{
  "schema": "AN04-restored-positive-ideals-proof-map/v1",
  "proofs": [
    {
      "id": "PI:G0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "For a real Lagrangian submanifold",
      "dependencies": [
        "G2:G0",
        "G2:G21",
        "U001:P3",
        "U001:P4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Real ideal, Poisson closure, independent generators, full Taylor division and locality"
    },
    {
      "id": "PI:G1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Conversely, suppose a real ideal",
      "dependencies": [
        "PI:G0",
        "U001:P3"
      ],
      "scope": "Full real converse, Hamilton orthogonality and complete vanishing ideal"
    },
    {
      "id": "PI:G2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "A **complex Lagrangian ideal**",
      "dependencies": [
        "QF:A0",
        "G2:G21"
      ],
      "scope": "Exact complex independence, conic locality and real-zero definition"
    },
    {
      "id": "PI:G3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "If \\(\\widetilde u=A u\\)",
      "dependencies": [
        "PI:G2",
        "G2:G0"
      ],
      "scope": "Generator invariance, complex Lagrangian tangent and real isotropy"
    },
    {
      "id": "PI:D0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "The one-variable division input",
      "dependencies": [
        "FN:D0",
        "FN:D1",
        "FN:D2",
        "FN:D3",
        "FN:D4"
      ],
      "scope": "Exact common-neighborhood complex division with earlier complete proof"
    },
    {
      "id": "PI:D1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "It gives the multiple-equation version",
      "dependencies": [
        "PI:D0",
        "QF:A0"
      ],
      "scope": "Full multivariable induction, pivot coefficients and Schur-complement identity"
    },
    {
      "id": "PI:D2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "All divisions use the common neighborhoods",
      "dependencies": [
        "PI:D1",
        "QF:A0"
      ],
      "scope": "Graph form, common domain and invertible generator changes"
    },
    {
      "id": "PI:D3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Multiply by a fixed real cutoff",
      "dependencies": [
        "U001:P4",
        "U001:P13.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Bounded-family almost-analytic extension, diagonal cutoff construction and every mixed derivative"
    },
    {
      "id": "PI:D4",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "On \\(w=0\\), the normal Taylor jets",
      "dependencies": [
        "PI:D3"
      ],
      "scope": "All antiholomorphic jets cancel and full uniform flat estimates hold"
    },
    {
      "id": "PI:D5",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Shrink once so that the segment",
      "dependencies": [
        "PI:D4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact complex straight-path identity with the positive flat-error sign"
    },
    {
      "id": "PI:D6",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Put \\(S=\\sum_j|v_j-T_j|^2",
      "dependencies": [
        "PI:D5",
        "U001:P13.1"
      ],
      "scope": "Conjugate-generator division, all reciprocal estimates and smooth zero extension across singular sets"
    },
    {
      "id": "PI:F0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Indeed, repeatedly divide the coefficients",
      "dependencies": [
        "PI:D6",
        "FD:E0"
      ],
      "scope": "Uniform ideal-power expansion and real-node evaluation at i"
    },
    {
      "id": "PI:F1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "All-power control transfers to derivatives",
      "dependencies": [
        "PI:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "FD:E0"
      ],
      "scope": "Lipschitz-controlled all-power derivative estimates with full tensor interpolation proof"
    },
    {
      "id": "PI:F2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Conversely, (3.1) implies",
      "dependencies": [
        "PI:F1",
        "PI:D6"
      ],
      "scope": "All-power flatness implies every ideal power on one fixed patch"
    },
    {
      "id": "PI:A0",
      "source": "contour-and-gaussian-foundations.md",
      "source_sha256": "cd01e26a0ad6cff9118fb8a5c9b0a9f09951584ed0cb98cfcfb0d9be188de5d2",
      "proof_locator": "## A0.",
      "dependencies": [
        "CP:L0",
        "CP:L1",
        "PC:L0",
        "QF:A0",
        "U001:P2"
      ],
      "scope": "Complex common transversal, real determinant polynomial and actual base cotangent map"
    },
    {
      "id": "PI:N0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "At a real zero of a conic Lagrangian ideal",
      "dependencies": [
        "PI:G3",
        "PI:A0"
      ],
      "scope": "Real adapted cotangent chart with exact second derivative"
    },
    {
      "id": "PI:N1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Complex division in \\(x\\) gives",
      "dependencies": [
        "PI:D2",
        "PI:N0",
        "G2:G41"
      ],
      "scope": "Homogeneous graph generators on a fixed angular patch"
    },
    {
      "id": "PI:N2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Define \\(H(\\xi)=\\sum_j\\xi_jh_j",
      "dependencies": [
        "PI:N1",
        "G2:G0"
      ],
      "scope": "Euler construction, exact Poisson-gradient congruence and degree-one normal form"
    },
    {
      "id": "PI:N3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "If \\(\\widetilde H\\) gives the same ideal",
      "dependencies": [
        "PI:N2",
        "PI:F0",
        "PI:F1",
        "PI:F2"
      ],
      "scope": "Full angular and radial representative ambiguity and converse"
    },
    {
      "id": "PI:N4",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "There is a degree discrepancy",
      "dependencies": [
        "PI:N2"
      ],
      "scope": "Source statement degree correction with an explicit nonzero homogeneous model"
    },
    {
      "id": "PI:P0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Suppose a representative in (4.3)",
      "dependencies": [
        "PI:N2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "G2:G41"
      ],
      "scope": "Nonnegative derivative inequality and exact homogeneous radial factor"
    },
    {
      "id": "PI:P1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "At these frequencies",
      "dependencies": [
        "PI:P0",
        "PI:N3"
      ],
      "scope": "Real-zero description and damping-sign invariance under all-power ambiguity"
    },
    {
      "id": "PI:P2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "A **positive nondegenerate phase**",
      "dependencies": [
        "PI:G2",
        "G2:G41"
      ],
      "scope": "Full positive phase, complex rank and marked nonzero covector hypotheses"
    },
    {
      "id": "PI:P3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "The \\(-I\\) block",
      "dependencies": [
        "PI:P2",
        "QF:A0"
      ],
      "scope": "Extended ideal independence and intrinsic base-coordinate transformation"
    },
    {
      "id": "PI:P4",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Its invertibility is equivalent",
      "dependencies": [
        "PI:P3",
        "G2:F0",
        "PI:A0"
      ],
      "scope": "Full Hessian/transversality equivalence, injective linear critical image and its actual Lagrangian proof"
    },
    {
      "id": "PI:P5",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Division gives",
      "dependencies": [
        "PI:P4",
        "PI:D2",
        "G2:G41"
      ],
      "scope": "Full critical graph with degrees zero and one"
    },
    {
      "id": "PI:P6",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Divide \\(f\\) by this graph",
      "dependencies": [
        "PI:P5",
        "PI:D2"
      ],
      "scope": "Exact critical-value ideal-square membership and homogeneous realization"
    },
    {
      "id": "PI:P7",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "We need a positive critical-value estimate",
      "dependencies": [
        "PI:P6",
        "U001:Q5",
        "U001:F0-COMP"
      ],
      "scope": "Full quantitative matrix bound, coefficient perturbations and semipositive critical-value inequality"
    },
    {
      "id": "PI:P8",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Differentiate (5.7)",
      "dependencies": [
        "PI:P6"
      ],
      "scope": "Actual parameter derivative generators in the extended ideal"
    },
    {
      "id": "PI:P9",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "To identify the eliminated ideal",
      "dependencies": [
        "PI:P8",
        "PI:P7",
        "PI:D6",
        "PI:F2",
        "G2:G41"
      ],
      "scope": "Exact elimination via all-power residue and square-root shift bound, uniformly for bounded families"
    },
    {
      "id": "PI:P10",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Conversely, (5.11) is parametrized",
      "dependencies": [
        "PI:P9",
        "PI:P1"
      ],
      "scope": "Positive normal phase, all adapted coordinate choices and converse"
    },
    {
      "id": "PI:P11",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "At a real zero, write",
      "dependencies": [
        "PI:P10",
        "QF:P0",
        "G2:G41"
      ],
      "scope": "Exact Hermitian tangent sign, symmetric graph and compulsory radial null direction"
    },
    {
      "id": "PI:E0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Split \\(\\theta=(\\theta',\\theta'')\\)",
      "dependencies": [
        "PI:P2",
        "G2:G41"
      ],
      "scope": "Nonzero remaining marked parameter from the radial Hessian identity"
    },
    {
      "id": "PI:E1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Apply complex division and the critical-value",
      "dependencies": [
        "PI:E0",
        "PI:D2",
        "PI:P6",
        "PI:P7"
      ],
      "scope": "Positive partially reduced critical value and exact ideal-square identity"
    },
    {
      "id": "PI:E2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Construct it on a \\(\\theta'\\) sphere",
      "dependencies": [
        "PI:E1",
        "PI:D1",
        "PI:P3"
      ],
      "scope": "Homogeneous reduction, full Schur-complement rank and equality of extended ideals"
    },
    {
      "id": "PI:E3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Mixed parameters must first be homogenized.",
      "dependencies": [
        "PI:E2",
        "PI:P4",
        "PI:P11",
        "U001:F0-DIFF"
      ],
      "scope": "Full partial Legendre construction, nonzero retained covector and exact homogeneous Hessian congruence"
    },
    {
      "id": "PI:A1",
      "source": "contour-and-gaussian-foundations.md",
      "source_sha256": "cd01e26a0ad6cff9118fb8a5c9b0a9f09951584ed0cb98cfcfb0d9be188de5d2",
      "proof_locator": "## A1.",
      "dependencies": [
        "QF:A0",
        "QF:A1",
        "U001:P13.1",
        "U001:P13.2-mertens"
      ],
      "scope": "Bilinear symmetric square factorization and all differentiated noncommuting matrix-series estimates"
    },
    {
      "id": "PI:A2",
      "source": "contour-and-gaussian-foundations.md",
      "source_sha256": "cd01e26a0ad6cff9118fb8a5c9b0a9f09951584ed0cb98cfcfb0d9be188de5d2",
      "proof_locator": "## A2.",
      "dependencies": [
        "U001:F0-ALG",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:M4",
        "U001:P15.1"
      ],
      "scope": "Exact determinant homotopy identity, oriented boundaries and all rapid-error estimates"
    },
    {
      "id": "PI:CS0",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 2. A complete nonstationary proof",
      "dependencies": [
        "U001:P4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:M4"
      ],
      "scope": "Complete complex nonstationary estimate including imaginary-value alternative and all fixed parameter derivatives"
    },
    {
      "id": "PI:CS1",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 3. Exact quadratic factorization",
      "dependencies": [
        "PI:P6",
        "PI:P7",
        "PI:D3",
        "PI:D4",
        "PI:D2",
        "PI:A1",
        "U001:P3"
      ],
      "scope": "Exact quadratic ideal factor, chosen almost-analytic phase, symmetric square map and common smooth inverse"
    },
    {
      "id": "PI:CS2",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 4. First contour:",
      "dependencies": [
        "PI:CS1",
        "PI:A2"
      ],
      "scope": "First contour with nonnegative phase, full gradient damping and uniform flat-error estimates"
    },
    {
      "id": "PI:CS3",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 5. Second contour:",
      "dependencies": [
        "PI:CS2",
        "U001:P3",
        "PI:A2",
        "PI:P7"
      ],
      "scope": "Second contour through the virtual point, exact convex damping identity, fixed boundary cutoffs and complete pullback amplitude"
    },
    {
      "id": "PI:A3",
      "source": "contour-and-gaussian-foundations.md",
      "source_sha256": "cd01e26a0ad6cff9118fb8a5c9b0a9f09951584ed0cb98cfcfb0d9be188de5d2",
      "proof_locator": "## A3.",
      "dependencies": [
        "PI:A2",
        "PI:CS0",
        "PI:CS3",
        "U001:Q2",
        "U001:Q5",
        "U001:Q6",
        "P3:M3",
        "P3:M4",
        "U001:F0-ALG"
      ],
      "scope": "Full matrix Gaussian by moments and path ODE, Abel endpoint, inherited orientation and normal-phase block branch"
    },
    {
      "id": "PI:A4",
      "source": "contour-and-gaussian-foundations.md",
      "source_sha256": "cd01e26a0ad6cff9118fb8a5c9b0a9f09951584ed0cb98cfcfb0d9be188de5d2",
      "proof_locator": "## A4.",
      "dependencies": [
        "PI:D6",
        "PI:F2",
        "PI:CS1",
        "PI:A1",
        "PI:A2",
        "U001:F0-DIFF"
      ],
      "scope": "Exact residue algebra, inverse-map jet induction, bounded extension linearity and all flat errors"
    },
    {
      "id": "PI:CS4",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 6. Gaussian expansion",
      "dependencies": [
        "PI:CS3",
        "PI:A4",
        "U001:Q2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All Gaussian moments, complete absolute Taylor remainder, residue operators and amplitude differential order"
    },
    {
      "id": "PI:CS5",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 7. Gaussian branch,",
      "dependencies": [
        "PI:CS4",
        "PI:A3"
      ],
      "scope": "Full Gaussian branch including null imaginary directions and actual oriented determinant"
    },
    {
      "id": "PI:CS6",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 8. Representatives,",
      "dependencies": [
        "PI:CS5",
        "PI:F2",
        "PI:P7",
        "PI:D6"
      ],
      "scope": "All representative errors, differentiated absolute estimates and actual symbol-family substitution"
    },
    {
      "id": "PI:CS7",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 1. The exact contract",
      "dependencies": [
        "PI:CS0",
        "PI:CS6"
      ],
      "scope": "Complete fixed-patch smooth complex stationary-phase contract including P1–P6 and null imaginary Hessian"
    },
    {
      "id": "PI:O0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "For a test function",
      "dependencies": [
        "PI:CS0",
        "PI:P2",
        "OSC:O1",
        "HT:H8"
      ],
      "scope": "Positive dyadic oscillatory extension, unique cutoff limit and exact finite distribution order"
    },
    {
      "id": "PI:O1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "At a real critical point",
      "dependencies": [
        "PI:P2",
        "PI:P9",
        "G2:G41"
      ],
      "scope": "Critical image consists of real covectors and lies in the eliminated ideal"
    },
    {
      "id": "PI:O2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Take a small spatial cutoff",
      "dependencies": [
        "PI:O1",
        "U001:F0-COMP"
      ],
      "scope": "Full conic gradient separation including extreme radial ratios"
    },
    {
      "id": "PI:O3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "On a shell",
      "dependencies": [
        "PI:O2",
        "PI:CS0",
        "PI:O0",
        "CH:W1"
      ],
      "scope": "Every Fourier decay order from the whole dyadic sum and exact wavefront containment"
    },
    {
      "id": "PI:R0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Work in the adapted graph",
      "dependencies": [
        "PI:P5",
        "PI:D6",
        "HT:H0"
      ],
      "scope": "Bounded normalized amplitude with all angular and logarithmic scale derivatives"
    },
    {
      "id": "PI:R1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Choose a smooth nonnegative bump",
      "dependencies": [
        "PI:R0",
        "U001:P4",
        "P3:M3"
      ],
      "scope": "Actual fixed annular Mellin partition and exact high-frequency identity"
    },
    {
      "id": "PI:R2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "where the integrands defining",
      "dependencies": [
        "PI:R1",
        "G2:G41",
        "HT:H0"
      ],
      "scope": "All rescaled coefficient orders and derivatives, including the degree-one generator loss"
    },
    {
      "id": "PI:R3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Conversely, given",
      "dependencies": [
        "PI:D6",
        "PI:R1",
        "PI:R2"
      ],
      "scope": "Reverse residue construction, actual identity frequency block and fixed support"
    },
    {
      "id": "PI:S0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Let a frequency-only",
      "dependencies": [
        "PI:R2",
        "PI:R3",
        "PI:F2",
        "PI:P0"
      ],
      "scope": "Uniform symbol-ideal hypothesis and every differentiated flat angular bound"
    },
    {
      "id": "PI:S1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Since \\(|e^{-iH}|",
      "dependencies": [
        "PI:S0",
        "P3:L2",
        "U001:P15.1"
      ],
      "scope": "All differentiated damping estimates and smooth inverse Fourier transform"
    },
    {
      "id": "PI:S2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "The same argument handles",
      "dependencies": [
        "PI:S1",
        "PI:N3",
        "PI:P1"
      ],
      "scope": "Uniform residue and homogeneous phase representative independence"
    },
    {
      "id": "PI:T0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "The determinant expression",
      "dependencies": [
        "PI:A3",
        "PI:P4",
        "PI:A4"
      ],
      "scope": "Exact leading-residue interpretation and locally fixed Gaussian branch"
    },
    {
      "id": "PI:T1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "We now prove all parts",
      "dependencies": [
        "PI:O3",
        "PI:P4",
        "PI:CS7"
      ],
      "scope": "Full Fourier rescaling, isolated marked critical patch and off-critical rapid remainder"
    },
    {
      "id": "PI:T2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "To track the radial order",
      "dependencies": [
        "PI:T1",
        "PI:R2",
        "PI:A4"
      ],
      "scope": "Complete Hessian block scaling and determinant degree n minus N"
    },
    {
      "id": "PI:T3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Each coefficient at stationary-phase level",
      "dependencies": [
        "PI:T2",
        "PI:CS6",
        "PI:R2"
      ],
      "scope": "Every coefficient has the full symbol order sigma minus j"
    },
    {
      "id": "PI:T4",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "The leading term is exactly",
      "dependencies": [
        "PI:T3",
        "PI:CS7"
      ],
      "scope": "Exact two-pi constant and absolute sigma minus L remainder"
    },
    {
      "id": "PI:T5",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "The complete earlier H6",
      "dependencies": [
        "PI:T4",
        "HT:H6"
      ],
      "scope": "Actual support-preserving asymptotic sum on one cone"
    },
    {
      "id": "PI:T6",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "This directly proves microlocal smoothness",
      "dependencies": [
        "PI:T5",
        "CH:W1",
        "PI:S1"
      ],
      "scope": "Whole damped Fourier comparison and correctly localized microlocal conclusion"
    },
    {
      "id": "PI:T7",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "For the **converse**,",
      "dependencies": [
        "PI:T0",
        "PI:T6",
        "PI:R3",
        "PI:S1"
      ],
      "scope": "Leading reverse amplitude with full determinant factor and one-order correction"
    },
    {
      "id": "PI:T8",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Repeat (10.10)",
      "dependencies": [
        "PI:T7",
        "HT:H6",
        "PI:T6"
      ],
      "scope": "All lower-order corrections, same closed conic support and complete converse asymptotic sum"
    },
    {
      "id": "PI:M0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "On \\(\\xi=(r,z)\\)",
      "dependencies": [
        "PI:N2"
      ],
      "scope": "Real degree-one graph with full gradient and Hessian calculation"
    },
    {
      "id": "PI:M1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "For a positive example,",
      "dependencies": [
        "PI:P11",
        "PI:A3"
      ],
      "scope": "Non-strict conic positive model, radial null direction and full Gaussian determinant factor"
    },
    {
      "id": "PI:M2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "In three frequency variables",
      "dependencies": [
        "PI:P11"
      ],
      "scope": "Singular crossing real-zero set and exact changing positive tangent rank"
    },
    {
      "id": "PI:M3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "By contrast,",
      "dependencies": [
        "PI:N3",
        "PI:P10"
      ],
      "scope": "Positive tangent planes do not imply a positive ideal, with the cubic sign obstruction"
    },
    {
      "id": "PI:M4",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "For partial parameter elimination let",
      "dependencies": [
        "PI:E2",
        "PI:A3",
        "PI:T2",
        "PI:T4"
      ],
      "scope": "Actual partially eliminated positive Gaussian, phase rank, branch and all normalizing factors"
    },
    {
      "id": "PI:B0",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "Let \\(E\\to X\\)",
      "dependencies": [
        "PI:T8",
        "PI:S2",
        "CH:W1",
        "CH:W3"
      ],
      "scope": "Full some-phase/every-phase equivalence, frame and coordinate invariance and open-cone locality"
    },
    {
      "id": "PI:X1",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 1 ",
      "dependencies": [
        "PI:G1"
      ],
      "scope": "Original Exercise 1 with its complete unchanged solution"
    },
    {
      "id": "PI:X2",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 2 ",
      "dependencies": [
        "PI:M0"
      ],
      "scope": "Original Exercise 2 with its complete unchanged solution"
    },
    {
      "id": "PI:X3",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 3 ",
      "dependencies": [
        "PI:D5",
        "PI:D6"
      ],
      "scope": "Original Exercise 3 with its complete unchanged solution"
    },
    {
      "id": "PI:X4",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 4 ",
      "dependencies": [
        "PI:F0",
        "PI:F2"
      ],
      "scope": "Original Exercise 4 with its complete unchanged solution"
    },
    {
      "id": "PI:X5",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 5 ",
      "dependencies": [
        "PI:M1",
        "PI:A3"
      ],
      "scope": "Original Exercise 5 with its complete unchanged solution"
    },
    {
      "id": "PI:X6",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 6 ",
      "dependencies": [
        "PI:M2"
      ],
      "scope": "Original Exercise 6 with its complete unchanged solution"
    },
    {
      "id": "PI:X7",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 7 ",
      "dependencies": [
        "PI:M3"
      ],
      "scope": "Original Exercise 7 with its complete unchanged solution"
    },
    {
      "id": "PI:X8",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 8 ",
      "dependencies": [
        "PI:P7",
        "QF:A8"
      ],
      "scope": "Original Exercise 8 with its complete unchanged solution"
    },
    {
      "id": "PI:X9",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 9 ",
      "dependencies": [
        "PI:M4",
        "PI:A3"
      ],
      "scope": "Original Exercise 9 with its complete unchanged solution"
    },
    {
      "id": "PI:X10",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 10 ",
      "dependencies": [
        "PI:R1",
        "PI:R2"
      ],
      "scope": "Original Exercise 10 with its complete unchanged solution"
    },
    {
      "id": "PI:X11",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 11 ",
      "dependencies": [
        "PI:S1"
      ],
      "scope": "Original Exercise 11 with its complete unchanged solution"
    },
    {
      "id": "PI:X12",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 12 ",
      "dependencies": [
        "PI:T2",
        "PI:T4",
        "PI:M4"
      ],
      "scope": "Original Exercise 12 with its complete unchanged solution"
    },
    {
      "id": "PI:X13",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 13 ",
      "dependencies": [
        "U001:P15.1",
        "PI:T6"
      ],
      "scope": "Original Exercise 13 with its complete unchanged solution"
    },
    {
      "id": "PI:X14",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 14 ",
      "dependencies": [
        "PI:T8"
      ],
      "scope": "Original Exercise 14 with its complete unchanged solution"
    },
    {
      "id": "PI:X15",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 15 ",
      "dependencies": [
        "PI:B0"
      ],
      "scope": "Original Exercise 15 with its complete unchanged solution"
    },
    {
      "id": "PI:X16",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 16 ",
      "dependencies": [
        "PI:O3",
        "P3:M3"
      ],
      "scope": "Original Exercise 16 with its complete unchanged solution"
    },
    {
      "id": "PI:X17",
      "source": "positive-lagrangian-ideals-and-distributions.md",
      "source_sha256": "f86e01bac44653e202c8d8ce9af8e314a11a8a8c4bd62966cbd9b201f3f49974",
      "proof_locator": "**Exercise 17 ",
      "dependencies": [
        "PI:S0",
        "PI:R2",
        "U001:P15.1"
      ],
      "scope": "Original Exercise 17 with its complete unchanged solution"
    },
    {
      "id": "PI:CM0",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "## 9. An exact model",
      "dependencies": [
        "PI:CS7",
        "PI:A3"
      ],
      "scope": "Actual semipositive virtual-point model, critical value and complete Gaussian constant M1–M3"
    },
    {
      "id": "PI:CM1",
      "source": "complex-stationary-contract.md",
      "source_sha256": "f2a09e2ecb4497a059f852545690cdad32a847de357f8ef845b2e9bd510b61ab",
      "proof_locator": "### 9.1. Exact contours",
      "dependencies": [
        "PI:CM0",
        "PI:A2",
        "PI:A3"
      ],
      "scope": "Exact translated and rotated contours, all damping signs, endpoint Jacobian and reproducible M4–M7 figure"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
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    "../20261005-restored-analytic-composition/proof-map.json",
    "../20261005-restored-subprincipal-transport/proof-map.json",
    "../20261005-restored-graph-egorov/proof-map.json",
    "../20261005-restored-corank-continuity/proof-map.json",
    "../20261005-restored-submanifolds/proof-map.json",
    "../20261005-restored-cubic-scaling/proof-map.json",
    "../20261005-restored-smooth-descent/proof-map.json",
    "../20261005-restored-simultaneous-reflections/proof-map.json",
    "../20261005-restored-folded-forms/proof-map.json",
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