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      "id": "FA:C0",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "0b64b7ca722d50f60eceb4028c505ebd3533e5ee0a9e3f89ce1fc010dd03a90c",
      "proof_locator": "## 1.",
      "dependencies": [
        "CF:H6",
        "CF:P3"
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      "scope": "Entire two-fold model, sign convention, genuine homogeneous phase and precise compact/conic supports"
    },
    {
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      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "0b64b7ca722d50f60eceb4028c505ebd3533e5ee0a9e3f89ce1fc010dd03a90c",
      "proof_locator": "**Proposition 1.1",
      "dependencies": [
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        "OSC:O3",
        "IR:R6",
        "FD:P0",
        "U001:F0-DIFF"
      ],
      "scope": "Both directions of full symbol equivalence under tau=rho s, radial Jacobian and exact normalized order"
    },
    {
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      "source": "fold-amplitudes-and-critical-densities.md",
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      "proof_locator": "The delta notation has a direct coordinate meaning.",
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        "U001:U001-A5",
        "U001:P2"
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      "scope": "Critical density from actual normal coordinates; variable-dependent constraint matrix rule"
    },
    {
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      "source": "fold-amplitudes-and-critical-densities.md",
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      "proof_locator": "**Proposition 2.1",
      "dependencies": [
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        "U001:F0-DIFF"
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      "scope": "Both independent radial factors, full normal determinant and exact amplitude/half-density cancellation"
    },
    {
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      "proof_locator": "**Proposition 3.1.",
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        "FA:D1",
        "U001:F0-DIFF"
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      "scope": "All mixed symbol estimates under critical substitution and homogeneous positive density degree"
    },
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      "proof_locator": "The fixed-phase principal-symbol formula",
      "dependencies": [
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        "GS:Z10",
        "GS:Z11",
        "GS:Z12"
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      "scope": "Exact principal symbol in its phase frame, total order m+n/2 and one-order quotient"
    },
    {
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      "proof_locator": "## 4.",
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        "FA:D0",
        "CF:H6",
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      "scope": "Reduced homogeneous phase, full critical independence, nonzero differential, product embedding and density"
    },
    {
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      "proof_locator": "**Lemma 4.1",
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        "GS:Z11",
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        "U001:P2"
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      "scope": "Exact degree-one hyperbolic elimination, ordered determinant and zero-signature Gaussian transition"
    },
    {
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      "proof_locator": "For a coefficient",
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        "FA:R1",
        "OSC:O1",
        "U001:Q3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact distributional first-frequency Fourier elimination with test-pair domination and all constants"
    },
    {
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      "source": "fold-amplitudes-and-critical-densities.md",
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      "proof_locator": "In particular the conically truncated amplitude",
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        "U001:U001-A4"
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      "proof_locator": "To make the rapid-decrease assertion explicit,",
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    },
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      "proof_locator": "**Proof.** By (4.1)–(4.4)",
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      "proof_locator": "Apply the imported support-preserving",
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        "IR:R4",
        "OSC:O3"
      ],
      "scope": "Exact support-preserving asymptotic summation in x,s;xi-prime, arbitrary-order remainder and smooth residual"
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      "proof_locator": "Finally Fourier inversion",
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        "FA:T2",
        "OSC:O2",
        "U001:Q3",
        "U001:U001-A4"
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      "proof_locator": "## 7.",
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        "FA:V3",
        "U001:P14.2-powers"
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      "scope": "Explicit ordinary symbol, ellipticity through fold and exact example orders without claiming continuity"
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      "proof_locator": "**Exercise 8.1 ",
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        "FA:R0"
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      "scope": "Original Exercise 8.1 and complete unchanged solution"
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      "proof_locator": "**Exercise 8.2 ",
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        "FA:D1",
        "FA:A1",
        "IR:R6"
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      "scope": "Original Exercise 8.2 and complete unchanged solution"
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    {
      "id": "FA:X3",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "0b64b7ca722d50f60eceb4028c505ebd3533e5ee0a9e3f89ce1fc010dd03a90c",
      "proof_locator": "**Exercise 8.3 ",
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      "scope": "Original Exercise 8.3 and complete unchanged solution"
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      "proof_locator": "**Exercise 8.4 ",
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      ],
      "scope": "Original Exercise 8.4 and complete unchanged solution"
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    {
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      "source_sha256": "0b64b7ca722d50f60eceb4028c505ebd3533e5ee0a9e3f89ce1fc010dd03a90c",
      "proof_locator": "**Exercise 8.5 ",
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      "scope": "Original Exercise 8.5 and complete unchanged solution"
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    {
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      "source_sha256": "0b64b7ca722d50f60eceb4028c505ebd3533e5ee0a9e3f89ce1fc010dd03a90c",
      "proof_locator": "**Exercise 8.6 ",
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        "FA:R2"
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      "scope": "Original Exercise 8.6 and complete unchanged solution"
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      "source_sha256": "0b64b7ca722d50f60eceb4028c505ebd3533e5ee0a9e3f89ce1fc010dd03a90c",
      "proof_locator": "**Exercise 8.7 ",
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    "GS:Z10",
    "GS:Z11",
    "GS:Z12",
    "GS:Z2",
    "GS:Z3",
    "GS:Z4",
    "GS:Z5",
    "GS:Z6",
    "GS:Z7",
    "GS:Z8",
    "GS:Z9",
    "HT:H1",
    "HT:H2",
    "HT:H3",
    "HT:H4",
    "HT:H5",
    "IR:R0",
    "IR:R1",
    "IR:R2",
    "IR:R3",
    "IR:R4",
    "IR:R5",
    "IR:R6",
    "IR:R7",
    "IR:R8",
    "K:K0",
    "K:K1",
    "K:K1a",
    "K:K2",
    "K:K3",
    "K:K4",
    "K:K5",
    "K:K6",
    "K:K7",
    "ML:M0",
    "ML:M0a",
    "ML:M1",
    "ML:M2",
    "ML:M3",
    "ML:M4",
    "ML:M5",
    "ML:M6",
    "OSC:O1",
    "OSC:O2",
    "OSC:O3",
    "OSC:O4",
    "OSC:O5",
    "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:OP0",
    "P2:OP1",
    "P2:OP2",
    "P2:OP3",
    "P2:OP4",
    "P2:OP5",
    "P2:OP6",
    "P2:QG1",
    "P2:QG2",
    "P2:QG3",
    "P2:QG4",
    "P2:QG5",
    "P2:QG6",
    "P2:QG7",
    "P2:QG8",
    "P3:L1",
    "P3:L2",
    "P3:L3",
    "P3:M0",
    "P3:M1",
    "P3:M2",
    "P3:M3",
    "P3:M4",
    "P3:M5",
    "P3:M6",
    "P3:M7",
    "P3:M8",
    "PH:F0",
    "PH:F1",
    "PH:F2",
    "PH:F3",
    "PH:F4",
    "PH:F5",
    "PH:F6",
    "PH:F6a",
    "PH:F7",
    "PS:PS0",
    "PS:PS1",
    "PS:PS2",
    "PS:PS3",
    "PS:PS4",
    "PS:PS5",
    "PS:PS6",
    "SF:C1",
    "SF:C2",
    "SF:C3",
    "SF:F1",
    "SF:F2",
    "SF:H0",
    "SF:H1",
    "SF:H2",
    "SF:H3",
    "SF:H4",
    "SF:H5",
    "SF:H6",
    "SF:H7",
    "SF:O1",
    "SF:O2",
    "SF:O3",
    "SF:O4",
    "SF:O5",
    "SF:R1",
    "SF:S0",
    "SF:S1",
    "SF:S2",
    "SI:C1",
    "SI:C2",
    "SI:C3",
    "SI:F1",
    "SI:H1",
    "SI:I0",
    "SI:I1",
    "SI:I2",
    "SI:I3",
    "SI:J1",
    "SI:J2",
    "SI:J3",
    "SI:J4",
    "SI:M0",
    "SI:S1",
    "SI:W1",
    "SN:F0",
    "SN:F1",
    "SN:F2",
    "TG:T0",
    "TG:T6"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}
