{
  "schema": "AN04-restored-hyperbolicity-necessity-proof-map/v1",
  "proofs": [
    {
      "id": "HN:D1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Flatten the first surface",
      "dependencies": [
        "U001:P3",
        "CH:T0",
        "TG:B2",
        "TG:B3",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact compact chart source extension and complete globally bounded-derivative Frechet space, actual compact test neighborhood"
    },
    {
      "id": "HN:D2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Here is the precise support estimate",
      "dependencies": [
        "HN:D1",
        "HN0:C5",
        "CH:T0"
      ],
      "scope": "Full same-order distribution estimate on the actual convex half-ball or cone; no cutoff inserted into the adjoint"
    },
    {
      "id": "HN:D3",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "We now make the two derivative budgets uniform.",
      "dependencies": [
        "HN:D1",
        "HN:D2",
        "TG:B1",
        "TG:B2",
        "P3:M4"
      ],
      "scope": "Complete Baire closed-cover argument, all source/test budgets, seminorm kernels and no solution selection"
    },
    {
      "id": "HN:I1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
        "HN0:F4",
        "HN0:F5",
        "CH:T0"
      ],
      "scope": "General nonzero constant-coefficient fixed smooth local inverse with all passive derivatives and actual compact difference set"
    },
    {
      "id": "HN:I2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "For \\(z=s+it\\), the other operator",
      "dependencies": [
        "HN:I1",
        "FN:E1",
        "FN:E2",
        "FN:E3",
        "FN:E4",
        "FN:E5",
        "FN:E6",
        "U001:U001-A4"
      ],
      "scope": "Exact L=-2dbar fundamental-solution sign, repeated cutoff inverse on one common patch and aLr division"
    },
    {
      "id": "HN:T1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "## 4.",
      "dependencies": [
        "HN:I1",
        "HN:I2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:U001-A4"
      ],
      "scope": "Full finite h Taylor expansion and finite recursive transport with common domains and all coefficient remainders"
    },
    {
      "id": "HN:T2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Choose \\(\\chi\\) supported in",
      "dependencies": [
        "HN:T1",
        "U001:P14.3",
        "U001:P15.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full compact adjoint residual, all derivative losses and actual support-cone annulus damping, exact finite J"
    },
    {
      "id": "HN:N1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "## 5.",
      "dependencies": [
        "QF:A1",
        "U001:P3",
        "CH:T0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Nonreal root normalization, actual real shear, nonzero tangential covector and homogeneous factor with coefficientwise adjoint conjugation"
    },
    {
      "id": "HN:N2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Choose an integer \\(\\nu>r\\), put",
      "dependencies": [
        "HN:N1",
        "U001:P21.4",
        "CH:T0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full anisotropic pullback, exact Jacobian, every monomial weight, principal coefficient remainder and adjoint lower terms"
    },
    {
      "id": "HN:N3",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Write \\(z=y_s+i y_t\\)",
      "dependencies": [
        "HN:N2",
        "HN:I2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact complex phase, L annihilation, positive-side quadratic bound and fixed small patch"
    },
    {
      "id": "HN:N4",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "The leading conjugated frozen operator",
      "dependencies": [
        "HN:N2",
        "HN:N3",
        "HN:I2",
        "HN:T1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact aLr leading operator and normalization; every perturbation power smooth at h0 including nu-r gap"
    },
    {
      "id": "HN:N5",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Pull HN8 back by HN18.",
      "dependencies": [
        "HN:D3",
        "HN:N2",
        "HN:N4",
        "HN:T2",
        "U001:P21.4"
      ],
      "scope": "Full rescaled dual estimate with reciprocal Jacobian and both derivative budgets"
    },
    {
      "id": "HN:N6",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Choose a compact smooth \\(F\\) supported",
      "dependencies": [
        "HN:N5",
        "HN:N4",
        "HN:T2",
        "U001:U001-A4",
        "U001:P21.4",
        "U001:P15.1",
        "P3:M3",
        "P3:M8"
      ],
      "scope": "Actual supported concentrated source, Hermitian phase limit and positive bump, finite budget contradiction; full Theorem1.1"
    },
    {
      "id": "HN:C1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "## 6.",
      "dependencies": [
        "HN:N1",
        "HN:D3",
        "CH:T0",
        "U001:P21.4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Cone theorem contradiction and n1 endpoint; homogeneous tangential-degree decomposition, actual isotropic shrinking and all coefficient remainders"
    },
    {
      "id": "HN:C2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "\\tag{HN28}",
      "dependencies": [
        "HN:C1",
        "HN:I1",
        "HN:T1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact constant nonelliptic tangential transport with i power, full normalized operator and smooth perturbation gap"
    },
    {
      "id": "HN:C3",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "The required phase positivity on the test cone",
      "dependencies": [
        "HN:C1",
        "HN:C2",
        "HN:T2",
        "U001:P14.3"
      ],
      "scope": "Sharp stated positive cone damping and actual cutoff-annulus bound; B0 obstruction retained"
    },
    {
      "id": "HN:C4",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "The pullback of HN8 has a safe loss",
      "dependencies": [
        "HN:C1",
        "HN:C2",
        "HN:C3",
        "HN:D3",
        "HN:T2",
        "U001:U001-A4",
        "U001:P21.4",
        "U001:P15.1",
        "P3:M3",
        "P3:M8"
      ],
      "scope": "Both isotropic derivative losses, cone-supported concentrated source and positive pairing, full noncharacteristic contradiction"
    },
    {
      "id": "HN:R1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "Only nonvanishing of \\(dp\\) at the root",
      "dependencies": [
        "U001:P3",
        "CH:T0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Local real fiber polynomial, dphi0 exception, exact finite multiplicity, normal-frequency shift and punctured domain"
    },
    {
      "id": "HN:R2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Lemma 7.2",
      "dependencies": [
        "HN:R1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full smooth-parameter finite expansion with complex polynomial variable and compact-uniform binomial scaling"
    },
    {
      "id": "HN:R3",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Complete circle-count bridge.**",
      "dependencies": [
        "QF:A1",
        "QF:A5",
        "U001:F0-COMP",
        "U001:L3.3.8",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact logarithmic derivative circle count, multiplicity, continuous integer homotopy and degree drops without root labels"
    },
    {
      "id": "HN:R4",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "For \\(s\\ne0\\), this binomial",
      "dependencies": [
        "HN:R2",
        "HN:R3",
        "QF:A1",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All local roots persist; two-sided parameters exclude every nonzero first derivative, arbitrary k not just finite diagnostics"
    },
    {
      "id": "HN:R5",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "At an interior point \\(x_n>0\\)",
      "dependencies": [
        "HN:R1",
        "HN:R4"
      ],
      "scope": "All available base and fiber directions force dp0; exact interior simple-root conclusion"
    },
    {
      "id": "HN:R6",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "At \\(x_n=0\\), both signs",
      "dependencies": [
        "HN:R1",
        "HN:R4",
        "HN:R5"
      ],
      "scope": "Only normal base derivative survives, one-sided binomial yields precisely k2 and ab<0"
    },
    {
      "id": "HN:R7",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "With the convention",
      "dependencies": [
        "HN:R6",
        "G2:G0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete Hamilton second derivative, all tangential terms justified zero and positive double boundary contact"
    },
    {
      "id": "HN:M1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "## 8.",
      "dependencies": [
        "HN:R7",
        "U001:P14.2-powers",
        "G2:G0"
      ],
      "scope": "Exact normal-noncharacteristic model with boundary merger, zero-covector and genuine-multiple-characteristic scope"
    },
    {
      "id": "HN:M2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "At a double root the sign conclusion survives",
      "dependencies": [
        "HN:R7",
        "HN:M1",
        "G2:G0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full nonconstant real multiplier invariance with vanishing terms and complex sign reversal"
    },
    {
      "id": "HN:H0",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "## 1.",
      "dependencies": [
        "HN:N6",
        "HN:C4",
        "HC:L3"
      ],
      "scope": "Full retained support theorems and compact-source strengthening, identically-zero exception, strict cone conditions and relation to earlier sufficiency"
    },
    {
      "id": "HN:X1",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 1 (",
      "dependencies": [
        "HN:D2",
        "HN0:C5",
        "P3:L3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Retained original Exercise 1 with complete unchanged solution"
    },
    {
      "id": "HN:X2",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 2 (",
      "dependencies": [
        "HN:D3"
      ],
      "scope": "Retained original Exercise 2 with complete unchanged solution"
    },
    {
      "id": "HN:X3",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 3 (",
      "dependencies": [
        "HN:N6",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Retained original Exercise 3 with complete unchanged solution"
    },
    {
      "id": "HN:X4",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 4 (",
      "dependencies": [
        "HN:N4",
        "HN:N6"
      ],
      "scope": "Retained original Exercise 4 with complete unchanged solution"
    },
    {
      "id": "HN:X5",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 5 (",
      "dependencies": [
        "HN:N2",
        "HN:N4",
        "CH:T0"
      ],
      "scope": "Retained original Exercise 5 with complete unchanged solution"
    },
    {
      "id": "HN:X6",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 6 (",
      "dependencies": [
        "HN:I2"
      ],
      "scope": "Retained original Exercise 6 with complete unchanged solution"
    },
    {
      "id": "HN:X7",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 7 (",
      "dependencies": [
        "HN:N5",
        "HN:N6",
        "HN:T2"
      ],
      "scope": "Retained original Exercise 7 with complete unchanged solution"
    },
    {
      "id": "HN:X8",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 8 (",
      "dependencies": [
        "HN:N6"
      ],
      "scope": "Retained original Exercise 8 with complete unchanged solution"
    },
    {
      "id": "HN:X9",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 9 (",
      "dependencies": [
        "HN:I1",
        "P3:L3",
        "P3:M4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Retained original Exercise 9 with complete unchanged solution"
    },
    {
      "id": "HN:X10",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 10 (",
      "dependencies": [
        "HN:C3",
        "HN:C4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Retained original Exercise 10 with complete unchanged solution"
    },
    {
      "id": "HN:X11",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 11 (",
      "dependencies": [
        "HN:C1"
      ],
      "scope": "Retained original Exercise 11 with complete unchanged solution"
    },
    {
      "id": "HN:X12",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 12 (",
      "dependencies": [
        "HN:C2"
      ],
      "scope": "Retained original Exercise 12 with complete unchanged solution"
    },
    {
      "id": "HN:X13",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 13 (",
      "dependencies": [
        "HN:R3",
        "HN:R4",
        "QF:A1"
      ],
      "scope": "Retained original Exercise 13 with complete unchanged solution"
    },
    {
      "id": "HN:X14",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 14 (",
      "dependencies": [
        "HN:R7",
        "HN:M1"
      ],
      "scope": "Retained original Exercise 14 with complete unchanged solution"
    },
    {
      "id": "HN:X15",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 15 (",
      "dependencies": [
        "HN:M2"
      ],
      "scope": "Retained original Exercise 15 with complete unchanged solution"
    },
    {
      "id": "HN:X16",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 16 (",
      "dependencies": [
        "HN:R1",
        "HN:R7"
      ],
      "scope": "Retained original Exercise 16 with complete unchanged solution"
    },
    {
      "id": "HN:X17",
      "source": "hyperbolicity-necessity-support-tests-and-double-roots.md",
      "source_sha256": "ed9768bd3c8cda95823c1889851bf526fe7e4800fe3192e04e1a222056f6de92",
      "proof_locator": "**Exercise 17 (",
      "dependencies": [
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