{
  "schema": "AN04-restored-function-normal-forms-proof-map/v1",
  "proofs": [
    {
      "id": "FN:R0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "## 1.",
      "dependencies": [
        "PC:S0",
        "PC:H0",
        "U001:P14.2-powers"
      ],
      "scope": "Hamilton signs and weights, allowed multipliers and degree reduction"
    },
    {
      "id": "FN:R1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Theorem 1.1",
      "dependencies": [
        "FN:R0",
        "PC:H5"
      ],
      "scope": "Full real principal-type model with exact mark and radial obstruction"
    },
    {
      "id": "FN:W0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Theorem 2.1",
      "dependencies": [
        "FN:R0",
        "U001:P14.2-powers",
        "U001:P3"
      ],
      "scope": "Weighted bracket scalar equation and exact normal dilation field"
    },
    {
      "id": "FN:W1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Choose submersion coordinates \\((p,q,z)\\)",
      "dependencies": [
        "FN:W0",
        "U001:P3",
        "SN:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Smooth tangential Taylor division and full backwards system"
    },
    {
      "id": "FN:W2",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Take nested relatively compact",
      "dependencies": [
        "FN:W1",
        "G2:NF7"
      ],
      "scope": "Entire backwards orbit in a common compact box, integrable drift and limiting fixed manifold"
    },
    {
      "id": "FN:W3",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "We need estimates for every derivative",
      "dependencies": [
        "FN:W2",
        "G2:NF2",
        "G2:NF3",
        "G2:NF4",
        "G2:NF5",
        "U001:P15.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All-order variational bounds with explicit integrable-coefficient inequality"
    },
    {
      "id": "FN:W4",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "\\tag{2.7}",
      "dependencies": [
        "FN:W3",
        "P3:M3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Positive infinite-time integral, every derivative and fixed-set value"
    },
    {
      "id": "FN:W5",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "To verify the equation,",
      "dependencies": [
        "FN:W4",
        "SN:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact scalar equation and uniqueness among smooth local germs"
    },
    {
      "id": "FN:W6",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "For the conic assertion let",
      "dependencies": [
        "FN:W5",
        "PC:H0",
        "PC:H5"
      ],
      "scope": "Uniqueness gives degrees and consistent conic extension"
    },
    {
      "id": "FN:S0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "## 3.",
      "dependencies": [
        "FN:R0"
      ],
      "scope": "Complete multiplier sign invariant at the zero set"
    },
    {
      "id": "FN:S1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
        "FN:R0",
        "SN:F0",
        "SN:F1",
        "PC:H5",
        "G2:G21"
      ],
      "scope": "Commuting positive momentum from transverse Hamilton flows, homogeneous datum and full marked coordinate completion"
    },
    {
      "id": "FN:S2",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Theorem 3.2",
      "dependencies": [
        "FN:S0",
        "FN:W6",
        "FN:S1"
      ],
      "scope": "Full simple complex model and both marked signs"
    },
    {
      "id": "FN:D0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Smooth complex division.**",
      "dependencies": [
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P3"
      ],
      "scope": "Almost-analytic extension with all parameter derivatives and prescribed jets"
    },
    {
      "id": "FN:D1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Apply this construction to",
      "dependencies": [
        "FN:D0",
        "U001:P2",
        "U001:P3"
      ],
      "scope": "Smooth complex root as a real two-dimensional contraction, including all parameters"
    },
    {
      "id": "FN:D2",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "We next divide by the known graph",
      "dependencies": [
        "FN:D1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact real segment identity with the antiholomorphic error"
    },
    {
      "id": "FN:D3",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Each derivative of this expression costs",
      "dependencies": [
        "FN:D2",
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All-order flat quotient at a possibly singular graph-zero set and smooth zero extension"
    },
    {
      "id": "FN:D4",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "For \\(a=F\\),",
      "dependencies": [
        "FN:D3",
        "U001:P2"
      ],
      "scope": "Nonvanishing factor and common-domain complex division with no uniqueness claim"
    },
    {
      "id": "FN:D5",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Theorem 4.1",
      "dependencies": [
        "FN:D4",
        "FN:R1",
        "PC:H5"
      ],
      "scope": "Homogeneous one-momentum elimination, nonzero multiplier and exact retained position"
    },
    {
      "id": "FN:F0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "## 5.",
      "dependencies": [
        "FN:R0",
        "SN:F0"
      ],
      "scope": "Full nearby fixed-order and existence hypothesis"
    },
    {
      "id": "FN:F1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Lemma 5.1",
      "dependencies": [
        "FN:F0",
        "SN:F0",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Smooth zero graph and full transverse kth-order factorization"
    },
    {
      "id": "FN:F2",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "For the bracket assertion put",
      "dependencies": [
        "FN:F1",
        "PC:S0"
      ],
      "scope": "Ideal filtration for both Hamilton fields and every bracket tree"
    },
    {
      "id": "FN:F3",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Theorem 5.2",
      "dependencies": [
        "FN:F2"
      ],
      "scope": "Full nonconstant multiplier law including all differentiated terms"
    },
    {
      "id": "FN:F4",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "If \\(F\\ne0\\),",
      "dependencies": [
        "FN:F3",
        "SN:F0",
        "U001:P3"
      ],
      "scope": "Transformed fixed-order existence on every nearby curve and odd sign invariance"
    },
    {
      "id": "FN:K0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Theorem 6.1",
      "dependencies": [
        "FN:F4",
        "FN:R0",
        "SN:F0"
      ],
      "scope": "Finite contact excludes radial real field"
    },
    {
      "id": "FN:K1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "A positive preliminary degree change",
      "dependencies": [
        "FN:K0",
        "FN:R1",
        "FN:D5",
        "FN:F3"
      ],
      "scope": "Actual division multiplier is admissible and preserves the leading contact sign"
    },
    {
      "id": "FN:K2",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Rename these coordinates \\((x,\\xi)\\).",
      "dependencies": [
        "FN:K1",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P14.2-powers"
      ],
      "scope": "Homogeneous zero graph, full Taylor factor and signed smooth kth root"
    },
    {
      "id": "FN:K3",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Apply Theorem 2.1 to",
      "dependencies": [
        "FN:K2",
        "FN:W6",
        "FN:S1"
      ],
      "scope": "Exact fractional weights, full canonical construction and complete model multiplier"
    },
    {
      "id": "FN:K4",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "For negative kth derivative,",
      "dependencies": [
        "FN:K3",
        "FN:F3"
      ],
      "scope": "Both parity operations, marked signs and obstruction in the negative odd case"
    },
    {
      "id": "FN:E0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Lemma 7.1",
      "dependencies": [
        "U001:P3",
        "G2:G21"
      ],
      "scope": "Uniformly elliptic two-real-variable operator with smooth transverse parameters"
    },
    {
      "id": "FN:E1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Proof.** A real linear change",
      "dependencies": [
        "FN:E0",
        "U001:Q3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:M4",
        "P3:M8"
      ],
      "scope": "Complete fundamental-solution calculation by polar substitution and periodic integration"
    },
    {
      "id": "FN:E2",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Choose a compact smooth cutoff",
      "dependencies": [
        "FN:E1",
        "U001:P14.3",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "P3:M4",
        "P3:M5",
        "P3:M7",
        "P1:B1"
      ],
      "scope": "Compact fundamental kernel, Fourier decay, actual convolution and all-real gain"
    },
    {
      "id": "FN:E3",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Let \\(\\chi\\) be a spatial cutoff",
      "dependencies": [
        "FN:E2",
        "P3:M6"
      ],
      "scope": "Uniform small-coefficient Neumann inverse, exact support margin and actual right inverse"
    },
    {
      "id": "FN:E4",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "It remains to justify joint smoothness;",
      "dependencies": [
        "FN:E3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All parameter operator derivatives and inverse identity on the same L2 space"
    },
    {
      "id": "FN:E5",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "For spatial derivatives use difference quotients",
      "dependencies": [
        "FN:E4",
        "P3:L1",
        "P3:L2",
        "P3:M3"
      ],
      "scope": "Exact difference-quotient equation, Fourier Fatou derivative and strong limit"
    },
    {
      "id": "FN:E6",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Inductively, after",
      "dependencies": [
        "FN:E5",
        "P1:B1",
        "P1:B2",
        "P3:L2",
        "P3:M4",
        "P3:M5"
      ],
      "scope": "All spatial/mixed estimates, joint weak derivatives, explicit Fourier smooth representative and seminorm continuity"
    },
    {
      "id": "FN:I0",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "**Theorem 8.1",
      "dependencies": [
        "FN:R0",
        "G2:G21",
        "SN:N2",
        "CP:T0"
      ],
      "scope": "Actual coisotropic zero set, nonzero restricted primitive, exact conic mark and dimension exclusion"
    },
    {
      "id": "FN:I1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Make \\(p\\) degree one.",
      "dependencies": [
        "FN:I0",
        "FN:E6"
      ],
      "scope": "Elliptic restricted Hamilton field and homogeneous common-domain parameter solver"
    },
    {
      "id": "FN:I2",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Let \\(I=(p_1,p_2)",
      "dependencies": [
        "FN:I1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full normal jet ideals and exact four-term exponential bracket"
    },
    {
      "id": "FN:I3",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Initially divide",
      "dependencies": [
        "FN:I2"
      ],
      "scope": "Anti-conjugate first coefficients and separate first-order correction"
    },
    {
      "id": "FN:I4",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "For \\(k>1\\), suppose",
      "dependencies": [
        "FN:I3"
      ],
      "scope": "All higher correction equations, degrees, omitted-term orders and even middle coefficient"
    },
    {
      "id": "FN:I5",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "All corrections (8.8) have degree zero",
      "dependencies": [
        "FN:I4",
        "FN:E6"
      ],
      "scope": "Fixed restricted operator and common parameter domain at every iteration"
    },
    {
      "id": "FN:I6",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "For completeness, realize the formal corrections",
      "dependencies": [
        "FN:I5",
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Uniform parameter smooth-jet realization and homogeneous flat bracket"
    },
    {
      "id": "FN:I7",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Rename \\(e^Wp=p_1+ip_2\\).",
      "dependencies": [
        "FN:I6",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact real bracket division into flat degree-zero coefficients"
    },
    {
      "id": "FN:I8",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "The real fields \\(H_{p_j}\\)",
      "dependencies": [
        "FN:I7",
        "SN:F0",
        "SN:F1",
        "PC:H0"
      ],
      "scope": "Both sequential real transport corrections with zero data, flatness and homogeneous uniqueness"
    },
    {
      "id": "FN:I9",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "The quotient",
      "dependencies": [
        "FN:I8",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every derivative of the complex flat quotient and its nonzero degree-zero extension"
    },
    {
      "id": "FN:I10",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "Finally \\(P_1,P_2\\)",
      "dependencies": [
        "FN:I9",
        "PC:H5"
      ],
      "scope": "Exact commuting momenta, final canonical map and full scalar multiplier"
    },
    {
      "id": "FN:M1",
      "source": "real-and-complex-symplectic-function-normal-forms.md",
      "source_sha256": "476b90ac7be8f68a1c95f8a6b4dd838d062bf71d9352e595cf70516e71aa9031",
      "proof_locator": "### 9.1.",
      "dependencies": [
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