{
  "title": "Kernels, adjoints and clean composition — restored proof edition",
  "original_source_sha256": "c0f7fa6c505d04f032da1b624e3b2498f734fd5ea5d6a9752958ecc96451ed24",
  "restored_source_sha256": "fe31d929d6c87971f271b12cdc09ca6f6849b40ab34d400e14d378f595abaca8",
  "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": [
        28,
        29,
        30,
        31,
        32,
        33,
        34,
        35,
        36
      ],
      "relevant_printed_pages": [
        17,
        18,
        19,
        20,
        21,
        22
      ]
    },
    {
      "edition_id": "ISBN-978-3-642-61497-2-pdf",
      "pdf_pages_read": [
        13,
        141,
        142,
        143,
        144,
        145,
        146,
        147
      ],
      "relevant_printed_pages": [
        126,
        127,
        128,
        129,
        130,
        131,
        132
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "kernel_companion": {
    "source": "scalar-kernels-and-strong-topology.md",
    "sha256": "c28f938eb3996ec910a279f07e5de266aafb79ca6cf868d9901f326c17b60e66",
    "licence": "GFDL-1.2-only",
    "source_lesson": "AN03-U012",
    "source_sha256": "062b34207ea15696d61f1d6df881d91cef51f28bb87988f6100189cd3275f014",
    "retained_body_sha256": "296c230fee18b8d936818f0983e1e49406df6f01f972c551438d2b31f03be02c",
    "adapter_sha256": "4a899d37670a51b72109a883dc74e1a46c9f1985506ee59a9b33a941fa5b250a",
    "imported_blocks_unchanged": true,
    "notices": [
      "notices/COPYING",
      "notices/TITLE_PAGE.md",
      "notices/HISTORY.md",
      "notices/RIGHTS.md"
    ]
  },
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_six_exercise_solutions_unchanged": true,
  "proof_review": [
    "Read the complete retained AN03 kernel proof and bundle adapter. Checked the test LF seminorm criterion and escaping-support evaluation, the strong dual and all support-dependent finite orders.",
    "Checked the full periodic Fourier construction: differential coefficient estimate, Fejer product approximate identity, every differentiated uniformly convergent series, common compact cutoffs and product-test density.",
    "Checked the Baire fixed-support joint estimate, absolute kernel series, exact summability threshold, agreement on products, common-refinement gluing and continuity into the strong distribution dual.",
    "Checked restriction, multiplication, derivatives, compact distributions, tensor iterated pairings, exact product supports and coordinate pullbacks with every Jacobian.",
    "Supplied the previously unbound compact exhaustion directly. Supplied strong compact-distribution approximation with one common support and the common polynomial Fourier bound. Checked finite-chart and bundle topology adapters and exact transpose seminorms.",
    "Checked the reflected input convention, finite anti-dual bundles, half-density order and closed full-punctured-cotangent hypothesis excluding both zero-covector axes.",
    "Checked all smooth and strong distribution mappings, compact approximation uniqueness, noncomparable Fourier tails and the finite spatial/angular cover proving the full wavefront inclusion.",
    "Checked adjoint order, inverse relation, conjugated Maslov factor, finite bundle-map order and preservation of proper support.",
    "Checked clean constant-rank geometry, the proper connected-fiber embeddedness proof and automatic cotangent properness for compact middle base.",
    "Checked the full transposed nonstationary field on unequal frequencies, negative amplitude orders, every fixed output derivative and the finite-seminorm smoothing bound.",
    "Checked the homogeneous middle-variable change, its exact q^(-nY) Jacobian, clean phase rank, amplitude degree and the half-excess order.",
    "Checked cutoff convergence in every slightly higher symbol order, uniform smooth-input convergence, equality with the actual operator product, finite microlocal assembly and closed proper support.",
    "Checked tangent Hessian reduction, exact excess densities, the single (2pi)^(-e/2) factor and compact fiber integration. Added the full ordinary-symbol chain-rule argument from A6, including nonhomogeneous symbols and every derivative.",
    "Checked the one-order remainder and cancellation criterion. Checked the compact-fiber model and all six unchanged solutions, including the nonzero lower term and failed noncompact-fiber hypotheses.",
    "Compared the admitted H IV pages 17-22 and H I pages 126-132. The retained lesson and kernel component have independent explanations and full programme proofs; the kernel uses a Fourier-series construction instead of the book regularization argument. No book text, figure or exercise is distributed.",
    "Both imported mathematical blocks, every original lesson display and all six original solutions remain unchanged. The separate kernel component retains GFDL 1.2, complete licence, original attribution, title information and history."
  ],
  "changes": [
    "Replaced stale original lesson and unpublished AN03 links by exact current programme proofs.",
    "Imported the complete scalar kernel proof and finite bundle adapter unchanged, retaining GFDL 1.2 and all notices.",
    "Supplied the explicit compact exhaustion and manifold bounded-test topology.",
    "Proved strong compact-distribution approximation with one support and a common Fourier bound.",
    "Specified strong distribution topologies and exact transpose continuity.",
    "Made ordinary symbol derivative estimates explicit via the actual A6 argument, and clarified symbol order without assumed homogeneity.",
    "Preserved every original display and all six solved exercises, plus all alternative proofs."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}