TNLean paper-gap notes

Mathematical notes recording each discrepancy between a cited source and the formal development: missing hypotheses, scalar corrections, scope restrictions, and replacement proof routes. 104 notes: 37 open (14 high-severity), 67 resolved or historical. Classified by status; source keys in the legend below. The conventions are stated in the policy note. Cite a note by its permanent URL https://sirui-lu.com/TNLean/paper-gaps/<name>.pdf or via paper-gaps.bib.

Open 37

2026-07-31The Mutual-Information Bound for Matrix Product Density Operators cpsv16 false-source · cited ×25 (tex)
2026-04-30The Perron–Frobenius Rank-One Step in Cirac–Perez-Garcia–Schuch–Verstraete 2017 Appendix C.2 cpsv16 false-source · cited ×67 (tex)
2026-05-10Canonical-Form Weight Normalization and the Proportional MPV Comparison cpsv16 unfaithful · cited ×3 (tex)
2026-08-10Normalization of the tensor printed in CPSV16 Example 4.12 cpsv16 false-source · cited ×15 (tex)
2026-08-12Finite-Ring Entropy Corrections for CPSV16 Examples 4.10 and 4.11 cpsv16 false-source · cited ×41 (tex)
2026-07-16Purification RFP in Cirac–Perez-Garcia–Schuch–Verstraete 2017 cpsv16 false-source · cited ×12 (tex)
2026-07-31Per-block RFP Isometry Form and the Source Isometry Condition in CPSV16 cpsv16 false-source · cited ×38 (tex)
2026-06-08Zero correlation length uses the physical-trace transfer in Cirac–Perez-Garcia–Schuch–Verstraete 2017 cpsv16 open-gap · cited ×20 (tex)
2026-04-30The Root-Channel Kraus-Rank Step in Theorem 4.1 dccsp17 unfaithful · cited ×1 (tex)
2026-06-13The Two-Dimensional Overlapping-Window Strengthening of the Normal PEPS Fundamental Theorem peps open-gap · cited ×185 (tex)
2026-05-01Conditional After-Blocking Fundamental Theorem versus the CPSV Statement rmp open-gap · cited ×1 (tex)
2026-06-16Source-Notation Gap: The Majumdar–Ghosh Example Tensor rmp open-gap · cited ×1 (tex)
2026-05-06Paper Comparison for BNT Inputs in the Non-periodic MPS Fundamental Theorem rmp open-gap · cited ×2 (tex)
2026-06-17Deferred Result: The Translation-Invariant Bond-Dimension Lower Bound for the W State rmp open-gap · cited ×2 (tex)
2026-07-30Active physical support compression in the CPSV16 commuting-form argument cpsv16 scope-restriction · cited ×11 (tex)
2026-08-12Exact arithmetic scope for CPSV16 Examples 4.10 and 4.11 cpsv16 scope-restriction · cited ×1 (tex)
2026-06-12The NNCPH ground-state clause in Cirac–Perez-Garcia–Schuch–Verstraete 2016 cpsv16 scope-restriction · cited ×61 (tex)
2026-06-15The parent commuting Hamiltonian condition in Cirac–Perez-Garcia–Schuch–Verstraete 2016 cpsv16 scope-restriction · cited ×3 (tex)
2026-06-10Pure zero correlation length and local orthogonality in Cirac–Perez-Garcia–Schuch–Verstraete 2017 cpsv16 scope-restriction · cited ×31 (tex)
2026-06-04; revised 2026-08-02Proof-Route Alignment for the Periodic Overlap Dichotomy (de las Cuevas–Cirac–Schuch–Pérez-García) dccsp17 scope-restriction · cited ×20 (tex)
2026-08-12Full Support in MPU Normality and Theorem III.8 mpu scope-restriction · cited ×48 (tex)
2026-08-10Fixed Virtual Bond Dimension in MPU Tensor Equivalence mpu scope-restriction · cited ×2 (tex)
2026-08-16The Supplied Fixed Pair in the Reduced-Representative Step of Proposition IV.5 mpu scope-restriction · cited ×4 (tex)
2026-05-06Normal PEPS Fundamental Theorem: Section 3 Route peps scope-restriction · cited ×95 (tex)
2026-05-05The Direct-Sum Input from the Older MPS Representation Paper pgvwc07 scope-restriction · cited ×5 (tex)
2026-08-22The Intertwiner Count in the Quadratic Reconstruction Argument pgvwc07 local-correction · cited ×1 (tex)
2026-08-12Scope of the Formal Periodic Decomposition Relative to PGVWC07 pgvwc07 scope-restriction · cited ×3 (tex)
2026-08-12Scope of the Formal Site-Independent Construction Relative to PGVWC07 pgvwc07 scope-restriction · cited ×2 (tex)
2026-05-12Scope of the Formal Block-Separation Lemmas Relative to PGVWC07 Uniqueness pgvwc07 scope-restriction (tex)
2026-06-11PGWSVC08 Physical String Order versus Virtual-Boundary Nondecay pgwsvc08 scope-restriction · cited ×9 (tex)
2026-06-04The Martingale Principal-Angle Estimate for MPS Parent Hamiltonians rmp scope-restriction · cited ×10 (tex)
2026-06-04Scope Note: The AKLT On-Site SO(3) Symmetry Is Formalized; Its Cohomological Classification Is Not rmp scope-restriction · cited ×3 (tex)
2026-06-04Example Parent Hamiltonians in Cirac–Perez-Garcia–Schuch–Verstraete 2021 rmp scope-restriction (tex)
2026-08-22The One-Step Subspace in the Quantum Wielandt Inequality spwc10 scope-restriction · cited ×3 (tex)
2026-08-10The Hidden Range Restriction in the Source-u Contraction mpu clarification · cited ×1 (tex)
2026-05-05Theorem Surface Audit for Canonical Form, BNT, and the Non-periodic MPS Fundamental Theorem tnlean clarification · cited ×8 (tex)
2026-05-08Dependency Audit for the Non-periodic MPS Fundamental Theorem Blueprint tnlean clarification (tex)

Resolved and historical 67

2026-07-11The SAL to Eta-Local Structure Step in Cirac–Perez-Garcia–Schuch–Verstraete 2017 Appendix C.2 cpsv16 open-gap resolved · cited ×83 (tex)
2026-07-11Diagonal Restriction and the Vertical Canonical Form in Proposition 4.13 cpsv16 unfaithful resolved · cited ×2 (tex)
2026-07-11The Vertical Direction in Proposition 4.13 Is the Index-Exchanged Tensor cpsv16 unfaithful resolved (tex)
2026-05-13Historical Note: Rate-Quantified BNT Cross-Overlap Decay for a Retired Projection Argument cpsv16 unfaithful historical · cited ×3 (tex)
2026-07-18BNT Uniqueness and Zero Coefficients in CPSV16 cpsv16 false-source resolved · cited ×38 (tex)
2026-07-30A Phase Obstruction to Canonical-Form Renormalization-Flow Convergence cpsv16 false-source resolved · cited ×7 (tex)
2026-05-10CPSV16 Lemma equalMPS Gauge-Phase Components and Resolution cpsv16 unfaithful resolved · cited ×4 (tex)
2026-05-11Historical Note: the Fixed-Block Cancellation Step in CPSV16 Theorem II.1 cpsv16 unfaithful historical · cited ×6 (tex)
2026-05-13Historical Note: Non-Dominant Per-Block Projection in CPSV16 § II Step 1 cpsv16 unfaithful historical · cited ×2 (tex)
2026-07-25Adjacent regions obstruct the unrestricted CID forward implication in Cirac–Perez-Garcia–Schuch–Verstraete 2017 cpsv16 false-source resolved · cited ×6 (tex)
2026-07-14A raw-weight counterexample to the pure-state ZCL equivalence in Cirac–Perez-Garcia–Schuch–Verstraete 2017 cpsv16 false-source resolved · cited ×4 (tex)
2026-08-10The Pure-State Blocking-up-to-Gauge Equivalence Is False cpsv16 false-source resolved · cited ×5 (tex)
2026-07-28Closure Record for the Strong-Subadditivity Equality Characterization and Hayashi–Markov Decomposition cpsv16 open-gap resolved · cited ×1 (tex)
2026-06-22Strong Subadditivity as a Standalone Axiom and Its Lieb-Concavity Elimination Route cpsv16 open-gap resolved (tex)
2026-07-23The Physical Complement in the Topological Gibbs Decomposition cpsv16 open-gap resolved · cited ×55 (tex)
2026-07-23The Arbitrary-Chain Topological Projector Recursion cpsv16 open-gap resolved · cited ×11 (tex)
2026-08-15The CPSV16 Unit-Weight Convention and Renormalization-Fixed-Point Scale cpsv16 false-source resolved · cited ×24 (tex)
2026-08-09Shifted Trace Powers and the Exact MPU Transfer Spectrum cpsv17 open-gap resolved · cited ×1 (tex)
2026-06-06Injective PEPS Gauge Consistency Needs a Connectivity Hypothesis peps false-source resolved · cited ×17 (tex)
2026-04-30PEPS Gauge Uniqueness and Balanced Edge Scalars peps false-source resolved · cited ×19 (tex)
2026-06-11Block-Diagonal Boundary Conditions in the Degenerate Parent-Hamiltonian Ground Space rmp open-gap resolved · cited ×44 (tex)
2026-07-20Fixed-Length Chain Equality Does Not Imply Combined-Tensor MPV Equality tnlean unfaithful resolved (tex)
2026-07-15Sharp Finite-Length Block Separation for Canonical-Form Tensors cpsv16 scope-restriction resolved · cited ×18 (tex)
2026-07-09CPGSV17 Blocked-Basis Chi Families and Uniform BNT Labels cpsv16 scope-restriction resolved · cited ×29 (tex)
2026-07-12Physical-Isometry Transport for the Blocked MPDO Fixed Point cpsv16 scope-restriction resolved · cited ×4 (tex)
2026-07-12The Final Traced Subspin in the Definition of S_0 cpsv16 local-correction resolved · cited ×7 (tex)
2026-07-12The Implication Label in the Proof of MPDO Theorem 4.9 cpsv16 local-correction resolved · cited ×7 (tex)
2026-07-12Zero-Weight Sector Completions in the MPDO Refinement Map cpsv16 local-correction resolved · cited ×24 (tex)
2026-07-13The Periodic-Sector Argument in Proposition 4.13 cpsv16 local-correction resolved · cited ×14 (tex)
2026-07-13The Vertical Canonical Form Retains Only the Nonzero Physical Sectors cpsv16 local-correction resolved · cited ×37 (tex)
2026-07-21The Trace-Ratio Exponent in the Blocked Vertical-Operator Formula cpsv16 local-correction resolved · cited ×1 (tex)
2026-07-18Active Canonical-Form Blocks in the BNT Characterization cpsv16 local-correction resolved · cited ×15 (tex)
2026-07-31The Commuting-Form-to-SAL Implication for Simple MPDOs cpsv16 scope-restriction resolved · cited ×213 (tex)
2026-07-15The marginal-support expansion in CPSV16 Appendix D.2 cpsv16 local-correction resolved · cited ×4 (tex)
2026-07-20The Figure 11 Fusion Map Is a Coisometry onto the Active Sectors cpsv16 local-correction resolved · cited ×28 (tex)
2026-05-10Basis-of-Representatives Scope Restriction in the Fundamental Theorem Formalization cpsv16 scope-restriction resolved · cited ×18 (tex)
2026-05-30Global and Per-Sector Unit-Weight Witnesses in the CPSV16 BNT Argument cpsv16 scope-restriction resolved · cited ×12 (tex)
2026-07-15The Sector Decomposition in the GSNNCH Definition of Cirac–Pérez-García–Schuch–Verstraete cpsv16 local-correction resolved · cited ×15 (tex)
2026-07-30The Index Typo in the Pure-State Renormalization-Flow Equation cpsv16 local-correction resolved · cited ×3 (tex)
2026-07-16Nilpotent Basis Elements in the Simple-Tensor Definition of Cirac–Pérez-García–Schuch–Verstraete cpsv16 local-correction resolved · cited ×1 (tex)
2026-07-23The Domain Boundary of the Topological Gibbs Decomposition cpsv16 local-correction resolved · cited ×29 (tex)
2026-05-13Historical Note: Two-Layer Sector Refinement of CPSV16 eq:II_ABasicTensors cpsv16 scope-restriction historical · cited ×2 (tex)
2026-08-02A Mixed-Prefix Repair of the Two-Site Sector Comparison cpsv16 local-correction resolved · cited ×45 (tex)
2026-07-19CPSV16 Vertical-Sector Invertibility and Sector Relabelling cpsv16 scope-restriction resolved · cited ×5 (tex)
2026-07-28Support scope of the HJPW product-reference Petz factorization hjpw04 scope-restriction resolved (tex)
2026-08-12The Endpoint Rank-Product Exponent in MPU Index Well-Definedness mpu local-correction resolved · cited ×4 (tex)
2026-08-08Nil-Matrix Product-Length Bound in the MPU Blocking Argument mpu local-correction resolved · cited ×11 (tex)
2026-08-09MPU One-Letter Mixed Residual Trace: Resolved Local Correction mpu local-correction resolved · cited ×5 (tex)
2026-08-12Two-Projection Decomposition: Repeated-Projector Typo mpu local-correction resolved · cited ×2 (tex)
2026-07-30Common Identity Coefficients in the Block Ground-Space Intersection pgvwc07 scope-restriction resolved · cited ×8 (tex)
2026-05-12Scope of the Formal Canonical-Form Reductions Relative to PGVWC07 pgvwc07 scope-restriction resolved · cited ×14 (tex)
2026-04-30Normal Parent-Hamiltonian Range Reduction in Cirac–Perez-Garcia–Schuch–Verstraete 2021 rmp scope-restriction resolved · cited ×15 (tex)
2026-07-26Schuch–Pérez-García–Cirac SPT Interpolation: Upper-Range Correction spc11 local-correction resolved · cited ×1 (tex)
2026-07-27Vertical Canonical-Form Grouping in Proposition 4.13 cpsv16 clarification resolved · cited ×46 (tex)
2026-07-21Per-Pair Support in the CPSV16 Figure 11 Decomposition cpsv16 clarification resolved · cited ×17 (tex)
2026-05-11Projective Reading of Proportional MPVs in CPSV16 cpsv16 clarification resolved · cited ×3 (tex)
2026-05-08Power Sums in the BNT Weight Comparison cpsv16 clarification historical · cited ×4 (tex)
2026-07-10The Projector-Closure Sufficient Condition for Canonical Form cpsv16 clarification resolved · cited ×5 (tex)
2026-07-26Length-Dependent Scalars Do Not Determine Copy Weights in the Proportional BNT Theorem cpsv16 clarification resolved · cited ×2 (tex)
2026-07-15Closure record for saturation of the area law from MPDO renormalization fixed points cpsv16 clarification resolved · cited ×5 (tex)
2026-05-10Common Sector Relabeling as an Unconditional Identity cpsv16 clarification resolved (tex)
2026-05-30Removing Length-Zero Bookkeeping from the Blocked Canonical-Form Decomposition cpsv16 clarification resolved · cited ×3 (tex)
2026-06-17Phase-Normalized Weights in the Normal-Canonical Implication to Irreducible Form dccsp17 clarification resolved · cited ×1 (tex)
2026-07-18The Product-Marginal Reference in the Relative-Entropy Form of Strong Subadditivity hjpw04 clarification resolved (tex)
2026-08-17The Finite-Range Cyclic Knabe Coefficient knabe88 clarification resolved · cited ×1 (tex)
2026-05-06Injective PEPS Fundamental Theorem: Section 3 Route peps clarification resolved · cited ×30 (tex)
2026-04-30Common Blocking and Span Equality in the Non-periodic MPS Fundamental Theorem rmp clarification resolved (tex)

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