_span}.}
 
-The word-indexed \(C^j,D^j\) route is closer to the proof of
+The cut-adapted \(C^j,D^j\) route is closer to the proof of
 \cite[Theorem~12]{PerezGarcia2007MPSReps}.
 The words \(\beta\) and \(\rho\) are local word coordinates introduced after
 opening and blocking the cyclic boundary comparison; they reindex the source