{
  "schema": "oa-mod-closed-form-source-notice/v2",
  "title": "Recovering operators from energy forms",
  "source": "src/closed-positive-forms.md",
  "source_sha256": "6a1387411dd13152645cc16d8ed3d6f73988af10b2f9b366b1c14217911202d2",
  "original_exposition": "OpenAI Codex (AI); exact original model variant unverified",
  "retained_license": "CC0-1.0",
  "new_text_license": "QF-NEW-TEXT-LICENSE.txt",
  "contributors": {
    "current_QF_mathematical_review": "GPT-6 Astra (OpenAI), Ultra",
    "prerequisite_proof_correspondence_and_clarifications": "GPT-6 Astra (OpenAI), Ultra",
    "source_and_input_binding_preparation": "GPT-6 Astra (OpenAI), Ultra",
    "root_exact_byte_proof_review": "GPT-6 Astra (OpenAI), Ultra",
    "reading_presentation": "GPT-6.1 Sol (OpenAI), Ultra, October 2026",
    "original_QF_exposition": "OpenAI Codex (AI); exact historical model variant unverified"
  },
  "free_primary_comparisons": [
    {
      "authors": "Zoltán Sebestyén and Zsigmond Tarcsay",
      "title": "Basic representation theorems of forms",
      "version": "arXiv:2505.09588v1, 14 May 2025",
      "url": "https://arxiv.org/pdf/2505.09588v1",
      "html": "https://arxiv.org/html/2505.09588v1",
      "raw_pdf_sha256": "87861af02617564e661e7c188c02fec6e5e3b0e1abf5f0759c42500da78d25c8",
      "bytes": 304811,
      "retrieved_at": "2026-10-04T02:47:08.6591161Z",
      "read_loci": [
        "Section 2 setup and Lemma 2.1 with proof, printed pp.2–3",
        "Corollary 2.7 with proof, printed p.7 (PDF page 7); web parsed lines 423–443"
      ],
      "correspondence": "Dense closed nonnegative forms, exact square-root domain and bounded-form-functional operator-domain criterion agree with QF-03 on arbitrary real or complex Hilbert spaces.",
      "boundary": "Their proof invokes closed Q*Q theory. QF independently proves representation from bounded inclusion and spectral calculus; QF-10 derives T*T afterward. This source is a comparator, never a back-edge construction input.",
      "reuse": "External human research reading only; no text, PDF or human structured component imported."
    },
    {
      "authors": "Barry Simon",
      "title": "A canonical decomposition for quadratic forms with applications to monotone convergence theorems",
      "publication": "Journal of Functional Analysis 28 (1978), 377–385",
      "url": "https://math.caltech.edu/SimonPapers/81.pdf",
      "doi": "10.1016/0022-1236(78)90094-0",
      "raw_pdf_sha256": "6bedf6ed356c6d4bff14477e4d7437a6f20b3c467ea792af724aae637bc14d60",
      "bytes": 473712,
      "retrieved_at": "2026-10-04T02:47:11.4532397Z",
      "read_loci": [
        "Section 1 finite-energy definition and form order, pp.377–379",
        "Theorem 3.1 with proof, p.382 (PDF page 6)",
        "Section 4 and Theorem 4.1, pp.383–384 (PDF pages 7–8)"
      ],
      "correspondence": "Theorem 3.1 gives increasing sequential closed-form convergence under a closed upper bound. Theorem 4.1 extends to nondense forms and removes that upper bound; resolvents vanish on the perpendicular complement.",
      "boundary": "The human paper proves sequential versions. QF-06–08 retain their own written directed-net and locally uniform power arguments. Do not claim Simon proves the net statement or use his Theorem 4.2 pointer to another book as a complete standalone proof provider.",
      "reuse": "External human research reading only; no PDF or text redistributed."
    }
  ],
  "complete_earlier_proofs": "closed-form-proof-bindings.json"
}