{
  "title": "Graph operators, continuity and Egorov — restored proof edition",
  "original_source_sha256": "99d388c57a84502fd3f9c6e10cce8efe10ba0cdfca2e4ed46f3c5a45f91670a9",
  "restored_source_sha256": "ccd02119f17ad4bef20a9d8795876385d50bb9979ae250585d69d32c2e83abf8",
  "original_authorship": "GPT-6.1 Sol (OpenAI), Ultra, September 2026",
  "restoration_review": "GPT-6 Astra (OpenAI), Ultra, 5 October 2026",
  "source_policy": "Independent exposition with complete programme proofs and exact scholarly references.",
  "source_references": [
    {
      "edition_id": "ISBN-978-3-642-00136-9-pdf",
      "pdf_pages_read": [
        36,
        37,
        38,
        39,
        40,
        69,
        70,
        71
      ],
      "relevant_printed_pages": "25–28 and 58–60; printed 29 was background only"
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "global_and_circle_companion": {
    "source": "global-inverses-and-periodic-multipliers.md",
    "sha256": "2e52c62b6bd21b2701d9a2ccd0b4da8a3ecdefd89a2c35927c4c8bd2e0491b82",
    "licence": "CC0-1.0",
    "independently_written": true
  },
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_eleven_exercise_solutions_unchanged": true,
  "proof_review": [
    "Read the entire retained graph lesson, including the complete ordered matrix gauge and all eleven exercise solutions. All original displays and the complete exercise section remain unchanged.",
    "Checked symplectic half volume, exact intrinsic and reduced orders, determinant normalization, Maslov graph units and ordered bundle contractions.",
    "Checked mixed graph coordinates from input-only second jets, frequency invertibility and the full left-amplitude reduction with supported asymptotic remainders.",
    "Checked actual T*T before extension, smooth-input mapping, finite ordinary PDO seminorms and finite graph-branch assembly without claiming that cross-branch products are pseudodifferential.",
    "Completed the TT* frequency change: fixed invertible derivative comparison on a convex cone proves global injectivity there, then the inverse theorem and homogeneity give every derivative bound.",
    "Checked small fixed jets using nested Taylor estimates, all high-frequency tail bounds, actual L2 kernel norms, finite product approximation and compact norm limits against CT T1-T2.",
    "Checked all real Sobolev exponents, proper reducers and inverse remainders, common compact-support approximation and equality with the original distributional action against P1 and WM W1/W6.",
    "Checked the proper base relation and cutoff of a global canonical graph. Proved the global ordinary matrix inverse by locally finite compact kernel sums of K3 inverses and K4 partitions, with separate source and target bundle cutoffs and both smoothing errors.",
    "Checked elliptic graph inverse, symbol surjectivity, exact inverse line and both ordered inverse identities. Retained the noninverse coefficient in Egorov and the noncommuting matrix product.",
    "Read the exact retained NF6-NF11 ordered matrix ODE and parameter proofs. Checked both inverse equations, factorial bounds, all parameter/frequency/time derivatives and fixed finite-cylinder ellipticity.",
    "Checked the full ordinary residual operator, exact differential commutator, negative-i correction integral, every differentiated residual order and global support justification for the fixed extension.",
    "Checked asymptotic summation leaving M unchanged, all time derivatives, two-sided matrix parametrix and smoothing conjugation. Return to the canonical graph retains the factor order and arbitrary real graph orders.",
    "Proved circle Fourier completeness from the existing Fejer proof and compact smooth density, Parseval and exact diagonal multiplier tail norms.",
    "Proved the circle regularizer is an ordinary order-minus-one PDO: distributional Fourier differentiation gives off-zero decay, compact-distribution plus Schwartz decomposition permits periodization, exact coefficients identify the multiplier and local kernels identify its principal symbol.",
    "Checked all eleven unchanged solutions, including the exponential half-density model and subprincipal sign, noninverse coefficient, noncommuting pulses and the actual first symbolic residual.",
    "Read the admitted H IV printed pages 25-28 and 58-60 (printed 29 only as background). The retained teaching explanations, ordered matrix extension and eleven solved exercises are independently written. No book text, figures or exercises are redistributed; all earlier component notices remain present."
  ],
  "changes": [
    "Exact current P1, P2, K, WM, CT, G2, GS, LR, RC and ST proof providers, preserving all original displayed equations and eleven complete solutions.",
    "Explicit injectivity of the TT* frequency change on a convex cone, with full inverse symbol bounds.",
    "Complete global proper matrix inverse by locally finite patching of existing conic inverses.",
    "Full circle basis, diagonal multiplier norm and periodization proof of its local pseudodifferential kernel.",
    "Retained the complete ordered matrix gauge and all differentiated residual and inverse estimates, with exact ordered ODE and calculus providers.",
    "citations and accurate independent provenance; all alternatives retained."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}