{
  "title": "Cubic scaling and necessary continuity — restored proof edition",
  "original_source_sha256": "83b50a18a3572ee43c6d6a263bd33441f1fe0215d6e3aa3ae4a2f04c7b1f2e3e",
  "restored_source_sha256": "95a085459a368e6d044da9e1c0d6c0056aa72227ac5e7eba2b5da49a4a93d3e4",
  "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": [
        41,
        42,
        43
      ],
      "relevant_printed_pages": [
        30,
        31,
        32
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "a5c1a36f47a2c55d750a8fb6c9207411e52ab8659d4d4d5e7bddf23601f17a20",
    "results": "model-check.json",
    "original_script_sha256": "0924627544f67c6a750137159c066716c7a9209b1adffe31b570a2bc29d71e55",
    "finite_checks": 6,
    "passed": true,
    "mathematical_checks_unchanged": true,
    "scope": "Finite models support the written proof; they do not establish the general theorem."
  },
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_nine_exercise_solutions_unchanged": true,
  "proof_review": [
    "Read the entire original U013 lesson and all nine complete solutions. Preserved every displayed equation and the full exercise section.",
    "Checked the pointwise tangent normalization and partial phase against the actual current corank proof, including the individual radial exclusions, positive shared dimension, zero two-jet and quarter-corank order shift. No rank-constant neighborhood is assumed.",
    "Checked transfer of universal boundedness through properly supported elliptic graph quantizations and both microlocal inverses. Made compact intermediate/exterior supports and smoothing errors on smaller angular sets explicit.",
    "Checked that the chosen amplitude is one fixed ordinary symbol for the whole argument: compact base factors, smooth angular factor, radial cutoff and the full real-order weight estimates. Verified exact equality of the input cutoff on every sufficiently concentrated test profile.",
    "Checked the actual kernel action, partial Fourier transform, central frequency T squared and spread T, every Jacobian and the exact half-density normalization. Added the frequency-cutoff and dominated-convergence justification of the absolutely convergent representation.",
    "Checked every third-order Taylor block and the fourth-order remainder. The pure nongraph cubic survives, graph/frequency factors decay, and the actual Fourier phase X-prime dot zeta remains after demodulation.",
    "Checked the whole-frequency amplitude bound for every real order, explicitly separating the region near total frequency cancellation. The polynomial majorant and compact-profile Fourier decrease are integrable uniformly in the parameters.",
    "Checked dominated convergence, the exact partial Fourier inversion factor and uniform convergence on compact output sets by splitting Fourier tails. No unproved global output L2 limit is used.",
    "Checked the explicit compact input cancelling Q at the marked output, its nonzero limiting value, continuity, positive local L2 lower bound and the cases of zero input or output nongraph dimension.",
    "Checked the different input and output Jacobians and the T-independent localized norm constant for the same fixed operator. Their comparison proves m at most minus corank divided by twelve at every point. Kept the universal quantifier and positive-bundle-rank qualification.",
    "Checked the exact changing-rank phase, both signed pullback forms, ranks, nonzero common canonical one-form, exact scaled identity and the distinction between sufficient, necessary and constant-rank sharp bounds.",
    "Completed the general kappa at least two-thirds argument: all mixed cubic weights are strictly negative, the remainder uses min(1,kappa), and a positive-mass profile supplies a nonzero limit when the cubic vanishes. This includes kappa greater than one and explains why zero shrinkage requires actual flattening.",
    "Read the exact admitted H IV printed pages 30–32 and compared the proof scope and normalizations.",
    "Preserved the independently derived correct linear phase subtraction, explicit cancellation estimate, nonzero-profile construction, detailed examples and nine solutions. The exposition and teaching organization are independent, with accurate credit; no book text, figure or exercise collection is redistributed.",
    "Reused the current corank, graph, composition, submanifold, Schwartz Fourier and measure proofs before adding details. Reused all six original finite-model calculations with destination-only changes. Finite computations support rather than replace the general argument."
  ],
  "changes": [
    "Restored every original displayed equation and nine complete solutions with exact current programme proof bindings.",
    "Explicit proper support and interior inverse-error localization, full ordinary-symbol cutoff estimates and actual Fourier-integral action.",
    "Explicit continuity of the limiting profile and the complete general shrinkage argument, including kappa greater than one and its correct fourth-order remainder.",
    "Accurate approved-edition credit and source comparison; all independent alternative proofs retained.",
    "Retained all six finite exact/numerical models with only destination changes; general proofs remain written in the lesson."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}