{
  "scope": "Exact selected intrinsic components with fully qualified dependency IDs; earlier U001 nodes are proved in the linked stationary edition.",
  "earlier_U001_proof_map": "../20261004-free-stationary-phase/proof-map.json",
  "proofs": [
    {
      "id": "GR:G5",
      "source": "intrinsic-frequency-graphs.md",
      "source_sha256": "779884e2a50f2396e23931f66964c049441c7bafa586151147121da21cb2c4b8",
      "proof_locator": "## 2. A real frequency phase with a properly supported kernel",
      "package": "GR",
      "dependencies": [
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:Q3",
        "U001:U001-A4"
      ]
    },
    {
      "id": "GR:G8",
      "source": "intrinsic-frequency-graphs.md",
      "source_sha256": "779884e2a50f2396e23931f66964c049441c7bafa586151147121da21cb2c4b8",
      "proof_locator": "## 3. Removing the frequency phase after a base localization",
      "package": "GR",
      "dependencies": [
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:U001-5.1",
        "U001:U001-6.1"
      ]
    },
    {
      "id": "GR:G12-forward",
      "source": "intrinsic-frequency-graphs.md",
      "source_sha256": "779884e2a50f2396e23931f66964c049441c7bafa586151147121da21cb2c4b8",
      "proof_locator": "**Forward proof.**",
      "package": "GR",
      "dependencies": [
        "GR:G5",
        "P1:B1",
        "P1:B2",
        "P1:B3",
        "P1:B4",
        "P1:B5",
        "P1:B6",
        "P2:F1",
        "P2:F2",
        "P2:F3",
        "P2:F4",
        "P2:F5",
        "P2:ML1",
        "P2:ML2",
        "P2:ML3",
        "P2:ML4",
        "P2:ML6",
        "P2:ML7",
        "P2:QG1",
        "P2:QG2",
        "P2:QG3",
        "P2:QG4",
        "P2:QG5",
        "P2:QG6",
        "P2:QG7",
        "P2:QG8",
        "P2:OP0",
        "P2:OP1",
        "P2:OP2",
        "P2:OP3",
        "P2:OP4",
        "P2:OP5",
        "P2:OP6",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "GR:G12-reverse",
      "source": "intrinsic-frequency-graphs.md",
      "source_sha256": "779884e2a50f2396e23931f66964c049441c7bafa586151147121da21cb2c4b8",
      "proof_locator": "**Reverse proof.**",
      "package": "GR",
      "dependencies": [
        "GR:G8",
        "P1:B1",
        "P1:B2",
        "P1:B3",
        "P1:B4",
        "P1:B5",
        "P1:B6",
        "P2:F1",
        "P2:F2",
        "P2:F3",
        "P2:F4",
        "P2:F5",
        "P2:ML1",
        "P2:ML2",
        "P2:ML3",
        "P2:ML4",
        "P2:ML6",
        "P2:ML7",
        "P2:QG1",
        "P2:QG2",
        "P2:QG3",
        "P2:QG4",
        "P2:QG5",
        "P2:QG6",
        "P2:QG7",
        "P2:QG8",
        "P2:OP0",
        "P2:OP1",
        "P2:OP2",
        "P2:OP3",
        "P2:OP4",
        "P2:OP5",
        "P2:OP6",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "P1:B1",
      "source": "prerequisites/dyadic-endpoint.md",
      "source_sha256": "b6f5fe1b279bb89111b9c4cdd6566342ced4a4da5c3f2b01e06665290b14c4a2",
      "proof_locator": "## B1. Dyadic norms and distributional convergence",
      "package": "P1",
      "dependencies": [
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:A0",
        "U001:F0-DIFF",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:P14.3",
        "U001:P21.3",
        "U001:P21.4",
        "U001:Q3",
        "U001:Q4"
      ]
    },
    {
      "id": "P1:B2",
      "source": "prerequisites/dyadic-endpoint.md",
      "source_sha256": "b6f5fe1b279bb89111b9c4cdd6566342ced4a4da5c3f2b01e06665290b14c4a2",
      "proof_locator": "## B2. The local frequency Sobolev estimate",
      "package": "P1",
      "dependencies": [
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:A0",
        "U001:F0-DIFF",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:P14.3",
        "U001:P21.3",
        "U001:P21.4",
        "U001:Q3",
        "U001:Q4"
      ]
    },
    {
      "id": "P1:B3",
      "source": "prerequisites/dyadic-endpoint.md",
      "source_sha256": "b6f5fe1b279bb89111b9c4cdd6566342ced4a4da5c3f2b01e06665290b14c4a2",
      "proof_locator": "## B3. Schur's estimate and the summable block matrix",
      "package": "P1",
      "dependencies": [
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "P1:B4",
      "source": "prerequisites/dyadic-endpoint.md",
      "source_sha256": "b6f5fe1b279bb89111b9c4cdd6566342ced4a4da5c3f2b01e06665290b14c4a2",
      "proof_locator": "## B4. The ordinary symbol endpoint",
      "package": "P1",
      "dependencies": [
        "P1:B1",
        "P1:B3",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "P1:B5",
      "source": "prerequisites/dyadic-endpoint.md",
      "source_sha256": "b6f5fe1b279bb89111b9c4cdd6566342ced4a4da5c3f2b01e06665290b14c4a2",
      "proof_locator": "## B5. Multiplication and changes of coordinates",
      "package": "P1",
      "dependencies": [
        "P1:B1",
        "P1:B2",
        "P1:B3",
        "P1:B4",
        "U001:A0",
        "U001:F0-DIFF",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:P14.3",
        "U001:P21.3",
        "U001:P21.4",
        "U001:Q3",
        "U001:Q4"
      ]
    },
    {
      "id": "P1:B6",
      "source": "prerequisites/dyadic-endpoint.md",
      "source_sha256": "b6f5fe1b279bb89111b9c4cdd6566342ced4a4da5c3f2b01e06665290b14c4a2",
      "proof_locator": "## B6. Local use with a supplied proper representation",
      "package": "P1",
      "dependencies": [
        "P1:B1",
        "P1:B4",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:A0",
        "U001:F0-DIFF",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:P14.3",
        "U001:P21.3",
        "U001:P21.4",
        "U001:Q3",
        "U001:Q4"
      ]
    },
    {
      "id": "P2:F1",
      "source": "prerequisites/quadratic-multiplier-foundations.md",
      "source_sha256": "78bd0d3bb696454878ea5af4d57dbdd524aa5daf39cb6a157a38a80d908b7825",
      "proof_locator": "## 1. Positive forms and coordinate maps",
      "package": "P2",
      "dependencies": [
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:F2",
      "source": "prerequisites/quadratic-multiplier-foundations.md",
      "source_sha256": "78bd0d3bb696454878ea5af4d57dbdd524aa5daf39cb6a157a38a80d908b7825",
      "proof_locator": "## 2. Lebesgue volume of each ellipsoid",
      "package": "P2",
      "dependencies": [
        "P2:F1",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:F3",
      "source": "prerequisites/quadratic-multiplier-foundations.md",
      "source_sha256": "78bd0d3bb696454878ea5af4d57dbdd524aa5daf39cb6a157a38a80d908b7825",
      "proof_locator": "## 3. Dual ellipsoids and phase distances",
      "package": "P2",
      "dependencies": [
        "P2:F1"
      ]
    },
    {
      "id": "P2:F4",
      "source": "prerequisites/quadratic-multiplier-foundations.md",
      "source_sha256": "78bd0d3bb696454878ea5af4d57dbdd524aa5daf39cb6a157a38a80d908b7825",
      "proof_locator": "## 4. Strict separation of an open convex set",
      "package": "P2",
      "dependencies": [
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "P2:F5",
      "source": "prerequisites/quadratic-multiplier-foundations.md",
      "source_sha256": "78bd0d3bb696454878ea5af4d57dbdd524aa5daf39cb6a157a38a80d908b7825",
      "proof_locator": "## 5. Support functions of Minkowski sums",
      "package": "P2",
      "dependencies": [
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:ML1",
      "source": "prerequisites/metric-localization.md",
      "source_sha256": "3a85c94342f209f3c25cfc130a86831cb874016666e806a1ca1ef648ea0929d2",
      "proof_locator": "## 1. Data and symbol seminorms",
      "package": "P2",
      "dependencies": [
        "P2:F1"
      ]
    },
    {
      "id": "P2:ML2",
      "source": "prerequisites/metric-localization.md",
      "source_sha256": "3a85c94342f209f3c25cfc130a86831cb874016666e806a1ca1ef648ea0929d2",
      "proof_locator": "## 2. A locally finite cover of moving ellipsoids",
      "package": "P2",
      "dependencies": [
        "P2:F1",
        "P2:F2",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:ML3",
      "source": "prerequisites/metric-localization.md",
      "source_sha256": "3a85c94342f209f3c25cfc130a86831cb874016666e806a1ca1ef648ea0929d2",
      "proof_locator": "## 3. A partition of unity with uniform derivatives",
      "package": "P2",
      "dependencies": [
        "P2:ML2",
        "P2:F1",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:ML4",
      "source": "prerequisites/metric-localization.md",
      "source_sha256": "3a85c94342f209f3c25cfc130a86831cb874016666e806a1ca1ef648ea0929d2",
      "proof_locator": "## 4. Localizing and reconstructing symbols",
      "package": "P2",
      "dependencies": [
        "P2:ML1",
        "P2:ML3"
      ]
    },
    {
      "id": "P2:ML6",
      "source": "prerequisites/metric-localization.md",
      "source_sha256": "3a85c94342f209f3c25cfc130a86831cb874016666e806a1ca1ef648ea0929d2",
      "proof_locator": "## 6. Compact comparison and the local topology",
      "package": "P2",
      "dependencies": [
        "P2:ML1",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:ML7",
      "source": "prerequisites/metric-localization.md",
      "source_sha256": "3a85c94342f209f3c25cfc130a86831cb874016666e806a1ca1ef648ea0929d2",
      "proof_locator": "## 7. Approximation on compact sets",
      "package": "P2",
      "dependencies": [
        "P2:ML2",
        "P2:ML3",
        "P2:ML4",
        "P2:ML6"
      ]
    },
    {
      "id": "P2:QG1",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 1. The phase and its dual distance",
      "package": "P2",
      "dependencies": [
        "P2:F1",
        "P2:F3",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:QG2",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 2. A Taylor estimate for the local multiplier",
      "package": "P2",
      "dependencies": [
        "P2:F1",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:QG3",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 3. Repeated integration by parts away from support",
      "package": "P2",
      "dependencies": [
        "P2:QG1",
        "P2:QG2",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:QG4",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 4. Decay at phase distance",
      "package": "P2",
      "dependencies": [
        "P2:QG3",
        "P2:F3",
        "P2:F4",
        "P2:F5"
      ]
    },
    {
      "id": "P2:QG5",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 5. Weights measured from the observation point",
      "package": "P2",
      "dependencies": [
        "P2:QG1",
        "P2:ML1"
      ]
    },
    {
      "id": "P2:QG6",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 6. How many localization balls are near?",
      "package": "P2",
      "dependencies": [
        "P2:QG4",
        "P2:QG5",
        "P2:ML2",
        "P2:ML3",
        "P2:F2"
      ]
    },
    {
      "id": "P2:QG7",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 7. Pointwise extension and derivative control",
      "package": "P2",
      "dependencies": [
        "P2:QG2",
        "P2:QG4",
        "P2:QG6",
        "P2:ML4",
        "P2:ML6",
        "P2:ML7"
      ]
    },
    {
      "id": "P2:QG8",
      "source": "prerequisites/gauss-transform-estimates.md",
      "source_sha256": "c5cf6ad934dd94d2eb5c5bcf415683cde7d847a864a185f5e6f65de40671d587",
      "proof_locator": "## 8. The remainder with a small parameter",
      "package": "P2",
      "dependencies": [
        "P2:QG2",
        "P2:QG5",
        "P2:QG6",
        "P2:QG7",
        "P2:ML4",
        "P2:ML7"
      ]
    },
    {
      "id": "P2:OP0",
      "source": "prerequisites/ordinary-operator-calculus.md",
      "source_sha256": "9d994f402f4503e0c2a3b540cf4030300d5df9aa69883d1aac58f829d87b0e60",
      "proof_locator": "### O0.1. Product tests determine a distributional kernel",
      "package": "P2",
      "dependencies": [
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:OP1",
      "source": "prerequisites/ordinary-operator-calculus.md",
      "source_sha256": "9d994f402f4503e0c2a3b540cf4030300d5df9aa69883d1aac58f829d87b0e60",
      "proof_locator": "## O1. Schwartz action and the exact multiplier formula",
      "package": "P2",
      "dependencies": [
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P2:OP2",
      "source": "prerequisites/ordinary-operator-calculus.md",
      "source_sha256": "9d994f402f4503e0c2a3b540cf4030300d5df9aa69883d1aac58f829d87b0e60",
      "proof_locator": "## O2. The exact ordinary metric verification",
      "package": "P2",
      "dependencies": [
        "P2:QG7",
        "P2:QG8",
        "P2:ML1",
        "P2:F1",
        "P2:F3"
      ]
    },
    {
      "id": "P2:OP3",
      "source": "prerequisites/ordinary-operator-calculus.md",
      "source_sha256": "9d994f402f4503e0c2a3b540cf4030300d5df9aa69883d1aac58f829d87b0e60",
      "proof_locator": "## O3. Adjoints, composition and tempered distributions",
      "package": "P2",
      "dependencies": [
        "P2:OP1",
        "P2:OP2",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "P2:OP4",
      "source": "prerequisites/ordinary-operator-calculus.md",
      "source_sha256": "9d994f402f4503e0c2a3b540cf4030300d5df9aa69883d1aac58f829d87b0e60",
      "proof_locator": "## O4. Compact amplitudes and smooth off-diagonal kernels",
      "package": "P2",
      "dependencies": [
        "P2:OP0",
        "P2:OP1",
        "P2:OP2",
        "P2:OP3",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "P2:OP5",
      "source": "prerequisites/ordinary-operator-calculus.md",
      "source_sha256": "9d994f402f4503e0c2a3b540cf4030300d5df9aa69883d1aac58f829d87b0e60",
      "proof_locator": "## O5. Proper operators and the actual local representation",
      "package": "P2",
      "dependencies": [
        "P2:OP0",
        "P2:OP3",
        "P2:OP4",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5",
        "P3:M0",
        "P3:M1",
        "P3:M2",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "P3:L1",
        "P3:L2",
        "P3:L3"
      ]
    },
    {
      "id": "P2:OP6",
      "source": "prerequisites/ordinary-operator-calculus.md",
      "source_sha256": "9d994f402f4503e0c2a3b540cf4030300d5df9aa69883d1aac58f829d87b0e60",
      "proof_locator": "## O6. Proper products, remainders and the principal symbol",
      "package": "P2",
      "dependencies": [
        "P2:OP0",
        "P2:OP3",
        "P2:OP4",
        "P2:OP5",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14",
        "U001:Q5",
        "U001:P21.3",
        "U001:P21.4",
        "U001:P18.5"
      ]
    },
    {
      "id": "P3:M0",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M0. The declared choice input",
      "package": "P3",
      "dependencies": [
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:M1",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M1. The generating-class lemma",
      "package": "P3",
      "dependencies": [
        "P3:M0"
      ]
    },
    {
      "id": "P3:M2",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M2. Completed Lebesgue measure",
      "package": "P3",
      "dependencies": [
        "P3:M0",
        "P3:M1",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:M3",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M3. Measurability and the integral",
      "package": "P3",
      "dependencies": [
        "P3:M0",
        "P3:M2",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:M4",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M4. Product integration",
      "package": "P3",
      "dependencies": [
        "P3:M1",
        "P3:M2",
        "P3:M3"
      ]
    },
    {
      "id": "P3:M5",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M5. The two required norms",
      "package": "P3",
      "dependencies": [
        "P3:M3",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:M6",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M6. Completeness",
      "package": "P3",
      "dependencies": [
        "P3:M3",
        "P3:M5",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:M7",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M7. Compact smooth density",
      "package": "P3",
      "dependencies": [
        "P3:M2",
        "P3:M3",
        "P3:M5",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:M8",
      "source": "prerequisites/measure-and-l2.md",
      "source_sha256": "89067fec2c31a02743bb6d5f10c4ea3d1ad7417459ba14310179198aaf956f77",
      "proof_locator": "## M8. Compatibility with the earlier integral",
      "package": "P3",
      "dependencies": [
        "P3:M2",
        "P3:M3",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:L1",
      "source": "prerequisites/fourier-l2.md",
      "source_sha256": "12d33935303665ed98310e733a295f13c0a5fa4e326c088a3a69c2b980e6ea4d",
      "proof_locator": "## L1. Both inverse maps",
      "package": "P3",
      "dependencies": [
        "P3:M6",
        "P3:M7",
        "P3:M8",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:L2",
      "source": "prerequisites/fourier-l2.md",
      "source_sha256": "12d33935303665ed98310e733a295f13c0a5fa4e326c088a3a69c2b980e6ea4d",
      "proof_locator": "## L2. Ordinary integrals and distributions",
      "package": "P3",
      "dependencies": [
        "P3:L1",
        "P3:M4",
        "P3:M5",
        "P3:M7",
        "P3:M8",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "P3:L3",
      "source": "prerequisites/fourier-l2.md",
      "source_sha256": "12d33935303665ed98310e733a295f13c0a5fa4e326c088a3a69c2b980e6ea4d",
      "proof_locator": "## L3. Pairing and measurable multipliers",
      "package": "P3",
      "dependencies": [
        "P3:L1",
        "P3:L2",
        "P3:M5",
        "U001:A0",
        "U001:P6.0",
        "U001:P6.1",
        "U001:P6.5",
        "U001:L2.4.5",
        "U001:P13.1",
        "U001:P13.2-series",
        "U001:P19.1",
        "U001:L10.1.12",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:F0-COMP",
        "U001:F0-ALG",
        "U001:F0-DIFF",
        "U001:L10.1.14"
      ]
    },
    {
      "id": "GEO:C0",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "## C0. Exact inputs and conventions",
      "package": "GEO",
      "dependencies": [
        "U001:L8.1.14",
        "U001:L8.1.17",
        "U001:L8.1.18",
        "U001:P9.1",
        "U001:P9.4",
        "U001:L7.1.4",
        "U001:P2",
        "U001:F0-DIFF",
        "U001:P4"
      ]
    },
    {
      "id": "GEO:C1",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "## C1. A horizontal graph transverse to a Lagrangian plane",
      "package": "GEO",
      "dependencies": [
        "GEO:C0"
      ]
    },
    {
      "id": "GEO:C2",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "## C2. The radial vector kills the canonical one-form",
      "package": "GEO",
      "dependencies": [
        "GEO:C0"
      ]
    },
    {
      "id": "GEO:C3",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "## C3. Construct the base coordinates and the conic chart",
      "package": "GEO",
      "dependencies": [
        "GEO:C0",
        "GEO:C1"
      ]
    },
    {
      "id": "GEO:C4",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "## C4. The homogeneous generating function and its extension",
      "package": "GEO",
      "dependencies": [
        "GEO:C2",
        "GEO:C3",
        "U001:U001-A4"
      ]
    },
    {
      "id": "GEO:C5",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "## C5. The geometry supplied by any nondegenerate phase",
      "package": "GEO",
      "dependencies": [
        "GEO:C0"
      ]
    },
    {
      "id": "GEO:C18",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "There is also an exact rank identity",
      "package": "GEO",
      "dependencies": [
        "GEO:C5"
      ]
    },
    {
      "id": "GEO:C-EX1",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "**Exercise C1: verify that this coordinate change is a base diffeomorphism.**",
      "package": "GEO",
      "dependencies": [
        "GEO:C0",
        "GEO:C3"
      ]
    },
    {
      "id": "GEO:C-EX2",
      "source": "prerequisites/conic-frequency-coordinates.md",
      "source_sha256": "c47e47f2d0f0f515683b4c0a176441ff1700a4e1d40c4d00550ea471feb3accb",
      "proof_locator": "**Exercise C2: identify the radial degeneracy of the model phase.**",
      "package": "GEO",
      "dependencies": [
        "GEO:C4",
        "GEO:C5",
        "GEO:C18"
      ]
    },
    {
      "id": "CH:T0",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## T0. Inputs, test functions and the distributional coordinate map",
      "package": "CH",
      "dependencies": [
        "P3:M8",
        "P3:L2",
        "U001:U001-A4",
        "U001:F0-DIFF",
        "U001:P2",
        "U001:P21.4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P17.2",
        "U001:P17.4",
        "U001:P17.5-order"
      ]
    },
    {
      "id": "CH:T1",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## T1. Transport an ordinary symbol with both Jacobians",
      "package": "CH",
      "dependencies": [
        "CH:T0",
        "P2:OP0",
        "P2:OP4",
        "P2:OP5",
        "P3:M3",
        "U001:F0-COMP"
      ]
    },
    {
      "id": "CH:T2",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## T2. Conic smoothing and composition",
      "package": "CH",
      "dependencies": [
        "CH:T1",
        "P2:OP3",
        "P2:OP4",
        "P2:OP6",
        "U001:F0-COMP"
      ]
    },
    {
      "id": "CH:T3",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## T3. Coordinate and frame invariance of iterated regularity",
      "package": "CH",
      "dependencies": [
        "CH:T0",
        "CH:T1",
        "P1:B4",
        "P1:B5",
        "P2:OP3",
        "P2:OP4",
        "P2:OP5",
        "U001:P8-smooth",
        "U001:U001-A4"
      ]
    },
    {
      "id": "CH:W1",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## W1. Fourier cutoffs and the definition of wavefront",
      "package": "CH",
      "dependencies": [
        "CH:T0",
        "P3:L2",
        "U001:Q3",
        "U001:Q4",
        "U001:F0-COMP",
        "U001:P17.5-order"
      ]
    },
    {
      "id": "CH:W2",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## W2. The complete nonstationary bound for a chart",
      "package": "CH",
      "dependencies": [
        "CH:T0",
        "U001:F0-CALC",
        "U001:P17.2",
        "U001:P17.4"
      ]
    },
    {
      "id": "CH:W3",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## W3. Diffeomorphisms transport exactly the cotangent direction",
      "package": "CH",
      "dependencies": [
        "CH:T0",
        "CH:W1",
        "CH:W2",
        "P3:M4",
        "U001:P21.4"
      ]
    },
    {
      "id": "CH:W4",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## W4. Ordinary operators and compact microlocal cutoffs",
      "package": "CH",
      "dependencies": [
        "CH:T2",
        "CH:W1",
        "P1:B4",
        "P2:OP4",
        "P2:OP5",
        "U001:U001-A4",
        "U001:P8-smooth"
      ]
    },
    {
      "id": "CH:W5",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "## W5. Iterated regularity has wavefront contained in the Lagrangian",
      "package": "CH",
      "dependencies": [
        "CH:T3",
        "CH:T2",
        "CH:W4",
        "P2:OP3",
        "P2:OP4",
        "P2:OP6",
        "P1:B1",
        "P1:B2",
        "P1:B6",
        "U001:F0-COMP"
      ]
    },
    {
      "id": "CH:T-EX1",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "**Exercise T1.**",
      "package": "CH",
      "dependencies": [
        "CH:T0",
        "CH:T1"
      ]
    },
    {
      "id": "CH:T-EX2",
      "source": "prerequisites/coordinate-and-wavefront-localization.md",
      "source_sha256": "bee282d8aceba0ca68ba3f12fe41d1382a2053c120f48c967c2a6ee16e381656",
      "proof_locator": "**Exercise T2.**",
      "package": "CH",
      "dependencies": [
        "CH:T1",
        "CH:W3",
        "U001:P2",
        "U001:P9.1"
      ]
    },
    {
      "id": "K:K0",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K0. Exact inputs and conic conventions",
      "package": "K",
      "dependencies": [
        "CH:T2",
        "P2:OP6"
      ]
    },
    {
      "id": "K:K1",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K1. Summation with every derivative and with support retained",
      "package": "K",
      "dependencies": [
        "K:K0",
        "U001:U001-A4",
        "U001:F0-DIFF",
        "U001:P6.5"
      ]
    },
    {
      "id": "K:K1a",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "### K1a. The properly supported operator sum actually needed here",
      "package": "K",
      "dependencies": [
        "K:K1",
        "P2:OP0",
        "P2:OP4",
        "P2:OP5",
        "P3:L2",
        "CH:T2"
      ]
    },
    {
      "id": "K:K2",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K2. Ordinary matrix inverses on an elliptic cone",
      "package": "K",
      "dependencies": [
        "K:K0",
        "U001:L8.1.18",
        "U001:P2",
        "U001:F0-DIFF",
        "CH:T1",
        "CH:T3",
        "U001:U001-A4"
      ]
    },
    {
      "id": "K:K3",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K3. A two-sided conic parametrix with one fixed error cone",
      "package": "K",
      "dependencies": [
        "K:K1a",
        "K:K2",
        "CH:T2",
        "P2:OP3",
        "P2:OP4",
        "P2:OP5",
        "P2:OP6",
        "CH:W4",
        "U001:U001-A4"
      ]
    },
    {
      "id": "K:K4",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K4. A finite conic partition reconstructs a compact localization",
      "package": "K",
      "dependencies": [
        "K:K0",
        "P2:OP4",
        "CH:T2",
        "U001:U001-A4",
        "U001:F0-COMP",
        "U001:P8-smooth"
      ]
    },
    {
      "id": "K:K5",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K5. Directional Besov tests give the actual local space",
      "package": "K",
      "dependencies": [
        "K:K3",
        "K:K4",
        "P1:B6",
        "P1:B1",
        "CH:T2",
        "U001:Q3"
      ]
    },
    {
      "id": "K:K6",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K6. The full intrinsic localization theorem",
      "package": "K",
      "dependencies": [
        "K:K5",
        "K:K3",
        "CH:W5",
        "CH:T3",
        "P2:OP6",
        "P1:B6"
      ]
    },
    {
      "id": "K:K7",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "## K7. Independence of sufficiently small cutoff and Lagrangian extension",
      "package": "K",
      "dependencies": [
        "K:K3",
        "K:K4",
        "K:K6",
        "CH:T2",
        "P2:OP3",
        "P2:OP5",
        "U001:U001-A4"
      ]
    },
    {
      "id": "K:K-EX1",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "**Exercise K1.**",
      "package": "K",
      "dependencies": [
        "K:K3",
        "U001:P8-root"
      ]
    },
    {
      "id": "K:K-EX2",
      "source": "prerequisites/conic-parametrices-and-localization.md",
      "source_sha256": "1bea6e00b676af3e2192da8b2cd667a68f8da83cf4efa95ecfe908ebb3ac2ce6",
      "proof_locator": "**Exercise K2.**",
      "package": "K",
      "dependencies": [
        "K:K4",
        "K:K5",
        "P2:OP4",
        "P2:OP5",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ]
    },
    {
      "id": "PH:F0",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F0. Inputs, orders and support conventions",
      "package": "PH",
      "dependencies": [
        "GEO:C0",
        "CH:T3",
        "K:K6",
        "U001:U001-DEF"
      ]
    },
    {
      "id": "PH:F1",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F1. Clean geometry and a transverse Fourier test",
      "package": "PH",
      "dependencies": [
        "PH:F0",
        "GEO:C3",
        "GEO:C4",
        "U001:P3",
        "U001:P4",
        "U001:P5",
        "U001:U001-A5",
        "U001:U001-A4"
      ]
    },
    {
      "id": "PH:F2",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F2. Distributional meaning and the noncompact-frequency tails",
      "package": "PH",
      "dependencies": [
        "PH:F0",
        "U001:Q1",
        "U001:Q3",
        "U001:U001-2.1",
        "U001:F0-COMP",
        "U001:P18.5",
        "P2:OP5",
        "CH:W1",
        "CH:T0"
      ]
    },
    {
      "id": "PH:F3",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F3. The full Fourier symbol and its exact leading fibre integral",
      "package": "PH",
      "dependencies": [
        "PH:F1",
        "PH:F2",
        "U001:U001-6.1",
        "U001:U001-7.2",
        "U001:U001-A6",
        "P3:L2",
        "CH:T0",
        "U001:P8-smooth"
      ]
    },
    {
      "id": "PH:F4",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F4. A symbol lift for the given phase",
      "package": "PH",
      "dependencies": [
        "PH:F1",
        "PH:F3",
        "U001:P3",
        "U001:U001-A4",
        "U001:P17.2",
        "U001:P14.2-powers"
      ]
    },
    {
      "id": "PH:F5",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F5. Constructing an amplitude to every order",
      "package": "PH",
      "dependencies": [
        "PH:F3",
        "PH:F4",
        "K:K1"
      ]
    },
    {
      "id": "PH:F6",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F6. The prescribed-phase representation theorem",
      "package": "PH",
      "dependencies": [
        "PH:F5",
        "K:K7",
        "CH:T3",
        "CH:W1",
        "CH:W3",
        "CH:W4",
        "GR:G12-forward",
        "GR:G12-reverse",
        "P3:L2",
        "U001:Q3",
        "U001:Q4"
      ]
    },
    {
      "id": "PH:F6a",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "### F6a. Reading a lower order on a smaller output cone",
      "package": "PH",
      "dependencies": [
        "PH:F6",
        "GR:G12-forward",
        "GR:G12-reverse",
        "CH:W1",
        "CH:W4",
        "K:K6",
        "K:K7"
      ]
    },
    {
      "id": "PH:F7",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "## F7. The first lower-order criterion and clean cancellation",
      "package": "PH",
      "dependencies": [
        "PH:F3",
        "PH:F6a"
      ]
    },
    {
      "id": "PH:F-EX1",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "**Exercise F1: a quadratic stabilization.**",
      "package": "PH",
      "dependencies": [
        "PH:F3",
        "PH:F6",
        "U001:P8-root",
        "U001:P5"
      ]
    },
    {
      "id": "PH:F-EX2",
      "source": "prerequisites/prescribed-phase-representation.md",
      "source_sha256": "dd9a8df31232e54122db43637341957d4c7c49de7d2ae76688637258faafa080",
      "proof_locator": "**Exercise F2: a redundant variable and cancellation.**",
      "package": "PH",
      "dependencies": [
        "PH:F2",
        "PH:F3",
        "PH:F6",
        "PH:F7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:U001-A4"
      ]
    },
    {
      "id": "SB:S0",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "## S0. Conventions and exact inputs",
      "package": "SB",
      "dependencies": [
        "P3:L2",
        "P3:L3",
        "P1:B1"
      ]
    },
    {
      "id": "SB:S1",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "## S1. A Fourier estimate with an adjustable split",
      "package": "SB",
      "dependencies": [
        "SB:S0",
        "P3:L2",
        "P3:M4",
        "U001:U001-A4"
      ]
    },
    {
      "id": "SB:S2",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "## S2. Weighted square integrability improves every symbol derivative",
      "package": "SB",
      "dependencies": [
        "SB:S1",
        "U001:U001-A4",
        "U001:F0-DIFF",
        "U001:P14.2-powers"
      ]
    },
    {
      "id": "SB:S3",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "## S3. Sobolev localization and coordinate transport",
      "package": "SB",
      "dependencies": [
        "SB:S0",
        "P1:B1",
        "P1:B3",
        "P1:B4",
        "P1:B5",
        "P1:B6",
        "P2:OP5",
        "CH:T0",
        "CH:T1",
        "K:K0",
        "K:K3"
      ]
    },
    {
      "id": "SB:S4",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "## S4. The intrinsic order improves, with a strict inequality",
      "package": "SB",
      "dependencies": [
        "SB:S2",
        "SB:S3",
        "GEO:C3",
        "GEO:C4",
        "CH:T3",
        "K:K7",
        "GR:G12-forward",
        "GR:G12-reverse"
      ]
    },
    {
      "id": "SB:S5",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "## S5. A counterexample at equality, including its microlocal location",
      "package": "SB",
      "dependencies": [
        "SB:S0",
        "SB:S4",
        "U001:U001-A4",
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:L2",
        "PH:F2",
        "K:K7",
        "CH:W4",
        "GR:G12-forward",
        "GR:G12-reverse"
      ]
    },
    {
      "id": "SB:S-EX1",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "**Exercise S1:",
      "package": "SB",
      "dependencies": [
        "SB:S0",
        "P1:B1",
        "GR:G12-reverse",
        "U001:P14.2-powers"
      ]
    },
    {
      "id": "SB:S-EX2",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "**Exercise S2:",
      "package": "SB",
      "dependencies": [
        "K:K6",
        "P2:OP6"
      ]
    },
    {
      "id": "SB:S-EX3",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "**Exercise S3:",
      "package": "SB",
      "dependencies": [
        "GR:G12-forward"
      ]
    },
    {
      "id": "SB:S-EX4",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "**Exercise S4:",
      "package": "SB",
      "dependencies": [
        "PH:F2",
        "PH:F3",
        "PH:F5",
        "PH:F6",
        "U001:U001-A4"
      ]
    },
    {
      "id": "SB:S-EX5",
      "source": "prerequisites/sobolev-order-refinement.md",
      "source_sha256": "a3d008a8c864183ea9e93eee741313ebd72c167c8a10db3a82ac08c7a7e3215f",
      "proof_locator": "**Exercise S5:",
      "package": "SB",
      "dependencies": [
        "SB:S5"
      ]
    },
    {
      "id": "ML:M0",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M0. Exact inputs and the sign convention",
      "package": "ML",
      "dependencies": [
        "GEO:C0",
        "GEO:C1",
        "U001:P13.1",
        "U001:P15.2",
        "U001:P16.3"
      ]
    },
    {
      "id": "ML:M0a",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M0a. Inertia with zero directions and local constancy",
      "package": "ML",
      "dependencies": [
        "ML:M0",
        "U001:Q5",
        "U001:P5",
        "U001:P9.5",
        "U001:P9.2-operator"
      ]
    },
    {
      "id": "ML:M1",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M1. Symplectic coordinates adapted to one Lagrangian",
      "package": "ML",
      "dependencies": [
        "ML:M0",
        "ML:M0a",
        "U001:P2"
      ]
    },
    {
      "id": "ML:M2",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M2. Reduce the intersection and define a signed form",
      "package": "ML",
      "dependencies": [
        "ML:M1",
        "GEO:C1",
        "ML:M0a"
      ]
    },
    {
      "id": "ML:M3",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M3. The signature of a phase Hessian",
      "package": "ML",
      "dependencies": [
        "ML:M2",
        "ML:M0a",
        "GEO:C5",
        "U001:P9.4"
      ]
    },
    {
      "id": "ML:M4",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M4. The four-plane integer stays continuous through rank changes",
      "package": "ML",
      "dependencies": [
        "ML:M1",
        "ML:M3",
        "ML:M0a"
      ]
    },
    {
      "id": "ML:M5",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M5. Construct the relative Maslov line globally",
      "package": "ML",
      "dependencies": [
        "ML:M4",
        "GEO:C0",
        "GEO:C2",
        "U001:P15.2",
        "U001:P16.3"
      ]
    },
    {
      "id": "ML:M6",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M6. Identify the line defined by phase charts",
      "package": "ML",
      "dependencies": [
        "ML:M3",
        "ML:M5",
        "GEO:C3",
        "GEO:C4",
        "U001:P4",
        "U001:F0-DIFF"
      ]
    },
    {
      "id": "ML:M7",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "## M7. Constant intersection dimension gives a canonical trivialization",
      "package": "ML",
      "dependencies": [
        "ML:M2",
        "ML:M5",
        "ML:M6",
        "ML:M0a",
        "U001:P2"
      ]
    },
    {
      "id": "ML:M-EX1",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "**Exercise M1:",
      "package": "ML",
      "dependencies": [
        "ML:M4",
        "ML:M5",
        "ML:M7",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P16.3"
      ]
    },
    {
      "id": "ML:M-EX2",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "**Exercise M2:",
      "package": "ML",
      "dependencies": [
        "ML:M3",
        "ML:M6",
        "U001:Q5"
      ]
    },
    {
      "id": "ML:M-EX3",
      "source": "prerequisites/relative-maslov-line.md",
      "source_sha256": "741ccb22cfeeec1157022dd415e48d499251135e9ace7d72fdec96842537a366",
      "proof_locator": "**Exercise M3:",
      "package": "ML",
      "dependencies": [
        "ML:M3",
        "ML:M4",
        "U001:Q5"
      ]
    },
    {
      "id": "PS:PS0",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "## PS0. Statement and exact inputs",
      "package": "PS",
      "dependencies": [
        "GEO:C0",
        "ML:M5",
        "ML:M6",
        "PH:F0",
        "K:K6",
        "CH:T3"
      ]
    },
    {
      "id": "PS:PS1",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "## PS1. Critical density and coordinate changes",
      "package": "PS",
      "dependencies": [
        "PS:PS0",
        "GEO:C5",
        "PH:F1",
        "ML:M3",
        "U001:P2",
        "U001:P3",
        "U001:P4",
        "U001:P8-root",
        "U001:P21.3",
        "U001:P21.4",
        "U001:U001-A5"
      ]
    },
    {
      "id": "PS:PS2",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "## PS2. Ordinary symbols with half-density values",
      "package": "PS",
      "dependencies": [
        "PS:PS1",
        "ML:M6",
        "GEO:C3",
        "GEO:C4",
        "U001:F0-DIFF",
        "U001:F0-COMP",
        "U001:P8-smooth",
        "U001:P14.2-powers"
      ]
    },
    {
      "id": "PS:PS3",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "## PS3. Fourier testing and the local symbol",
      "package": "PS",
      "dependencies": [
        "PS:PS2",
        "PH:F2",
        "PH:F3",
        "PH:F6",
        "PH:F6a",
        "PH:F7",
        "K:K7",
        "CH:W4",
        "ML:M3",
        "ML:M6",
        "U001:P15.2",
        "U001:P16.3"
      ]
    },
    {
      "id": "PS:PS4",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "## PS4. Extending a local symbol to an amplitude",
      "package": "PS",
      "dependencies": [
        "PS:PS3",
        "PH:F4",
        "PH:F6",
        "PH:F2",
        "U001:U001-A4"
      ]
    },
    {
      "id": "PS:PS5",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "## PS5. A partition with locally finite base supports",
      "package": "PS",
      "dependencies": [
        "PS:PS4",
        "GEO:C3",
        "GEO:C4",
        "U001:F0-COMP",
        "U001:U001-A4",
        "U001:P3",
        "U001:P8-smooth"
      ]
    },
    {
      "id": "PS:PS6",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "## PS6. Global gluing, surjectivity and the kernel",
      "package": "PS",
      "dependencies": [
        "PS:PS3",
        "PS:PS4",
        "PS:PS5",
        "PH:F6",
        "K:K6",
        "K:K7",
        "P1:B1",
        "ML:M6"
      ]
    },
    {
      "id": "PS:PS-EX1",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "**Exercise PS1:",
      "package": "PS",
      "dependencies": [
        "PS:PS1",
        "PS:PS3"
      ]
    },
    {
      "id": "PS:PS-EX2",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "**Exercise PS2:",
      "package": "PS",
      "dependencies": [
        "PS:PS3",
        "PS:PS4",
        "PS:PS6",
        "PH:F5",
        "ML:M6",
        "U001:Q5"
      ]
    },
    {
      "id": "PS:PS-EX3",
      "source": "prerequisites/global-principal-symbol.md",
      "source_sha256": "938746bdb2b23f9f6fea9cf07c4e7464ae29d74c06b0e2112d8a347d11d18a64",
      "proof_locator": "**Exercise PS3:",
      "package": "PS",
      "dependencies": [
        "PS:PS3",
        "PS:PS6",
        "ML:M7",
        "CH:T0",
        "U001:Q4"
      ]
    }
  ],
  "local_binding_checks": {
    "checkpoint-validation": {
      "sha256": "526078a67aae8a17d55f5d434e6ea9d587aa1a2090caaf3618a7ac4e0db6f8ff",
      "passed": true
    },
    "geometry-validation": {
      "sha256": "91f8f2c9b7e9a0b5bd06fab6115b6cd82c948cbf5a2107ee0cfb72562466d421",
      "passed": true
    },
    "chart-validation": {
      "sha256": "b4d1908fea84087d1bbcf3018ac4d3d584a7c24530c3e81ec645ebb4dd965137",
      "passed": true
    },
    "parametrix-validation": {
      "sha256": "194624bb5522e4f64a6c4623c849a9e71ac8ecde74372b1bc9e27088076cede5",
      "passed": true
    },
    "phase-validation": {
      "sha256": "849b26d2677d6b69ce33715fd451f228f082de871a65f2753245728c7425f6cf",
      "passed": true
    },
    "sobolev-validation": {
      "sha256": "ec632354b08a056c574d7619772ee07efb295af21534c55fe8e322ccf28bd0c5",
      "passed": true
    },
    "maslov-validation": {
      "sha256": "aa7a9c6ed1f2eff605d60b9293da9c6e834af8744bd890d137add3caaf53fa0e",
      "passed": true
    },
    "principal-symbol-validation": {
      "sha256": "dbc4c8a48b912c08a6879857ac2ed75ba60fe999a4bdab80235f0483ba5771b0",
      "passed": true
    }
  },
  "full_course_complete": false,
  "public_release_authorized": false,
  "all_dependency_ids_resolve": true,
  "acyclic_selected_chain": true,
  "selected_transitive_U001_nodes": 219
}