{
  "schema": "AN04-restored-maslov-topology-proof-map/v1",
  "proofs": [
    {
      "id": "MT:D0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "We use",
      "dependencies": [
        "G2:G0",
        "U001:P9.4"
      ],
      "scope": "Exact symplectic and complex Gram-Schmidt conventions"
    },
    {
      "id": "MT:C0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Lemma 1.1",
      "dependencies": [
        "MT:D0",
        "U001:F0-DIFF",
        "GS:Z0"
      ],
      "scope": "Section independence, symmetry, fixed-symplectic invariance and full graph kernel form"
    },
    {
      "id": "MT:C1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "The positive generator we will use",
      "dependencies": [
        "MT:C0",
        "U001:F0-CALC"
      ],
      "scope": "Exact positive generator, crossing form and reverse orientation"
    },
    {
      "id": "MT:T0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Lemma 2.1",
      "dependencies": [
        "P3:M0",
        "P3:M2",
        "U001:F0-COMP",
        "U001:Q5",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete smaller-image and equal-dimensional C1 critical-value null-cover arguments"
    },
    {
      "id": "MT:T1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Lemma 2.2",
      "dependencies": [
        "MT:G0",
        "MT:G1",
        "MT:G3",
        "U001:P3",
        "U001:P9.4",
        "U001:F0-COMP"
      ],
      "scope": "Normal bundle, compact uniform retraction and uniqueness"
    },
    {
      "id": "MT:T2",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "View the continuous loop as",
      "dependencies": [
        "MT:T1",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:M4"
      ],
      "scope": "Complete periodic smoothing with exact base correction and based retraction homotopy"
    },
    {
      "id": "MT:T3",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Lemma 2.3",
      "dependencies": [
        "U001:P2",
        "U001:P3",
        "G2:NF7",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Actual compact horizontal ODE lift, bounded-speed endpoint continuation, smooth parameter dependence and fixed loop/base data"
    },
    {
      "id": "MT:G0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Proposition 3.1.",
      "dependencies": [
        "MT:D0",
        "GS:Z0",
        "U001:P9.4",
        "U001:P2"
      ],
      "scope": "Actual unitary frames, smooth U(n)/O(n) quotient and determinant-square map"
    },
    {
      "id": "MT:G1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "The unitary groups used here",
      "dependencies": [
        "U001:P3",
        "U001:F0-ALG",
        "U001:F0-COMP"
      ],
      "scope": "Smooth matrix groups by full level derivatives, determinant derivative and compactness"
    },
    {
      "id": "MT:G2",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "On an arc of \\(S^1\\)",
      "dependencies": [
        "MT:G0",
        "MT:G1",
        "U001:F0-CALC"
      ],
      "scope": "Exact SU/SO fiber, determinant-sign repair and smooth local products"
    },
    {
      "id": "MT:G3",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "The space \\(\\mathcal L(S)\\) is compact.",
      "dependencies": [
        "GS:Z0",
        "MT:G0",
        "U001:F0-COMP"
      ],
      "scope": "Closed bounded projection model and compact fiber"
    },
    {
      "id": "MT:G4",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "On the loop (1.4)",
      "dependencies": [
        "MT:G0",
        "MT:C1"
      ],
      "scope": "Generator determinant square makes one positive turn"
    },
    {
      "id": "MT:W0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For a loop \\(b(t)\\in S^1\\)",
      "dependencies": [
        "U001:F0-COMP",
        "U001:F0-CALC"
      ],
      "scope": "Full lifted arguments, winding independence/additivity, compact-square homotopy invariance and zero-winding contraction"
    },
    {
      "id": "MT:W1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Proof.** Both",
      "dependencies": [
        "MT:G1",
        "U001:F0-CALC"
      ],
      "scope": "SO(n) connected by actual successive plane rotations including n=1 and opposite vectors"
    },
    {
      "id": "MT:W2",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "Every sphere \\(S^d\\)",
      "dependencies": [
        "MT:T0",
        "MT:T2",
        "U001:P3"
      ],
      "scope": "Smooth sphere loop misses a point, based stereographic contraction and path connectedness"
    },
    {
      "id": "MT:W3",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For \\(n\\geq2\\), the first-column map",
      "dependencies": [
        "MT:G1",
        "MT:D0",
        "U001:P9.4"
      ],
      "scope": "Actual SU(n) first-column local frame sections and determinant adjustment; explicit SU2 sphere inverse"
    },
    {
      "id": "MT:W4",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "so \\(SU(2)\\) is \\(S^3\\).",
      "dependencies": [
        "MT:W3",
        "MT:W2",
        "MT:T3"
      ],
      "scope": "Full connectedness and simple-connectedness induction with lifted smooth base contraction and residual fiber loop"
    },
    {
      "id": "MT:W5",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "The quotient \\(SU(n)\\to F\\)",
      "dependencies": [
        "MT:G2",
        "MT:W1",
        "MT:W4",
        "MT:T3",
        "MT:T2"
      ],
      "scope": "Actual quotient-loop lift, fiber endpoint repair, disk projection and continuous-loop reduction"
    },
    {
      "id": "MT:W6",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Theorem 4.2",
      "dependencies": [
        "MT:W0",
        "MT:G4",
        "MT:G2",
        "MT:G3",
        "MT:W5",
        "MT:T3",
        "MT:T2"
      ],
      "scope": "Entire integer isomorphism, based injectivity and surjectivity, connected plane space and base-path independence"
    },
    {
      "id": "MT:P0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Theorem 5.1",
      "dependencies": [
        "MT:T2",
        "MT:T0",
        "GS:Z1",
        "MT:C0",
        "U001:U001-A4",
        "U001:P3",
        "U001:F0-COMP"
      ],
      "scope": "All finite generic perturbations, exact codimensions, critical regularity, supported symplectic shears, compact first-jet openness and finite crossings"
    },
    {
      "id": "MT:B0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "We first compute a complete branch",
      "dependencies": [
        "U001:Q5",
        "U001:P2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "MT:G0"
      ],
      "scope": "Full transverse-chart phase branch and smooth spectral arctangent integral"
    },
    {
      "id": "MT:B1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For a crossing plane transverse",
      "dependencies": [
        "MT:B0",
        "U001:Q5",
        "U001:P3"
      ],
      "scope": "Exact unitary graph frame, determinant ratio and positive square-root Sylvester inverse"
    },
    {
      "id": "MT:B2",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "Indeed logarithmic determinant differentiation",
      "dependencies": [
        "MT:B1",
        "MT:G1"
      ],
      "scope": "Full noncommuting trace derivative and exact resolvent difference"
    },
    {
      "id": "MT:B3",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Lemma 6.1",
      "dependencies": [
        "MT:B0",
        "MT:B1",
        "U001:P3",
        "U001:P9.4",
        "MT:C0"
      ],
      "scope": "Marked eigenvector implicit map, exact eigenvalue derivative and full branch jump"
    },
    {
      "id": "MT:B4",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Proof.** First take",
      "dependencies": [
        "GS:Z1",
        "MT:C0",
        "MT:W0"
      ],
      "scope": "One common complement at every crossing and fixed-frame winding preserved by an explicit shear homotopy"
    },
    {
      "id": "MT:B5",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "Let \\(\\theta(t)\\) be a continuous argument",
      "dependencies": [
        "MT:B3",
        "MT:B4",
        "MT:W6",
        "MT:G0",
        "MT:P0"
      ],
      "scope": "Global branch gluing, signed crossing formula, reference/perturbation independence and full symplectic invariance without circularity"
    },
    {
      "id": "MT:B6",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "An important local calculation keeps",
      "dependencies": [
        "U001:P4",
        "U001:P2",
        "MT:C0"
      ],
      "scope": "Complete Schur determinant and inverse derivatives, kernel vector and signature convention"
    },
    {
      "id": "MT:B7",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For a regular crossing with",
      "dependencies": [
        "MT:B6",
        "MT:B5",
        "U001:Q5",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Actual supported graph homotopy, invertible block interpolation, distinct simple splitting and full regular signature contribution"
    },
    {
      "id": "MT:I0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "Subdivide a loop into chart segments",
      "dependencies": [
        "GS:Z5",
        "GS:Z6",
        "U001:F0-COMP"
      ],
      "scope": "Exact integer chart-switch cocycle, refinement and compact-square homotopy cancellation"
    },
    {
      "id": "MT:I1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "To evaluate it on the generator",
      "dependencies": [
        "MT:I0",
        "MT:C1",
        "MT:W6",
        "GS:Z4"
      ],
      "scope": "Actual ordered two-switch signs, higher-dimensional spectators and equality of ordered integer and winding"
    },
    {
      "id": "MT:I2",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Theorem 7.1",
      "dependencies": [
        "MT:I1",
        "GS:Z6"
      ],
      "scope": "Specified forward coefficient transport, dual transport and reciprocal local-frame dictionary"
    },
    {
      "id": "MT:I3",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "The ordered integer cocycle \\(-\\sigma\\)",
      "dependencies": [
        "MT:I0",
        "MT:I1",
        "MT:B5"
      ],
      "scope": "Explicit degree-one integer cocycle class, refinement and coboundary independence"
    },
    {
      "id": "MT:I4",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For a continuous map \\(f:Y\\to\\mathcal L(S)\\)",
      "dependencies": [
        "MT:I2",
        "GS:Z6",
        "U001:F0-COMP"
      ],
      "scope": "Pulled-back flat line, connected manifold path argument, necessary and sufficient flat-section criterion and actual tangent trivialization scope"
    },
    {
      "id": "MT:M0",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For the first coordinate of \\(\\gamma_+\\)",
      "dependencies": [
        "MT:C1",
        "MT:B0",
        "MT:B3",
        "MT:I2"
      ],
      "scope": "Complete exact transverse-branch example and integer/coefficient signs"
    },
    {
      "id": "MT:M1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For a nonconstant two-dimensional graph",
      "dependencies": [
        "MT:B6"
      ],
      "scope": "Full nonconstant crossing matrix with exact derivatives"
    },
    {
      "id": "MT:M2",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "A regular double crossing can carry",
      "dependencies": [
        "MT:B7"
      ],
      "scope": "Indefinite regular crossing and supported simple split"
    },
    {
      "id": "MT:M3",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "For exact direct-sum loops",
      "dependencies": [
        "MT:W0",
        "MT:G0",
        "MT:B5",
        "MT:I2"
      ],
      "scope": "All integer direct sums, exact rational crossing inventory and nonzero index with trivial fourth-root transport"
    },
    {
      "id": "MT:M4",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "An ordinary Lagrangian curve can exhibit",
      "dependencies": [
        "MT:C0",
        "MT:G0",
        "MT:I4"
      ],
      "scope": "Exact embedded ordinary curve, tangent Gauss map, two positive crossings and nontrivial pulled-back holonomy"
    },
    {
      "id": "MT:X1",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "MT:C1",
        "MT:I2"
      ],
      "scope": "Original Exercise 9.1 with its full retained solution; manifold hypothesis clarified in 9.12"
    },
    {
      "id": "MT:X2",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "MT:G0"
      ],
      "scope": "Original Exercise 9.2 with its full retained solution; manifold hypothesis clarified in 9.12"
    },
    {
      "id": "MT:X3",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "MT:B6"
      ],
      "scope": "Original Exercise 9.3 with its full retained solution; manifold hypothesis clarified in 9.12"
    },
    {
      "id": "MT:X4",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "MT:B6"
      ],
      "scope": "Original Exercise 9.4 with its full retained solution; manifold hypothesis clarified in 9.12"
    },
    {
      "id": "MT:X5",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "MT:M2"
      ],
      "scope": "Original Exercise 9.5 with its full retained solution; manifold hypothesis clarified in 9.12"
    },
    {
      "id": "MT:X6",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "MT:M3"
      ],
      "scope": "Original Exercise 9.6 with its full retained solution; manifold hypothesis clarified in 9.12"
    },
    {
      "id": "MT:X7",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.7 ",
      "dependencies": [
        "MT:B2"
      ],
      "scope": "Original Exercise 9.7 with its full retained solution; manifold hypothesis clarified in 9.12"
    },
    {
      "id": "MT:X8",
      "source": "maslov-index-crossings-and-global-phase.md",
      "source_sha256": "e30070f370437b3214c2383fc88d24370421ea25e6c316600a9b283705d2c3e1",
      "proof_locator": "**Exercise 9.8 ",
      "dependencies": [
        "MT:B0",
        "MT:I2"
      ],
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