{
  "source": "Extracted ids, titles, and status labels from the hosted omega_proofs.json. These are the bundle's labels, not this archive's review.",
  "theorems": [
    {
      "id": "P00",
      "title": "Meaning and soundness of the algebraic certificates",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "G01",
      "title": "Whole-geometry similarity and invariant selected readouts",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "G02",
      "title": "Exact conjugacy between scale and the affine contraction",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "G03",
      "title": "Canonical algebraic landmarks",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "G04",
      "title": "Transcendental readouts and nonvanishing real residuals",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "G05",
      "title": "Gram equality, intrinsic dimension, and Hilbert embedding",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "C01",
      "title": "Six-point polynomial code and exact error/erasure correction",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "C02",
      "title": "Five weighted interval measurements form a polynomial code",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "C03",
      "title": "Analog noise conditioning and invisible profiles",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "A01",
      "title": "Exact finite certificate for real/rational floor agreement",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "A02",
      "title": "Ordered Euclidean witnesses give exact integer-history reconstruction",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "A03",
      "title": "First-residue inversion on the canonical interval",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "A04",
      "title": "Exact separation of geometric, slope, and quantization errors",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "V01",
      "title": "Intrinsic convex bodies and exact polar-volume products",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "T01",
      "title": "Tensor-product reconstruction in multiple coordinates",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "H01",
      "title": "Positive cubature gives harmonic reconstruction and a noise bound",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "H02",
      "title": "Exact 12-node icosahedral instance and its first moment obstruction",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "O01",
      "title": "The center-relative moment ellipse and its family inverse",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "O02",
      "title": "Affine metric recovery and the feet-outside-ellipse correction",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "O03",
      "title": "Constructive numerical-range representation of the ellipse",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "O04",
      "title": "Exact operator norms and an elementary polynomial bound",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "O05",
      "title": "Two infinite-dimensional operator extensions with different behavior",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "O06",
      "title": "A quantitative nonlinear solvability class",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "S01",
      "title": "Correct symplectic normalization",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "I01",
      "title": "Finite probes measure a compression, not an arbitrary whole operator",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "I02",
      "title": "What a six-port series-chain impedance measurement can recover",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "I03",
      "title": "A dense-probe limit and its conditional inverse-problem consequence",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "I04",
      "title": "Finite-compression transfer of a uniform operator bound",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "M01",
      "title": "A local OMEGA placement on a smooth manifold and its exact metric contract",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "F01",
      "title": "Similarity covariance of forced incompressible Navier-Stokes and observers",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "B01",
      "title": "Exact reversible quantization of full-rank integer linear layers",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "B02",
      "title": "Boolean implementation and reversible gate embedding",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "B03",
      "title": "Byte towers, retained length, and information costs",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    },
    {
      "id": "U01",
      "title": "A composable geometric-to-computational reconstruction theorem",
      "mathematical_status": "PROVED_UNDER_STATED_ASSUMPTIONS",
      "formal_kernel_status": "NOT_FORMALIZED_IN_THIS_BUNDLE"
    }
  ],
  "not_established": [
    {
      "id": "EXT_017",
      "status": "NOT_ESTABLISHED",
      "claim": "A claimed rational irrationality-exponent result for pi can be transferred directly to lambda."
    },
    {
      "id": "EXT_087",
      "status": "DOES_NOT_FOLLOW",
      "claim": "The all-body Mahler inequality and equality classifications follow from the OMEGA triangle and rectangle examples."
    },
    {
      "id": "EXT_098",
      "status": "DOES_NOT_FOLLOW",
      "claim": "Embedding a single planar OMEGA in a Hilbert space gives independent infinite-dimensional geometry or a universal finite-dimensional reduction."
    },
    {
      "id": "EXT_107_130",
      "status": "NOT_ESTABLISHED",
      "claim": "The fixed tensor decoder proves a new matrix multiplication exponent or an FFT on the nonuniform OMEGA nodes."
    },
    {
      "id": "EXT_122_140_229",
      "status": "DOES_NOT_FOLLOW",
      "claim": "The finite polynomial code proves the release results on trace reconstruction, learning-memory lower bounds, or broadcasting."
    },
    {
      "id": "EXT_325",
      "status": "NOT_VERIFIED_IN_THIS_BUNDLE",
      "claim": "A universal Crouzeix matrix-polynomial bound with constant two has been proved and executed here."
    },
    {
      "id": "EXT_333_SYMPLECTIC",
      "status": "NOT_ESTABLISHED",
      "claim": "A nonlinear symplectic embedding/ball-packing theorem follows from the ellipse chart or a Fredholm derivative."
    },
    {
      "id": "EXT_365_372",
      "status": "FALSE_WITHOUT_ADDITIONAL_HYPOTHESES",
      "claim": "One finite six-port measurement identifies an arbitrary host metric, connection, conductivity, or manifold."
    },
    {
      "id": "EXT_376",
      "status": "NOT_ESTABLISHED",
      "claim": "The OMEGA floor transition implements the released universal fluid computer or proves its reachability result."
    },
    {
      "id": "APP_NEURAL_PERFORMANCE",
      "status": "NOT_ESTABLISHED",
      "claim": "The proposed remainder-preserving architecture improves training, accuracy, generalization, compression, energy use, or intelligence."
    },
    {
      "id": "APP_PHYSICS",
      "status": "NOT_ESTABLISHED",
      "claim": "The mathematical interfaces prove holography, quantum gravity, or physical unification."
    },
    {
      "id": "CORRECTION_ELLIPSE",
      "status": "REFUTED_FOR_THE_DECLARED_CHART",
      "claim": "The unnormalized canonical ellipse chart is symplectic and all mapped feet lie inside its numerical range."
    }
  ]
}