{
  "schema": "AN04-restored-principal-boundary-proof-map/v1",
  "proofs": [
    {
      "id": "PT:H0",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "**Weak subsequences in the actual Hilbert space.**",
      "dependencies": [
        "CT:T1",
        "CE:H5",
        "MH:Q1",
        "MH:A1",
        "P3:L1",
        "P3:M3",
        "U001:F0-COMP",
        "U001:Q5"
      ],
      "scope": "Complete weak subsequence construction, finite/infinite/zero cases, weak lower norm inequality, distributional equations and support through actual mixed pairings"
    },
    {
      "id": "PT:R1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "## 1.",
      "dependencies": [
        "HN:R1",
        "HN:R5",
        "HN:R6",
        "HN:R7",
        "HN:M2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full noncharacteristic simple/double alternative, exact real principal-type and conic hypotheses"
    },
    {
      "id": "PT:R2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Fix a double root",
      "dependencies": [
        "PT:R1",
        "HN:R3",
        "QF:A1",
        "QF:A5",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P2"
      ],
      "scope": "Complete smooth circle moments, polynomial factor division, real coefficients, homogeneity and no smooth branch assumption"
    },
    {
      "id": "PT:R3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "At the double root,",
      "dependencies": [
        "PT:R2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full sign -2btq2 and uniform positive b>=ct|xi|2 on smaller fixed cone"
    },
    {
      "id": "PT:P1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The exact earlier [scalar sharp",
      "dependencies": [
        "CE:G9",
        "CE:G10",
        "FP0:M1",
        "FP0:M3",
        "FP0:M4",
        "MP:F6",
        "MP:F7",
        "MP:F8",
        "WY:Q6"
      ],
      "scope": "Actual scalar order2, fixed2x2 order0 and scalar Fefferman-Phong interfaces with finite seminorms and quantization signs"
    },
    {
      "id": "PT:P2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Replacing \\(A\\)",
      "dependencies": [
        "PT:P1",
        "P2:OP3",
        "P2:OP6",
        "CE:G1",
        "WY:W4",
        "WY:Q6"
      ],
      "scope": "Permitted lower-order symmetrization, positive B operator and actual symbol correction, correct forward/backward absorption weights"
    },
    {
      "id": "PT:P3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Conjugating by",
      "dependencies": [
        "PT:P2",
        "CE:G1",
        "FP0:M4",
        "P2:OP3",
        "P2:OP6"
      ],
      "scope": "All-real Sobolev conjugation, principal constant and order-dependent threshold"
    },
    {
      "id": "PT:E1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "## 3.",
      "dependencies": [
        "PT:P2",
        "PT:P3",
        "CE:H7",
        "CE:H10",
        "P2:OP3",
        "WY:W4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Forward energy identities with original interval, inner-product convention, positive terminal terms and exact commutator sign"
    },
    {
      "id": "PT:E2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The order-one term includes",
      "dependencies": [
        "PT:E1",
        "G2:G0",
        "G2:NF1",
        "G2:NF5",
        "G2:NF6",
        "G2:NF7",
        "FC:F2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full Hamilton-time Taylor remainder and uniform second-flow derivative bound with lower symbol orders"
    },
    {
      "id": "PT:E3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Combining this with",
      "dependencies": [
        "PT:E1",
        "PT:E2",
        "PT:P1",
        "PT:P2",
        "CE:H2"
      ],
      "scope": "Normalized sharp inequality, parameter-uniform seminorms, Young forcing and error split"
    },
    {
      "id": "PT:E4",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For the remaining singular error,",
      "dependencies": [
        "PT:E3",
        "CE:H2",
        "U001:P2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete singular Hardy multiplier, exact polynomial majorants, terminal sign and absorption of C/lambda error"
    },
    {
      "id": "PT:E5",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "This is the stronger estimate",
      "dependencies": [
        "PT:E4",
        "PT:P2",
        "PT:P3",
        "CE:G1",
        "P3:M3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All-real stronger forward estimate, initial zero endpoint limit with actual traces and coefficient-order absorption"
    },
    {
      "id": "PT:E6",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "We shall also need the terminal terms",
      "dependencies": [
        "PT:E1",
        "PT:E4",
        "PT:E5",
        "PT:P2",
        "PT:P3"
      ],
      "scope": "Full terminal kinetic and Hardy bounds at order -1/2, corrected conjugate positivity and original forcing interval"
    },
    {
      "id": "PT:B1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "## 4.",
      "dependencies": [
        "PT:E1",
        "PT:E2",
        "PT:P2",
        "G2:NF7",
        "CE:H2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Backward multiplier and initial boundary sign, distinct flow Taylor comparison at2t and correct principal t2 weight"
    },
    {
      "id": "PT:B2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For precision in the positivity step,",
      "dependencies": [
        "PT:B1",
        "PT:P1",
        "MP:F8",
        "CE:H2"
      ],
      "scope": "Explicit normalized scalar Fefferman-Phong receiver, finite bounded seminorm family and exact C(1+lambda) zero-order error"
    },
    {
      "id": "PT:B3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The terminal zero trace gives",
      "dependencies": [
        "PT:B1",
        "PT:B2",
        "PT:P2",
        "PT:P3",
        "CE:H2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Terminal Hardy control and full backward estimate with t2 on top energies and forcing, uniform initial limit"
    },
    {
      "id": "PT:J1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "**The quotient inverse, including a double root.**",
      "dependencies": [
        "PT:R2",
        "QF:A1",
        "U001:P2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete simple/distinct/double modular inverse proof, full quotient basis, smooth adjugate inverse without internal root-gap division"
    },
    {
      "id": "PT:J2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Put \\(L=\\langle D'\\rangle\\)",
      "dependencies": [
        "PT:J1",
        "PT:R3",
        "HC:D1",
        "HC:D2",
        "FP0:M1",
        "P2:OP3",
        "P2:OP5",
        "P2:OP6",
        "HC:E1"
      ],
      "scope": "All normal jet rows and column normalization, finite cone/time cover, ordered total/normal error bounds, parameter matrix inverse and low-frequency remainder"
    },
    {
      "id": "PT:J3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For a quadratic factor the forward estimate",
      "dependencies": [
        "PT:J2",
        "PT:E5",
        "FC:E1",
        "FC:E3",
        "CE:H2",
        "HC:M1"
      ],
      "scope": "Complete forward reconstruction, highest-normal monic recovery, lower-jet Hardy absorption and fixed-interval uniformity"
    },
    {
      "id": "PT:J4",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "## 6.",
      "dependencies": [
        "PT:J2",
        "PT:B3",
        "FC:C2",
        "CE:H2",
        "HC:M1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full weighted backward reconstruction, empty m1 lower sum, unweighted m-2jet Hardy and actual source-reference correction"
    },
    {
      "id": "PT:C1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Here are the local steps and their orders.",
      "dependencies": [
        "PT:R1",
        "PT:R2",
        "PT:R3",
        "PT:J3",
        "PT:H0",
        "PT:C5",
        "HC:L1",
        "HC:L2",
        "HC:L3",
        "HC:L7",
        "HC:L8",
        "HC:M6",
        "NE:U2",
        "P3:M7"
      ],
      "scope": "Supported positive-start construction, fixed cones and uniform weak/density limit with zero delta terms"
    },
    {
      "id": "PT:C2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For forcing in",
      "dependencies": [
        "PT:C1",
        "HC:M1",
        "HC:M6",
        "MH:E1",
        "HC:L3"
      ],
      "scope": "All nonnegative real normal orders and full differential normal recovery retaining total order"
    },
    {
      "id": "PT:C3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Negative integer normal orders follow",
      "dependencies": [
        "PT:C2",
        "MH:N3",
        "HC:L6",
        "HC:M1",
        "HC:M6",
        "P2:OP3"
      ],
      "scope": "Finite supported downward induction with exact commutator normal degree and all solution orders"
    },
    {
      "id": "PT:C4",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For arbitrary isotropic",
      "dependencies": [
        "PT:C3",
        "HC:L6",
        "HC:M6",
        "MH:E1"
      ],
      "scope": "Actual ambient isotropic regularity across initial surface, integer normal seed and whole-space recovery with preserved support"
    },
    {
      "id": "PT:C5",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "To prove uniqueness,",
      "dependencies": [
        "PT:J4",
        "HC:L7",
        "HC:L8",
        "HC:M6",
        "HN0:C4",
        "NE:U2",
        "MH:TR3"
      ],
      "scope": "Backward adjoint cone, unweighted lower seed, all normal recovery, smooth extension and flat boundary pairing for all supported distributions"
    },
    {
      "id": "PT:C6",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For nonzero Cauchy data",
      "dependencies": [
        "PT:C2",
        "HC:M1",
        "MH:TR1",
        "MH:TR2",
        "MH:TR3",
        "P3:L1",
        "P3:M4",
        "U001:U001-A4",
        "U001:Q3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact scaled Fourier trace lift with every normal jet and half-order norm, actual strict trace thresholds"
    },
    {
      "id": "PT:C7",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The ordinary differential image",
      "dependencies": [
        "PT:C6",
        "PT:C1",
        "PT:C2",
        "HC:M6",
        "MH:E1",
        "MH:TR3",
        "NE:U2"
      ],
      "scope": "Correct zero extension in L2_tH^s, restricted recovery and all zero correction traces by ordered boundary delta coefficients"
    },
    {
      "id": "PT:T1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Let \\(C(t)\\)",
      "dependencies": [
        "HC:M1",
        "HC:M2",
        "MH:N2",
        "MH:T1",
        "MH:T2",
        "BW:T1",
        "BW:T5",
        "BW:T6",
        "CE:G1"
      ],
      "scope": "Complete all-integer tangential family mixed maps, time derivative coefficients, dual/restricted/support receivers"
    },
    {
      "id": "PT:T2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Expand the model",
      "dependencies": [
        "PT:T1",
        "HC:M3",
        "MH:N1",
        "BW:T10",
        "P2:OP3",
        "HC:B3"
      ],
      "scope": "Explicit normal-order sign, commutator orders and two normal-step applications to model beyond differential recovery"
    },
    {
      "id": "PT:T3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "We next solve the model",
      "dependencies": [
        "PT:B3",
        "CE:H1",
        "CT:T1",
        "CE:H5",
        "MH:A1",
        "MH:A2",
        "BW:T6"
      ],
      "scope": "Adjoint Hahn-Banach existence on positive-half norm with tests acrosszero and actual ambient equation"
    },
    {
      "id": "PT:T4",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The regularity needed to read its traces",
      "dependencies": [
        "PT:T3",
        "PT:T1",
        "PT:T2",
        "MH:N1",
        "MH:TR3",
        "NE:U2",
        "BW:T5"
      ],
      "scope": "Full H(2,s-1) model recovery and both actual traces, ambient delta derivative coefficients force zero jets"
    },
    {
      "id": "PT:T5",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For general \\(f\\in H_{(0,s)}\\)",
      "dependencies": [
        "PT:T3",
        "PT:T4",
        "PT:T6",
        "PT:E5",
        "P3:M7",
        "CE:H6"
      ],
      "scope": "Rough source approximation, difference estimates and mixed graph/trace convergence"
    },
    {
      "id": "PT:T6",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "For completeness, the estimate extends",
      "dependencies": [
        "PT:T4",
        "PT:E5",
        "P2:OP3",
        "HC:B8",
        "MH:T1",
        "MH:T2",
        "MH:N2",
        "P3:M7",
        "CE:H6"
      ],
      "scope": "Complete graph-domain density with uniform commutator convergence, zero-jet time convolution and diagonal limit"
    },
    {
      "id": "PT:L1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "## 9.",
      "dependencies": [
        "PT:E1",
        "PT:E5",
        "PT:E6",
        "P2:OP3",
        "WY:W4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Decreasing-cutoff signs, selfadjoint Q, both time-dependent commutator identities and endpoint limits"
    },
    {
      "id": "PT:L2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Here is the nonnegative-function estimate",
      "dependencies": [
        "FP0:S1",
        "PT:R3",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full global and local nonnegative-gradient proof, scaled S2 receiver, uniform Hessian and vanishing gradient at zeros"
    },
    {
      "id": "PT:L3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Thus \\(|b_{\\xi'}|",
      "dependencies": [
        "PT:L1",
        "PT:L2",
        "PT:P1",
        "FP0:M4",
        "CE:H2"
      ],
      "scope": "Derivative-cone positivity, exact Hermitian matrix signs, zero-diagonal case and all order-minus-one errors"
    },
    {
      "id": "PT:L4",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Write \\(G=\\int_0^T",
      "dependencies": [
        "PT:L3",
        "PT:E4",
        "PT:E5",
        "PT:E6",
        "PT:P1",
        "PT:P2",
        "CE:H2"
      ],
      "scope": "Complete additional w/t2 singular spatial multiplier, retained terminal terms, actual original interval, conjugation and interpolation"
    },
    {
      "id": "PT:L5",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Reordering the last term",
      "dependencies": [
        "PT:L1",
        "PT:L4",
        "P2:OP3",
        "P2:OP6",
        "CE:G1",
        "CE:H2"
      ],
      "scope": "Exact double-commutator identity and order zero bound, full lambda-half interpolation and scalar polynomial"
    },
    {
      "id": "PT:L6",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The scalar commutator on",
      "dependencies": [
        "PT:L1",
        "PT:L3",
        "PT:L4",
        "PT:L5",
        "PT:E3",
        "PT:E4",
        "PT:E5"
      ],
      "scope": "Final localized scalar Hardy absorption, forcing pair and shifted derivative commutator with exact lower orders"
    },
    {
      "id": "PT:L7",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Conjugation by \\(L^s\\)",
      "dependencies": [
        "PT:L6",
        "PT:P3",
        "P2:OP3",
        "CE:G1",
        "PT:E5"
      ],
      "scope": "All-real localized estimate, detailed lower-term/conjugation absorption and finite-seminorm family scope"
    },
    {
      "id": "PT:L8",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The zero-jet solution of Section 8 gains",
      "dependencies": [
        "PT:L7",
        "PT:T5",
        "PT:T6",
        "PT:H0",
        "P2:OP3",
        "CE:G1",
        "P3:M7"
      ],
      "scope": "Exact forcing mollification, uniform minus-one Qcommutator, weak half-step limit and actual normal derivative of Qu"
    },
    {
      "id": "PT:W1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "First consider the quadratic model.",
      "dependencies": [
        "BW:N1",
        "BW:N2",
        "BW:N3",
        "BW:W2",
        "BW:W3",
        "BW:W11",
        "BW:T8",
        "BW:T10",
        "BW:T11",
        "BW:S5",
        "HC:B3"
      ],
      "scope": "Full intrinsic extension, common input collar excludes pure normal caps, elliptic pure-normal removal and genuine smooth residual scope"
    },
    {
      "id": "PT:W2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "A compact distribution has a finite",
      "dependencies": [
        "PT:W1",
        "PT:T2",
        "PT:T4",
        "PT:L8",
        "HC:B3",
        "BW:T10",
        "MH:TR3",
        "U001:U001-A4"
      ],
      "scope": "Finite negative mixed seed, repeated normal recovery, actual smooth jet subtraction and localized forcing"
    },
    {
      "id": "PT:W3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "We can iterate on one fixed final collar.",
      "dependencies": [
        "PT:W2",
        "PT:L1",
        "U001:U001-A4",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full nested monotone profile construction, low frequency behavior, exact derivative condition and one fixed positive-width collar"
    },
    {
      "id": "PT:W4",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Successive half steps give",
      "dependencies": [
        "PT:W2",
        "PT:W3",
        "PT:L8",
        "PT:T2",
        "BW:T11",
        "BW:S5",
        "HC:M6",
        "P3:L1"
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      "scope": "All tangential orders on common final collar, every finite normal recovery and genuine smooth elliptic tester"
    },
    {
      "id": "PT:W5",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "The reverse inclusion uses",
      "dependencies": [
        "BW:W12",
        "BW:S3",
        "BW:S5",
        "NE:J1",
        "NE:J2",
        "BW:N1",
        "HC:B5"
      ],
      "scope": "Full forcing/jet reverse inclusion on same canonical extension, intrinsic delta correction and transverse-field triangular change"
    },
    {
      "id": "PT:F1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "## 11.",
      "dependencies": [
        "PT:J1",
        "PT:J2",
        "HC:F1",
        "FC:Q4",
        "P2:OP3",
        "P2:OP6",
        "BW:T10"
      ],
      "scope": "Complete ordered symbol factorization to all orders, parameter asymptotic sum and finite-normal-degree tangential-smoothing remainder"
    },
    {
      "id": "PT:F2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "A refined linear factor",
      "dependencies": [
        "PT:F1",
        "PT:P2",
        "PT:T2"
      ],
      "scope": "Refined simple/quadratic model normal ordering, full imaginary lower classes and unchanged principal positivity"
    },
    {
      "id": "PT:F3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "Set \\(w_1=",
      "dependencies": [
        "PT:F1",
        "PT:F2",
        "PT:W1",
        "PT:W2",
        "PT:W3",
        "PT:W4",
        "PT:W5",
        "HC:B6",
        "HC:B7",
        "BW:T12",
        "NE:J2"
      ],
      "scope": "All finite factor removals with actual intrinsic jet orders, smooth residuals and finite final collar shrink"
    },
    {
      "id": "PT:G1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
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      "proof_locator": "### Exact geometry of the model",
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        "PT:R3",
        "PT:W3",
        "G2:G0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact Hamilton trajectory, smooth merger versus singular root branches and exact finite-frequency nested collar geometry"
    },
    {
      "id": "PT:X1",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 1 —",
      "dependencies": [
        "PT:R2",
        "PT:R3"
      ],
      "scope": "Complete unchanged original solution 1"
    },
    {
      "id": "PT:X2",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 2 —",
      "dependencies": [
        "PT:R2"
      ],
      "scope": "Complete unchanged original solution 2"
    },
    {
      "id": "PT:X3",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 3 —",
      "dependencies": [
        "PT:R1"
      ],
      "scope": "Complete unchanged original solution 3"
    },
    {
      "id": "PT:X4",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 4 —",
      "dependencies": [
        "PT:E4"
      ],
      "scope": "Complete unchanged original solution 4"
    },
    {
      "id": "PT:X5",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 5 —",
      "dependencies": [
        "PT:J4"
      ],
      "scope": "Complete unchanged original solution 5"
    },
    {
      "id": "PT:X6",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 6 —",
      "dependencies": [
        "PT:J1"
      ],
      "scope": "Complete unchanged original solution 6"
    },
    {
      "id": "PT:X7",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 7 —",
      "dependencies": [
        "PT:J1"
      ],
      "scope": "Complete unchanged original solution 7"
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    {
      "id": "PT:X8",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 8 —",
      "dependencies": [
        "PT:J3",
        "PT:J4"
      ],
      "scope": "Complete unchanged original solution 8"
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      "id": "PT:X9",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 9 —",
      "dependencies": [
        "PT:P1",
        "PT:B2",
        "PT:B3"
      ],
      "scope": "Complete unchanged original solution 9"
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    {
      "id": "PT:X10",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 10 —",
      "dependencies": [
        "PT:L4",
        "PT:E6"
      ],
      "scope": "Complete unchanged original solution 10"
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      "id": "PT:X11",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 11 —",
      "dependencies": [
        "PT:L3"
      ],
      "scope": "Complete unchanged original solution 11"
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    {
      "id": "PT:X12",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 12 —",
      "dependencies": [
        "PT:W3"
      ],
      "scope": "Complete unchanged original solution 12"
    },
    {
      "id": "PT:X13",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 13 —",
      "dependencies": [
        "PT:T2"
      ],
      "scope": "Complete unchanged original solution 13"
    },
    {
      "id": "PT:X14",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 14 —",
      "dependencies": [
        "PT:T4"
      ],
      "scope": "Complete unchanged original solution 14"
    },
    {
      "id": "PT:X15",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 15 —",
      "dependencies": [
        "PT:L8"
      ],
      "scope": "Complete unchanged original solution 15"
    },
    {
      "id": "PT:X16",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 16 —",
      "dependencies": [
        "PT:W3",
        "PT:W4"
      ],
      "scope": "Complete unchanged original solution 16"
    },
    {
      "id": "PT:X17",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 17 —",
      "dependencies": [
        "PT:F1",
        "PT:F3"
      ],
      "scope": "Complete unchanged original solution 17"
    },
    {
      "id": "PT:X18",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 18 —",
      "dependencies": [
        "PT:C6",
        "PT:C7"
      ],
      "scope": "Complete unchanged original solution 18"
    },
    {
      "id": "PT:X19",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 19 —",
      "dependencies": [
        "PT:C3"
      ],
      "scope": "Complete unchanged original solution 19"
    },
    {
      "id": "PT:X20",
      "source": "principal-type-hyperbolic-boundaries-and-cauchy-problems.md",
      "source_sha256": "fd2525b07483e70fc56bb982add7cffa61de33811f602056df9a39f34b35ac5c",
      "proof_locator": "### **Exercise 20 —",
      "dependencies": [
        "PT:C5",
        "P3:L3"
      ],
      "scope": "Complete unchanged original solution 20"
    }
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