{
  "schema": "AN04-restored-cubic-scaling-proof-map/v1",
  "proofs": [
    {
      "id": "CS:P0",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "The box in the original output variables",
      "dependencies": [
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        "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "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": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "CS:N2",
        "CR:A2",
        "SN:A1"
      ],
      "scope": "Exercise 7.1 and solution"
    },
    {
      "id": "CS:E2",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "CS:S1"
      ],
      "scope": "Exercise 7.2 and solution"
    },
    {
      "id": "CS:E3",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "CS:C1"
      ],
      "scope": "Exercise 7.3 and solution"
    },
    {
      "id": "CS:E4",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "CS:A1"
      ],
      "scope": "Exercise 7.4 and solution"
    },
    {
      "id": "CS:E5",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "CS:N1"
      ],
      "scope": "Exercise 7.5 and solution"
    },
    {
      "id": "CS:E6",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "CS:N2"
      ],
      "scope": "Exercise 7.6 and solution"
    },
    {
      "id": "CS:E7",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "CS:M1"
      ],
      "scope": "Exercise 7.7 and solution"
    },
    {
      "id": "CS:E8",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "CS:S2",
        "CS:L1"
      ],
      "scope": "Exercise 7.8 and solution"
    },
    {
      "id": "CS:E9",
      "source": "cubic-scaling-and-necessary-continuity.md",
      "source_sha256": "24dcabbd8657273a447f0222c0bdb0e0872da2e61b93fb8e572d9dca407b4489",
      "proof_locator": "**Exercise 7.9 ",
      "dependencies": [
        "CS:K1"
      ],
      "scope": "Exercise 7.9 and solution"
    }
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    "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"
  ],
  "external_dependency_policy": {
    "record": "../20261004-free-stationary-phase/external-proof-dependencies.json",
    "notice": "The elementary proof providers have exact earlier-programme routes in C10/C20, with public PDF destinations and editable TeX passages. The author English edition remains available. Human source text and figures stay external to this CC0 export and retain their own terms.",
    "mathematical_proofs_rewritten": false
  }
}