{
  "schema": "AN04-restored-positive-symmetrizer-proof-map/v1",
  "proofs": [
    {
      "id": "SM:R001",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Fix a compact time interval",
      "dependencies": [
        "SY:H1",
        "SY:H2",
        "SY:H7",
        "U001:F0-COMP"
      ],
      "scope": "Full ordinary symbol and all-time parameter hypotheses. SM1 keeps the complete complex nonclassical S0 matrix, global bounded seminorms and each time derivative."
    },
    {
      "id": "SM:R002",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Assume a homogeneous degree-zero Hermitian matrix",
      "dependencies": [
        "SM:R001",
        "SY:H16",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Uniform positive principal metric. SM2 includes a globally bounded condition ratio, all derivatives and SH=H* S; a merely local metric is not a global theorem."
    },
    {
      "id": "SM:R003",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "We shall prove, for every real",
      "dependencies": [
        "SM:R001",
        "SM:R002",
        "SM:R023",
        "SM:R027"
      ],
      "scope": "Both time directions and integrable forcing. SM3--SM4 declare every real Sobolev order, initial time, integrated H(s-1) equation and s-dependent constants."
    },
    {
      "id": "SM:R004",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "No gap is required for this energy statement.",
      "dependencies": [
        "SM:R003",
        "SM:R037",
        "BR:R046"
      ],
      "scope": "Gap-free energy versus separated kernel scope. Energy requires the stated smooth positive metric; the kernel additionally requires separated fixed single-eigenvalue multiplicities on whole trajectory neighborhoods."
    },
    {
      "id": "SM:R005",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "First extend \\(S\\) smoothly through low frequency",
      "dependencies": [
        "SM:R002",
        "U001:P14.3",
        "SY:H7"
      ],
      "scope": "Low-frequency convex metric extension. SM5 preserves positivity and global S0 bounds; its low-frequency symmetrization defect is order zero."
    },
    {
      "id": "SM:R006",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Here are the scalar identities behind this construction.",
      "dependencies": [
        "SM:R005",
        "BR:R003",
        "U001:P13.2-series",
        "U001:P13.2-mertens",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Scalar binomial coefficients and matrix root. The two differential recursions identify absolutely convergent square-root and inverse-root series; finite Hermitian diagonalization verifies the identities."
    },
    {
      "id": "SM:R007",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "After any fixed total of \\(d\\)",
      "dependencies": [
        "SM:R006",
        "BR:S1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every differentiated root seminorm. SM7 controls ordered differentiated products by k^d q^(k-d); distributed frequency derivatives lose exactly their total order."
    },
    {
      "id": "SM:R008",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The last identity follows from",
      "dependencies": [
        "SM:R007",
        "SM:R002",
        "U001:P2"
      ],
      "scope": "Homogeneous principal metric distinguished from its extension. SM8 uses Q_h for the homogeneous Hermitian similarity. Extended Q need not symmetrize at low frequency."
    },
    {
      "id": "SM:R009",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "At one finite-dimensional point, a positive symmetrizer",
      "dependencies": [
        "SM:R008",
        "BR:R003",
        "U001:P2"
      ],
      "scope": "Pointwise metric and real diagonalizability equivalence. SM9 constructs V^(-*)V^(-1); the converse uses Hermitian similarity. Uniform smooth family bounds are not inferred pointwise."
    },
    {
      "id": "SM:R010",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Write \\(\\Lambda=\\langle D\\rangle\\)",
      "dependencies": [
        "SM:R001",
        "SY:H2",
        "SY:H7",
        "SY:H8"
      ],
      "scope": "Sobolev conjugation changes only order zero. The scalar weight commutes with the pointwise principal matrix and every differentiated composition loses one order under the exact calculus."
    },
    {
      "id": "SM:R011",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The leading symbol of \\(Q_0^*Q_0\\)",
      "dependencies": [
        "SM:R010",
        "SM:R007",
        "SM:R002",
        "SY:H7",
        "SY:H8"
      ],
      "scope": "Quantized principal energy cancellation. SM10 uses the full adjoint and ordered composition; its principal form cancels and all low-frequency/lower terms are S0."
    },
    {
      "id": "SM:R012",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The pointwise inverse \\(Q^{-1}\\)",
      "dependencies": [
        "SM:R007",
        "SY:H7",
        "SY:H8",
        "SY:H10"
      ],
      "scope": "Coercivity from an approximate inverse. SM11--SM12 retain the order-minus-one defect and use its H(-1)->L2 mapping, without assuming actual quantized invertibility."
    },
    {
      "id": "SM:R013",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Choose a fixed \\(\\gamma\\ge1\\).",
      "dependencies": [
        "SM:R012",
        "SY:H10",
        "U001:P2"
      ],
      "scope": "Equivalent corrected Sobolev energy. SM13 includes the lower-order norm and explicit lower/upper comparisons, uniformly in time."
    },
    {
      "id": "SM:R014",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For a spatially smooth solution, differentiation gives",
      "dependencies": [
        "SM:R011",
        "SM:R013",
        "SY:H10",
        "SM:AC1"
      ],
      "scope": "Complete energy derivative. SM14 retains time derivatives of Q and both forcing terms; Lambda^(-1) A_s is order zero."
    },
    {
      "id": "SM:R015",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Apply this to \\(\\sqrt{{\\mathcal E}_s+\\delta}\\)",
      "dependencies": [
        "SM:R014",
        "CE:H7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P15.1"
      ],
      "scope": "Scalar inequality, zero energy and time reversal. SM15 is integrated on sqrt(E+delta) before delta tends to zero; reversing A retains principal cancellation and bounded errors."
    },
    {
      "id": "SM:R016",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Choose a real compactly supported smooth Fourier multiplier",
      "dependencies": [
        "U001:P14.3",
        "SY:H1",
        "SY:H2",
        "SY:H7"
      ],
      "scope": "Uniform scalar Fourier regularizers. SM16 keeps epsilon-dependent derivative supports at comparable frequencies, giving uniform S0 cutoff bounds."
    },
    {
      "id": "SM:R017",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For fixed \\(\\epsilon\\), this is bounded",
      "dependencies": [
        "SM:R016",
        "SY:H10",
        "SY:H1",
        "CE:H4",
        "U001:P3"
      ],
      "scope": "Bounded regularized ODE in every Sobolev space. Successive substitutions contract on each short interval in the complete continuous-function space; finitely many intervals cover both time directions."
    },
    {
      "id": "SM:R018",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The symbol family of \\(A_\\epsilon\\)",
      "dependencies": [
        "SM:R016",
        "SM:R011",
        "SM:R013",
        "SM:R014",
        "SM:R015",
        "SY:H7",
        "SY:H8"
      ],
      "scope": "Uniform energy for regularized operators. The exact symbol of J A J is i j^2 chi H modulo uniformly bounded S0; the scalar j^2 preserves the metric cancellation."
    },
    {
      "id": "SM:R019",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The scalar multiplier inequality",
      "dependencies": [
        "SM:R016",
        "SY:H2",
        "SY:H10"
      ],
      "scope": "Order-losing convergence estimate. SM17--SM18 give norm H(s+1)->H(s-1) of size epsilon, retaining the two distinct cutoff terms."
    },
    {
      "id": "SM:R020",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "shows convergence in \\(C(I;H^{s-1})\\).",
      "dependencies": [
        "SM:R017",
        "SM:R018",
        "SM:R019",
        "SY:H1",
        "SY:H2",
        "CE:H4"
      ],
      "scope": "Strong convergence of smooth-data solutions. SM19 gives C H(s-1) convergence; the stronger uniform bound and Fourier Cauchy--Schwarz interpolate to C Hs."
    },
    {
      "id": "SM:R021",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For general \\(g\\in H^s\\)",
      "dependencies": [
        "SM:R020",
        "SM:R015",
        "SY:H1",
        "SY:H2",
        "SY:H3",
        "CE:H4",
        "CE:H7"
      ],
      "scope": "Data and Bochner forcing approximation. Finite time-step values, spatial density and scalar indicator smoothing give L1 Hs approximation; the integrated equation passes in H(s-1)."
    },
    {
      "id": "SM:R022",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The scalar multiplier commutator is",
      "dependencies": [
        "SM:R016",
        "SY:H7",
        "SY:H8",
        "SY:H10",
        "SY:H15",
        "U001:F0-COMP"
      ],
      "scope": "Commutator convergence uniformly in time. SM20 has a uniformly S0 commutator. Smooth-vector uniform convergence and a finite net in the compact trajectory of u control L1 Hs residuals."
    },
    {
      "id": "SM:R023",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The smoothed solution has every spatial Sobolev order",
      "dependencies": [
        "SM:R021",
        "SM:R022",
        "SM:R015",
        "SM:AC1",
        "SY:H2",
        "CE:H4"
      ],
      "scope": "Energy and uniqueness for every rough solution. The energy calculation is applied to J_epsilon u with its integrable derivative, then the residual tends to zero; no nonexistent Hs time derivative is used."
    },
    {
      "id": "SM:R024",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Define \\(U_P(t,r)g\\)",
      "dependencies": [
        "SM:R021",
        "SM:R023"
      ],
      "scope": "Evolution composition and invertibility. SM21 follows from uniqueness for arbitrary initial times; the inverse is the opposite-time solution."
    },
    {
      "id": "SM:R025",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The estimate bounds each operator on",
      "dependencies": [
        "SM:R024",
        "SM:R015",
        "SY:H3",
        "SY:H10",
        "CE:H4",
        "U001:F0-COMP"
      ],
      "scope": "Joint strong endpoint continuity. Smooth vectors give the short-time bound, density gives strong continuity, and the group law handles both varying endpoints and the diagonal."
    },
    {
      "id": "SM:R026",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For smooth vectors, the equation and the group law",
      "dependencies": [
        "SM:R024",
        "SM:R025",
        "SM:N1",
        "SY:H10"
      ],
      "scope": "Both ordered endpoint derivatives. SM22 derives the initial-time sign from the group law; every repeated derivative has only a finite Sobolev order loss."
    },
    {
      "id": "SM:R027",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The forcing solution, with the integral oriented from",
      "dependencies": [
        "SM:R023",
        "SM:R025",
        "SM:R026",
        "CE:H4",
        "CE:H7"
      ],
      "scope": "Oriented forcing integral. SM23 is Bochner under the strong continuity and bounds; smooth forcing then L1 approximation proves its integrated equation in both directions."
    },
    {
      "id": "SM:R028",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Let \\(T_0=Q^{-1}\\)",
      "dependencies": [
        "SM:R007",
        "K:K1",
        "K:K2",
        "K:K3",
        "SY:H7",
        "SY:H8"
      ],
      "scope": "Full parameter-aware two-sided inverse. SM24 receives the complete ordinary matrix parametrix, all time derivatives and Borel tails; no exact operator inverse is claimed."
    },
    {
      "id": "SM:R029",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Indeed \\(E=(I_N-TR)(T_t+AT)\\).",
      "dependencies": [
        "SM:R028",
        "SM:R008",
        "SY:H7",
        "SY:H8",
        "SY:H10",
        "SY:R1",
        "SY:U1"
      ],
      "scope": "Spatial intertwining with a Hermitian principal part. SM25 defines B=R(T_t+AT), computes E=(I-TR)(T_t+AT) and keeps every spatial smoothing/time parameter bound."
    },
    {
      "id": "SM:R030",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "It is obtained from \\(r\\#T_0=I_N\\)",
      "dependencies": [
        "SM:R028",
        "SY:H7",
        "U001:P2"
      ],
      "scope": "Ordered inverse first correction. SM26 is computed from r#T0=I with the correction multiplied on the right by q."
    },
    {
      "id": "SM:R031",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "To check the last term, the two inverse/composition contributions",
      "dependencies": [
        "SM:R030",
        "SM:R029",
        "SY:H7",
        "U001:P2"
      ],
      "scope": "Complete order-zero transformed coefficient. SM27 retains C, the time metric, the H_xi/T_x term and q_xi/T0/B1_x term; inverse/composition cancellation is explicitly derived."
    },
    {
      "id": "SM:R032",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Using the full symbol inverse in (SM25)",
      "dependencies": [
        "SM:R031",
        "SM:R029",
        "BR:R019"
      ],
      "scope": "Ordinary remainder and time-coefficient qualification. The congruence is modulo S-1 for a full nonclassical C; a smoothing coefficient of partial_t is not identified with a spatial order-zero error."
    },
    {
      "id": "SM:R033",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "where \\(K=TR-I_N\\) is smoothing.",
      "dependencies": [
        "SM:R028",
        "SM:R029",
        "SM:R026",
        "SY:U1",
        "EC:H3",
        "EC:H4"
      ],
      "scope": "Separate equation and initial defects. SM28 retains PV=E U_D R and V(r,r)=I+K."
    },
    {
      "id": "SM:R034",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Both initial and equation defects are retained.",
      "dependencies": [
        "SM:R033",
        "SM:R027",
        "SM:N1",
        "EC:C1",
        "EC:C2",
        "EC:C3",
        "EC:C4",
        "EC:C5",
        "CE:H4"
      ],
      "scope": "Actual-system Duhamel correction. SM29 corrects both defects; arbitrary smoothing gain absorbs every finite endpoint differentiation loss in global Sobolev spaces."
    },
    {
      "id": "SM:R035",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For a compactly supported input localization,",
      "dependencies": [
        "SM:R034",
        "EC:C6",
        "EC:C7"
      ],
      "scope": "Joint smooth compact-input kernel. The exact preceding delta-column reconstruction applies to all differentiated H(-M)->H(L) bounds after compact input localization."
    },
    {
      "id": "SM:R036",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The compact input operator \\(R(r)\\Psi\\)",
      "dependencies": [
        "SM:R028",
        "BR:R029",
        "BR:R030",
        "K:K3",
        "K:K4",
        "SY:H10"
      ],
      "scope": "Transformed input localization with full exterior tails. Xi is one on the whole conic microsupport of R Psi; separated-cone composition and compact-input exterior PDO estimates make (I-Xi)R Psi globally smoothing."
    },
    {
      "id": "SM:R037",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Thus the input in the Hermitian theorem actually contains",
      "dependencies": [
        "SM:R036",
        "SM:R035",
        "BR:R023",
        "BR:R027",
        "BR:R040",
        "BR:R041",
        "BR:R042",
        "RC:C9",
        "EC:G3"
      ],
      "scope": "Separated non-Hermitian branch kernels. SM30 composes the complete Hermitian branch theorem and SM29 with elliptic order-zero endpoint operators, retaining zero excess, order and all support/parameter contracts."
    },
    {
      "id": "SM:R038",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The principal \\(\\rho_j\\) are generally",
      "dependencies": [
        "SM:R028",
        "BR:R035",
        "BR:R037",
        "SM:R008",
        "U001:P2"
      ],
      "scope": "Nonorthogonal spectral metrics and full projectors. SM31 distinguishes principal rho from the full Pi; rho* S=S rho follows with homogeneous Q_h."
    },
    {
      "id": "SM:R039",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The full idempotence, complementarity and sum identities",
      "dependencies": [
        "SM:R038",
        "SM:R029",
        "BR:R035",
        "BR:R036",
        "SY:H7"
      ],
      "scope": "Full projector identities and correct time sign. SM32 uses differentiated RT and the actual intertwining; the expanded commutator retains R_t+BR-RA."
    },
    {
      "id": "SM:R040",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The leading branch map is the Hermitian endpoint",
      "dependencies": [
        "SM:R037",
        "SM:R038",
        "BR:R043",
        "BR:I1",
        "GS:Z12"
      ],
      "scope": "Elliptic maps of original eigenbundles. The endpoint map is conjugated by Q_h^(-1)(t) and Q_h(r), with rank d_j and no global eigenbasis."
    },
    {
      "id": "SM:R041",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Multiplying by invertible endpoint matrices preserves",
      "dependencies": [
        "SM:R040",
        "SM:R039",
        "BR:R044",
        "EC:G7",
        "EC:G8",
        "CH:W1"
      ],
      "scope": "Exact full-matrix branch wavefront union. Full projections isolate an elliptic branch even at graph intersections; no every-entry wavefront claim is made."
    },
    {
      "id": "SM:R042",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For compactly supported data whose normalized initial wavefront",
      "dependencies": [
        "SM:R041",
        "SM:R039",
        "BR:R045",
        "BR:I1",
        "EC:G9",
        "K:K3"
      ],
      "scope": "Precisely localized projected data equivalence. SM33 uses the full projections and normalized initial WF inside K; elliptic endpoint operators transport the complete prior two-direction equivalence."
    },
    {
      "id": "SM:R043",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "This bound is necessary for a uniform positive symmetrizer:",
      "dependencies": [
        "SM:R038",
        "SM:R008",
        "BR:R003",
        "U001:P2"
      ],
      "scope": "Necessary uniform projector bound. SM34 derives the norm bound from the uniform positive condition ratio, independently of an eigenframe choice."
    },
    {
      "id": "SM:R044",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Under the stated gap and bound, the projectors actually",
      "dependencies": [
        "SM:R001",
        "BR:R004",
        "BR:R005",
        "U001:P13.2-series",
        "U001:P2"
      ],
      "scope": "Stable nonnormal resolvent contours. SM35 and the uniform projector bound give resolvent control; the Neumann test bounds spectral motion on fixed small circles."
    },
    {
      "id": "SM:R045",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Its rank is stable: pointwise diagonalization",
      "dependencies": [
        "SM:R044",
        "BR:R005",
        "BR:R006",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All differentiated nonorthogonal projectors. SM36 retains trace/rank constancy, one eigenvalue per circle, trace eigenvalue formula and repeated ordered resolvent derivatives; homogeneity restores all symbol orders."
    },
    {
      "id": "SM:R046",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "There is consequently a global basis-free choice",
      "dependencies": [
        "SM:R045",
        "SM:R044",
        "U001:P2"
      ],
      "scope": "Global metric without a global frame. SM37 sums rho* rho; Cauchy--Schwarz gives 1/k positivity, k L^2 gives the upper bound and real eigenvalues give SH=H* S."
    },
    {
      "id": "SM:R047",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "Real semisimple eigenvalues at each point alone",
      "dependencies": [
        "SM:R043",
        "SM:R046",
        "SM:R003",
        "SM:R004",
        "SM:R051",
        "SM:R052"
      ],
      "scope": "Pointwise, uniform and crossing distinctions. Semisimple real roots alone imply no uniform constants; a permitted smooth metric through a crossing supplies energy but not the separated FIO theorem there."
    },
    {
      "id": "SM:R048",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "### 8.1.",
      "dependencies": [
        "SM:R009",
        "SM:R006",
        "BR:R003",
        "U001:P2",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Sheared square root and nonorthogonal eigenvectors. Exercise1 computes the exact positive matrix, its root via its quadratic identity, both projectors, metric orthonormality and area-preserving ellipse."
    },
    {
      "id": "SM:R049",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For the displayed instance \\(b=1\\)",
      "dependencies": [
        "SM:R048",
        "U001:P16.1",
        "U001:P2"
      ],
      "scope": "Exact component-energy ellipse. The b=1 curve is the stated exact shear parameterization, sampled361 times; all panels and labels were actually inspected. It is not a physical characteristic plane."
    },
    {
      "id": "SM:R050",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "### 8.2.",
      "dependencies": [
        "SM:R048",
        "SM:R024",
        "SM:R038",
        "SM:R040",
        "EC:G8",
        "SY:H2",
        "P3:L2"
      ],
      "scope": "Changing frame exact Cauchy kernel. Exercise2 cancels T_t T^(-1), solves both translations, proves SM39, initial identity, rank-one maps, metric conservation and signed cone ordering."
    },
    {
      "id": "SM:R051",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "### 8.3.",
      "dependencies": [
        "SM:R043",
        "SM:R009",
        "SM:R024",
        "SY:H2",
        "P3:L2",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P15.1"
      ],
      "scope": "Uniform-condition failure at a closing gap. Exercise3 computes projector growth, the exact sine evolution, an essential-sup norm lower bound and the sharp one-derivative nilpotent limit."
    },
    {
      "id": "SM:R052",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "### 8.4.",
      "dependencies": [
        "SM:R009",
        "SM:R024",
        "SY:H2",
        "P3:L2",
        "U001:P15.1",
        "U001:P2"
      ],
      "scope": "Jordan roots and sharp derivative loss. Exercise4 uses normalized high-frequency packets to prove failure of Hs boundedness, H(s+1)->Hs sufficiency and the positive-metric contradiction."
    },
    {
      "id": "SM:R053",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "### 8.5.",
      "dependencies": [
        "SM:R048",
        "SM:R008",
        "SM:R024",
        "SM:R037",
        "SM:R041",
        "EC:G8",
        "SY:H2",
        "SY:H10",
        "U001:P2"
      ],
      "scope": "Frequency metric with no constant replacement. Exercise5 constructs a global bounded directional metric, excludes any fixed positive matrix, solves high-frequency evolution and identifies both graph phases with the low-frequency smoothing qualification."
    },
    {
      "id": "SM:AC1",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "The differentiation also holds almost everywhere",
      "dependencies": [
        "SY:H1",
        "SY:H4",
        "CE:H4",
        "CE:H7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Norm-square rule for integrably differentiated Hilbert paths, vanishing squared-increment sum and norm-differentiable operator product rule."
    },
    {
      "id": "SM:N1",
      "source": "variable-positive-symmetrizers-preparation.md",
      "source_sha256": "226aeb5ef8e3b998e546d8d703517ed238350acbb64b6b07760e4d216f76071b",
      "proof_locator": "For later smoothing corrections, these derivatives are needed",
      "dependencies": [
        "SM:R024",
        "SM:R025",
        "SM:R015",
        "BR:N1",
        "SY:H10",
        "CE:H4",
        "CE:H7"
      ],
      "scope": "Both oriented extra-order operator-norm Taylor estimates for the actual symmetrized evolution, all endpoint derivatives and smoothing integral majorants."
    }
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