{
  "schema": "AN04-restored-clean-pairs-proof-map/v1",
  "proofs": [
    {
      "id": "CP:D0",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "## 1.",
      "dependencies": [
        "U001:F0-DIFF"
      ],
      "scope": "Actual clean intersection and tangent equality on a neighborhood"
    },
    {
      "id": "CP:D1",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Lemma 1.1",
      "dependencies": [
        "CP:D0",
        "U001:P3",
        "U001:F0-DIFF"
      ],
      "scope": "Shared intersection parameterization, explicit extensions and full simultaneous smooth chart"
    },
    {
      "id": "CP:D2",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "The common functions",
      "dependencies": [
        "CP:D1"
      ],
      "scope": "Independent common normals and exact tangent-sum annihilator"
    },
    {
      "id": "CP:R0",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "## 2.",
      "dependencies": [
        "CP:D0",
        "G2:G21",
        "G2:G41"
      ],
      "scope": "Radial tangent and exact conic dimension range"
    },
    {
      "id": "CP:R1",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "Choose a positive degree-one ray coordinate",
      "dependencies": [
        "CP:R0",
        "SN:F0",
        "G2:G41"
      ],
      "scope": "Actual conic products, clean ray slice and all tangent dimension splittings"
    },
    {
      "id": "CP:R2",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "Lemma 1.1 on \\(\\Sigma\\)",
      "dependencies": [
        "CP:D2",
        "CP:R1"
      ],
      "scope": "Exactly k common degree-one defining functions and their independent differentials"
    },
    {
      "id": "CP:R3",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "Here each Lagrangian plane equals",
      "dependencies": [
        "CP:R2",
        "G2:G21",
        "PC:S0",
        "SN:F0"
      ],
      "scope": "Common Hamilton span, tangency to both submanifolds and nonradial choice for k>1"
    },
    {
      "id": "CP:T0",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "We will use the full cotangent lift",
      "dependencies": [
        "U001:P2",
        "U001:P3",
        "G2:F0",
        "G2:G21"
      ],
      "scope": "Smooth full cotangent map and inverse, exact primitive and marked nonzero covector normalization"
    },
    {
      "id": "CP:H0",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Proof, base case \\(k=1\\)",
      "dependencies": [
        "PC:I5",
        "CP:T0"
      ],
      "scope": "First conic fiber straightening with marked point (0,e1)"
    },
    {
      "id": "CP:H1",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "Therefore \\(\\pi|_{V_2}\\) has rank",
      "dependencies": [
        "CP:H0",
        "CP:R0",
        "U001:F0-DIFF"
      ],
      "scope": "Genuine constant rank n-1 near the marked point from minor and radial bounds"
    },
    {
      "id": "CP:H2",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "The conormal-recognition theorem",
      "dependencies": [
        "CP:H1",
        "PC:C2",
        "CP:T0",
        "U001:P3"
      ],
      "scope": "Conormal germ, signed defining-function normalization and full simultaneous base lift"
    },
    {
      "id": "CP:H3",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Induction step \\(k>1\\)",
      "dependencies": [
        "CP:R3",
        "PC:H5"
      ],
      "scope": "Exact common momentum normalization with zero value and positive distinct p1"
    },
    {
      "id": "CP:H4",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "Now \\(p_n=0\\) on both",
      "dependencies": [
        "CP:H3",
        "SN:F0"
      ],
      "scope": "Both actual flow cylinders and embedded transverse sections"
    },
    {
      "id": "CP:H5",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "The section \\(S_0=",
      "dependencies": [
        "CP:H4",
        "CP:R1",
        "G2:G21",
        "U001:P3"
      ],
      "scope": "Reduced conic Lagrangians, smooth actual intersection and cleanliness at every nearby point"
    },
    {
      "id": "CP:H6",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "Induct on \\(k\\)",
      "dependencies": [
        "CP:H2",
        "CP:H5",
        "PC:I2"
      ],
      "scope": "Full reduced product extension, inverse, conic domain, marked point and both auxiliary models"
    },
    {
      "id": "CP:H7",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Simultaneous homogeneous model change.**",
      "dependencies": [
        "CP:H6",
        "PC:I5",
        "U001:F0-DIFF"
      ],
      "scope": "Entire projective exchange with exact inverse, degrees and primitive cancellation"
    },
    {
      "id": "CP:H8",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "On both models (3.6)",
      "dependencies": [
        "CP:H7"
      ],
      "scope": "Both model equalities, reverse containments, paired permutation and k=n endpoint"
    },
    {
      "id": "CP:L0",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
        "G2:G21",
        "PC:L0"
      ],
      "scope": "Perfect quotient pairing, full alternating-subspace extension and both linear endpoints"
    },
    {
      "id": "CP:L1",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Corollary 4.2",
      "dependencies": [
        "CP:L0",
        "G2:F0"
      ],
      "scope": "Explicit common transverse Lagrangian plane including both empty blocks"
    },
    {
      "id": "CP:O0",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Proof.** First obtain an ordinary fiber chart",
      "dependencies": [
        "PC:O2",
        "SN:F0",
        "G2:G21"
      ],
      "scope": "Complete ordinary fiber induction, cylinder reduction, exact exchange and dimension-zero base"
    },
    {
      "id": "CP:O1",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "When \\(k=0\\)",
      "dependencies": [
        "CP:O0",
        "U001:P3",
        "G2:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Transverse graph, proved closed-form primitive and full ordinary marked momentum translation"
    },
    {
      "id": "CP:O2",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "For \\(k>0\\)",
      "dependencies": [
        "CP:D2",
        "PC:O2",
        "CP:O1",
        "SN:F0",
        "CP:H5",
        "CP:H8"
      ],
      "scope": "Full ordinary clean-pair induction with exact primitive correction and index permutation"
    },
    {
      "id": "CP:E0",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Example 5.2",
      "dependencies": [
        "CP:H2",
        "CP:T0",
        "U001:F0-DIFF"
      ],
      "scope": "Curved conormal, clean radial intersection and entire nonlinear cotangent model"
    },
    {
      "id": "CP:E1",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Example 5.3",
      "dependencies": [
        "CP:D0",
        "U001:F0-DIFF"
      ],
      "scope": "Smooth but nonclean intersection and invariant tangent-dimension obstruction"
    },
    {
      "id": "CP:X1",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.1 ",
      "dependencies": [
        "CP:D2",
        "CP:R2"
      ],
      "scope": "Exercise 6.1 with solution"
    },
    {
      "id": "CP:X2",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.2 ",
      "dependencies": [
        "CP:R3"
      ],
      "scope": "Exercise 6.2 with solution"
    },
    {
      "id": "CP:X3",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.3 ",
      "dependencies": [
        "CP:H1"
      ],
      "scope": "Exercise 6.3 with solution"
    },
    {
      "id": "CP:X4",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.4 ",
      "dependencies": [
        "CP:H4",
        "CP:H5"
      ],
      "scope": "Exercise 6.4 with solution"
    },
    {
      "id": "CP:X5",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.5 ",
      "dependencies": [
        "CP:H7"
      ],
      "scope": "Exercise 6.5 with solution"
    },
    {
      "id": "CP:X6",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.6 ",
      "dependencies": [
        "CP:H8"
      ],
      "scope": "Exercise 6.6 with solution"
    },
    {
      "id": "CP:X7",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.7 ",
      "dependencies": [
        "CP:L1"
      ],
      "scope": "Exercise 6.7 with solution"
    },
    {
      "id": "CP:X8",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.8 ",
      "dependencies": [
        "CP:E0"
      ],
      "scope": "Exercise 6.8 with solution"
    },
    {
      "id": "CP:X9",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.9 ",
      "dependencies": [
        "CP:O1"
      ],
      "scope": "Exercise 6.9 with solution"
    },
    {
      "id": "CP:X10",
      "source": "clean-lagrangian-pairs-and-common-transversals.md",
      "source_sha256": "eadf92b81fa0689277c3f4ed436a124ff29455dd989d118d2b6f2162e201ad73",
      "proof_locator": "**Exercise 6.10 ",
      "dependencies": [
        "CP:E1"
      ],
      "scope": "Exercise 6.10 with solution"
    }
  ],
  "earlier_maps": [
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    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json",
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    "../20261005-restored-analytic-composition/proof-map.json",
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    "../20261005-restored-graph-egorov/proof-map.json",
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    "../20261005-restored-canonical-folds/proof-map.json",
    "../20261005-restored-fold-densities/proof-map.json",
    "../20261005-restored-airy-models/proof-map.json",
    "../20261005-restored-invariant-folds/proof-map.json",
    "../20261005-restored-prescribed-coordinates/proof-map.json"
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    "L8.1.14",
    "L8.1.16",
    "L8.1.17",
    "L8.1.18",
    "L8.2.4",
    "L8.2.5",
    "L8.2.6",
    "L8.2.7",
    "L8.2.8",
    "L8.2.9",
    "L8.3.2",
    "L8.3.3",
    "L8.3.5",
    "L8.3.6",
    "L8.3.7",
    "L8.3.9",
    "L8.4.1",
    "L8.4.2",
    "L8.4.6",
    "L8.5.1",
    "L8.5.6",
    "L8.6.2",
    "L9.1.1",
    "P1",
    "P10.1",
    "P10.2",
    "P10.2-typo",
    "P10.3",
    "P10.4",
    "P10.5-rules",
    "P10.6",
    "P10.6-zero",
    "P11.1",
    "P11.2-bridge",
    "P11.2-orders",
    "P12.1",
    "P12.2-criterion",
    "P12.2-upper",
    "P12.3",
    "P12.4",
    "P12.5-endpoints",
    "P12.5-extension",
    "P12.6",
    "P12.7",
    "P13.1",
    "P13.2-mertens",
    "P13.2-series",
    "P14.2-inverse",
    "P14.2-powers",
    "P14.3",
    "P15.1",
    "P15.2",
    "P16.1",
    "P16.2",
    "P16.3",
    "P17.1",
    "P17.2",
    "P17.3",
    "P17.4",
    "P17.4-grid",
    "P17.5-order",
    "P17.5-upper",
    "P18.1",
    "P18.2",
    "P18.3",
    "P18.4",
    "P18.5",
    "P19.1",
    "P19.2",
    "P19.3",
    "P19.4",
    "P2",
    "P20.1",
    "P20.2",
    "P20.3",
    "P20.4",
    "P20.5",
    "P21.1",
    "P21.2",
    "P21.3",
    "P21.4",
    "P21.5",
    "P3",
    "P4",
    "P5",
    "P6.0",
    "P6.1",
    "P6.2",
    "P6.3-inf",
    "P6.3-subtract",
    "P6.3-tail",
    "P6.4-ball",
    "P6.4-complete",
    "P6.5",
    "P6.6",
    "P7",
    "P8-root",
    "P8-smooth",
    "P9.1",
    "P9.2",
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