{
  "canonical_lesson_id": "AN04-U048",
  "identity_basis": "All51 original private coverage atoms bind this exact source to AN04-U048-private.",
  "all_original_targets": 51,
  "targets": [
    {
      "original_id": "AN04-U048-R001",
      "original_target": "Uniform gap and fixed single-eigenvalue multiplicities",
      "current_proofs": [
        "BR:R001"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R002",
      "original_target": "Separate global energy and local reduction hypotheses",
      "current_proofs": [
        "BR:R002"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R003",
      "original_target": "Finite Hermitian spectral theorem",
      "current_proofs": [
        "BR:R003"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R004",
      "original_target": "Stable resolvent contours",
      "current_proofs": [
        "BR:R004"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R005",
      "original_target": "Smooth full-rank spectral projections",
      "current_proofs": [
        "BR:R005"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R006",
      "original_target": "Smooth eigenvalues with fixed multiplicity",
      "current_proofs": [
        "BR:R006"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R007",
      "original_target": "Polynomial projection formula",
      "current_proofs": [
        "BR:R007"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R008",
      "original_target": "Reciprocal gap bounds",
      "current_proofs": [
        "BR:R008"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R009",
      "original_target": "Local Gram frames",
      "current_proofs": [
        "BR:R009"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R010",
      "original_target": "Unitary block frame and domain",
      "current_proofs": [
        "BR:R010"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R011",
      "original_target": "Full ordinary finite-matrix composition",
      "current_proofs": [
        "BR:R011"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R012",
      "original_target": "Conjugation with the time derivative retained",
      "current_proofs": [
        "BR:R012"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R013",
      "original_target": "Signed gap correction",
      "current_proofs": [
        "BR:R013"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R014",
      "original_target": "Conjugation remainder bookkeeping",
      "current_proofs": [
        "BR:R014"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R015",
      "original_target": "Ordinary block recursion",
      "current_proofs": [
        "BR:R015"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R016",
      "original_target": "Parameter-aware asymptotic sum",
      "current_proofs": [
        "BR:R016"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R017",
      "original_target": "Infinite-order intertwiner",
      "current_proofs": [
        "BR:R017"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R018",
      "original_target": "Both ordered inverses",
      "current_proofs": [
        "BR:R018"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R019",
      "original_target": "Precise inverse-conjugation defect",
      "current_proofs": [
        "BR:R019"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R020",
      "original_target": "Leading polarization coefficient",
      "current_proofs": [
        "BR:R020"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R021",
      "original_target": "Signed inverse correction and cancellation",
      "current_proofs": [
        "BR:R021"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R022",
      "original_target": "Ordered frame-change law",
      "current_proofs": [
        "BR:R022"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R023",
      "original_target": "Localized receiving theorem",
      "current_proofs": [
        "BR:R023"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R024",
      "original_target": "Unitary matrix logarithm",
      "current_proofs": [
        "BR:R024"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R025",
      "original_target": "Global auxiliary extension",
      "current_proofs": [
        "BR:R025"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R026",
      "original_target": "Global smoothing intertwining estimates",
      "current_proofs": [
        "BR:R026"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R027",
      "original_target": "Scalar-principal block evolutions",
      "current_proofs": [
        "BR:R027"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R028",
      "original_target": "Auxiliary branch kernels and normalization",
      "current_proofs": [
        "BR:R028"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R029",
      "original_target": "Vanishing conic composition terms",
      "current_proofs": [
        "BR:R029"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R030",
      "original_target": "Exterior full-PDO tails",
      "current_proofs": [
        "BR:R030"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R031",
      "original_target": "Distinct initial and PDE defects",
      "current_proofs": [
        "BR:R031"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R032",
      "original_target": "Actual-system energy correction",
      "current_proofs": [
        "BR:R032"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R033",
      "original_target": "Jointly smooth actual remainder kernel",
      "current_proofs": [
        "BR:R033"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R034",
      "original_target": "Finite compact trajectory cover",
      "current_proofs": [
        "BR:R034"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R035",
      "original_target": "Full commuting spectral projections",
      "current_proofs": [
        "BR:R035"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R036",
      "original_target": "Uniqueness from idempotence and the gap",
      "current_proofs": [
        "BR:R036"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R037",
      "original_target": "Overlap compatibility and partition gluing",
      "current_proofs": [
        "BR:R037"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R038",
      "original_target": "Projected local input normalization",
      "current_proofs": [
        "BR:R038"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R039",
      "original_target": "Endpoint block membership",
      "current_proofs": [
        "BR:R039"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R040",
      "original_target": "Eliminate every switched-branch word",
      "current_proofs": [
        "BR:R040"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R041",
      "original_target": "Finite same-branch graph composition",
      "current_proofs": [
        "BR:R041"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R042",
      "original_target": "Two directions and all endpoint families",
      "current_proofs": [
        "BR:R042"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R043",
      "original_target": "Elliptic endpoint bundle transport",
      "current_proofs": [
        "BR:R043"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R044",
      "original_target": "Exact full-matrix kernel wavefront",
      "current_proofs": [
        "BR:R044"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R045",
      "original_target": "Precisely localized projected data propagation",
      "current_proofs": [
        "BR:R045"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R046",
      "original_target": "Crossing and internal-cluster scope guard",
      "current_proofs": [
        "BR:R046"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R047",
      "original_target": "Rotating-frame correction and exact spectrum",
      "current_proofs": [
        "BR:R047"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R048",
      "original_target": "Energy with a closing gap",
      "current_proofs": [
        "BR:R048"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R049",
      "original_target": "Uniform gap with a nontrivial complex eigenline",
      "current_proofs": [
        "BR:R049"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R050",
      "original_target": "Exact rotating-polarization branch kernel",
      "current_proofs": [
        "BR:R050"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    },
    {
      "original_id": "AN04-U048-R051",
      "original_target": "Exact rays and polarization figure",
      "current_proofs": [
        "BR:R051"
      ],
      "disposition": "Complete independently written proof at the original separated single-eigenvalue uniform-gap scope, with exact current programme providers."
    }
  ],
  "all_original_solutions_preserved": true,
  "full_course_complete": false
}