{
  "schema": "AN04-restored-cubic-scaling-proof-map/v1",
  "proofs": [
    {
      "id": "CS:P0",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "Fix \\(c_0\\).",
      "dependencies": [
        "CR:R1",
        "CR:R2",
        "CR:P1",
        "CR:P2"
      ],
      "scope": "Pointwise tangent normalization, positive shared dimension, actual partial phase/order and zero two-jet without rank constancy"
    },
    {
      "id": "CS:T1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Lemma 1.2",
      "dependencies": [
        "CS:P0",
        "GE:D9",
        "GE:D11",
        "RC:C1",
        "RC:C2",
        "RC:C9",
        "CT:T2",
        "SN:A1"
      ],
      "scope": "Transfer universal boundedness through proper graph inverses on fixed compact base and angular supports"
    },
    {
      "id": "CS:A0",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "## 2.",
      "dependencies": [
        "CS:P0",
        "OSC:O1",
        "OSC:O3",
        "U001:Q3",
        "U001:P14.2-powers",
        "HT:H2"
      ],
      "scope": "One fixed ordinary amplitude with complete cutoff bounds, compact kernel localization and exact action on shrinking inputs"
    },
    {
      "id": "CS:S1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "Put \\(\\kappa=2/3\\)",
      "dependencies": [
        "CS:A0",
        "P3:M4",
        "U001:Q3"
      ],
      "scope": "Exact input Jacobian, partial Fourier transform, central frequency and spread for arbitrary positive shrinkage"
    },
    {
      "id": "CS:S2",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "Evaluate the output at",
      "dependencies": [
        "CS:S1",
        "P3:M3",
        "P3:M4",
        "U001:Q3",
        "CR:P1"
      ],
      "scope": "Actual absolutely convergent output action and exact frequency/parameter changes of variables"
    },
    {
      "id": "CS:C1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
        "CS:P0",
        "CS:S2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All weighted third-order monomials, pure nongraph cubic limit and uniform compact fourth-order remainder"
    },
    {
      "id": "CS:A1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
        "CS:A0",
        "U001:P14.2-powers"
      ],
      "scope": "Whole-frequency normalized majorant for every real amplitude order, including near total frequency cancellation"
    },
    {
      "id": "CS:L1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "The function \\(\\widehat u(\\zeta,Y'')\\)",
      "dependencies": [
        "CS:C1",
        "CS:A1",
        "CS:S2",
        "U001:Q3",
        "U001:Q4",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Integrable uniform Fourier tails, dominated limit, exact inverse factor and uniform convergence on compact output sets"
    },
    {
      "id": "CS:N1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "## 5.",
      "dependencies": [
        "CS:L1",
        "P3:M3",
        "P3:M5",
        "U001:P4"
      ],
      "scope": "Explicit compact profile cancelling the cubic at the marked output, nonzero continuous limit and positive local L2 lower bound"
    },
    {
      "id": "CS:N2",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "The box in the original output variables",
      "dependencies": [
        "CS:N1",
        "CS:S1",
        "CS:S2",
        "CS:T1",
        "P3:M4",
        "P3:M5",
        "HT:H3"
      ],
      "scope": "Distinct output/input Jacobians and one fixed operator force the full pointwise necessary bound, including positive-rank bundles"
    },
    {
      "id": "CS:M1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "## 6.",
      "dependencies": [
        "CS:P0",
        "CR:E4",
        "CS:C1",
        "CS:N2",
        "CR:A2",
        "SN:A1"
      ],
      "scope": "Exact rank-changing canonical relation, common form and radial hypotheses, scaled identity and the gap between sufficient/necessary bounds"
    },
    {
      "id": "CS:K1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "More generally, the cubic-controlled family",
      "dependencies": [
        "CS:C1",
        "CS:A1",
        "CS:L1",
        "CS:N1",
        "CS:N2",
        "CR:M1",
        "SN:R1"
      ],
      "scope": "All shrinkages kappa>=2/3, correct remainder for kappa>1, nonzero limits, strongest cubic obstruction and justified flat zero-shrinkage case"
    },
    {
      "id": "CS:E1",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "CS:N2",
        "CR:A2",
        "SN:A1"
      ],
      "scope": "Original Exercise 7.1 and complete unchanged solution"
    },
    {
      "id": "CS:E2",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "CS:S1"
      ],
      "scope": "Original Exercise 7.2 and complete unchanged solution"
    },
    {
      "id": "CS:E3",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "CS:C1"
      ],
      "scope": "Original Exercise 7.3 and complete unchanged solution"
    },
    {
      "id": "CS:E4",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "CS:A1"
      ],
      "scope": "Original Exercise 7.4 and complete unchanged solution"
    },
    {
      "id": "CS:E5",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "CS:N1"
      ],
      "scope": "Original Exercise 7.5 and complete unchanged solution"
    },
    {
      "id": "CS:E6",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "CS:N2"
      ],
      "scope": "Original Exercise 7.6 and complete unchanged solution"
    },
    {
      "id": "CS:E7",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "CS:M1"
      ],
      "scope": "Original Exercise 7.7 and complete unchanged solution"
    },
    {
      "id": "CS:E8",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "CS:S2",
        "CS:L1"
      ],
      "scope": "Original Exercise 7.8 and complete unchanged solution"
    },
    {
      "id": "CS:E9",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "56673a8b95fdc95201f877440b69a2c50f3e1065f9d9d9490821135fd8018b0e",
      "proof_locator": "**Exercise 7.9 ",
      "dependencies": [
        "CS:K1"
      ],
      "scope": "Original Exercise 7.9 and complete unchanged solution"
    }
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    "P2:QG6",
    "P2:QG7",
    "P2:QG8",
    "P3:L1",
    "P3:L2",
    "P3:L3",
    "P3:M0",
    "P3:M1",
    "P3:M2",
    "P3:M3",
    "P3:M4",
    "P3:M5",
    "P3:M6",
    "P3:M7",
    "P3:M8",
    "PH:F0",
    "PH:F1",
    "PH:F2",
    "PH:F3",
    "PH:F4",
    "PH:F5",
    "PH:F6",
    "PH:F6a",
    "PH:F7",
    "PS:PS0",
    "PS:PS1",
    "PS:PS2",
    "PS:PS3",
    "PS:PS4",
    "PS:PS5",
    "PS:PS6",
    "RC:C0",
    "RC:C1",
    "RC:C2",
    "RC:C3",
    "RC:C5",
    "RC:C6",
    "RC:C7",
    "RC:C8",
    "RC:C9",
    "RC:K0",
    "RC:K1",
    "RC:K2",
    "RC:K3",
    "RC:K4",
    "RC:K5",
    "RC:K6",
    "RC:K7",
    "RC:K8",
    "SN:A1",
    "SN:F0",
    "SN:F1",
    "SN:H1",
    "SN:H2",
    "SN:L1",
    "SN:N1",
    "SN:R1",
    "ST:T4",
    "ST:T6",
    "TG:B0",
    "TG:B1",
    "TG:B2",
    "TG:B3",
    "TG:T0",
    "TG:T5",
    "TG:T6",
    "TG:U1",
    "TG:Z2",
    "TZ:E5",
    "TZ:Z0",
    "TZ:Z4",
    "TZ:Z5",
    "TZ:Z6",
    "WM:W0",
    "WM:W1",
    "WM:W2"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}