{
  "labeler": "Codex AI assistant (this chat; exact deployed model identifier unavailable)",
  "labeler_type": "AI; not human gold",
  "frozen_at": "2026-10-28T10:42:10.032234+00:00",
  "items_sha256": "487ecd22919a5e3b0174a2846ab8fe7c0daf4d3feccccf8ea6571aec04c41bed",
  "key_sha256": "f827cac92ee4e907554fadcc7c0e006ef2b829350f74e7f8b54ca11945e62836",
  "method": "Single-pass semantic annotation in shuffled blind-ID order; per-item compiler outcomes and detector outputs hidden. Pure syntax/proof defects are ignored when meaning is recoverable. No compiler/API-name audit. Full extracted-span coverage is assessed; unspecified design tasks or unresolved contextual references are not formalizable.",
  "limitations": [
    "Single AI annotator; no human adjudication or inter-annotator agreement.",
    "The annotator reviewed detector implementation and aggregate compilation counts before labeling; only per-item outputs were hidden.",
    "AI labels replace the originally planned human/Kimi labels by explicit owner instruction in this chat; the original human-ground-truth protocol is therefore not fully satisfied."
  ],
  "labels": {
    "P01": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The field axioms and nontriviality match the stated definition."
    },
    "P02": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The whole carrier and identity singleton are the two specified subgroups; API spelling is not assessed."
    },
    "P03": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Every integer unit is constrained to the two specified values."
    },
    "P04": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Generated subgroup is equated with all integer powers; escaped notation is treated as a syntax defect."
    },
    "P05": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Existence of a bijective multiplication-preserving map matches the definition."
    },
    "P06": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The theorem statement gives exactly the inverse-of-product identity; proof validity is not assessed."
    },
    "P07": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion",
        "degenerate"
      ],
      "notes": "Only already-assumed prime factors are asserted to exist; no recovery from the totient is specified."
    },
    "P08": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Both identity preservation and inverse preservation are stated under a multiplicative equivalence."
    },
    "P09": {
      "verdict": "wrong",
      "errors": [
        "wrong_type",
        "wrong_conclusion"
      ],
      "notes": "The multiplicative ring carrier does not encode the additive cyclic group, and the second extracted exercise is omitted."
    },
    "P10": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Both left and right equations have unique solutions."
    },
    "P11": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The group cardinality is the exponent, with the correct identity and quantification."
    },
    "P12": {
      "verdict": "not_formalizable",
      "errors": [
        "other"
      ],
      "notes": "The extracted design question provides no formal security or authentication specification for a fidelity comparison."
    },
    "P13": {
      "verdict": "minor_drift",
      "errors": [
        "wrong_type"
      ],
      "notes": "The intended modular identity is retained, but the integer variable is supplied to a natural-number coprimality predicate."
    },
    "P14": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion"
      ],
      "notes": "Only existence of a ring structure is stated; the identity and unit characterization are omitted."
    },
    "P15": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Unique solvability in both factors expresses the row and column permutation property."
    },
    "P16": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Passing to the product representation preserves the order-divisibility claim; capitalization and namespace errors are not semantic changes."
    },
    "P17": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion"
      ],
      "notes": "An independent non-field example in the extracted passage is omitted."
    },
    "P18": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The norm-one complex carrier and multiplication describe the specified subgroup; bundling/API validity is not assessed."
    },
    "P19": {
      "verdict": "minor_drift",
      "errors": [
        "wrong_type"
      ],
      "notes": "The natural-valued order convention needs an infinite-order convention absent from the cardinality definition."
    },
    "P20": {
      "verdict": "faithful",
      "errors": [],
      "notes": "A single left-identity action on one group element forces equality to the identity."
    },
    "P21": {
      "verdict": "minor_drift",
      "errors": [
        "wrong_type"
      ],
      "notes": "The intended commutativity equivalence is visible, but the left operation lacks the carrier annotation needed to identify G."
    },
    "P22": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The two quantified unique-solvability claims express the specified table property."
    },
    "P23": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Prime factors, modulus, lcm exponent and coprimality reproduce the extracted claim; no unstated distinctness assumption is introduced."
    },
    "P24": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion",
        "degenerate"
      ],
      "notes": "An arbitrary group satisfying an order condition is not identified with the geometric symmetry group."
    },
    "P25": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The two declarations cover both cancellation directions."
    },
    "P26": {
      "verdict": "wrong",
      "errors": [
        "extra_hypothesis",
        "missing_hypothesis"
      ],
      "notes": "The inserted divisibility assumption does not express the RSA relationship required by the argument."
    },
    "P27": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion"
      ],
      "notes": "The isomorphism is asserted for a quotient of the ambient group instead of the stabilizer subgroup."
    },
    "P28": {
      "verdict": "wrong",
      "errors": [
        "wrong_type",
        "wrong_conclusion"
      ],
      "notes": "Natural exponent division fails to express a square root, and the homomorphism target does not encode the specified automorphism."
    },
    "P29": {
      "verdict": "faithful",
      "errors": [],
      "notes": "A bijective homomorphism supplies the isomorphism assumption and both power assertions are retained."
    },
    "P30": {
      "verdict": "wrong",
      "errors": [
        "wrong_type"
      ],
      "notes": "An unordered finite product is used for an arbitrary semigroup, losing the ordered nonempty product required by the exercise."
    },
    "P31": {
      "verdict": "not_formalizable",
      "errors": [
        "other"
      ],
      "notes": "The isolated instruction refers to an earlier map whose definition is absent; the actual proposition cannot be recovered from this span."
    },
    "P32": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion",
        "degenerate"
      ],
      "notes": "The result merely supplies an unspecified ring homomorphism and does not constrain it to evaluation at the matrix parameter."
    },
    "P33": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The cardinality assumption and the nonidentity involution conclusion match; the missing instance binder is a compilation defect."
    },
    "P34": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The contrasting abelian properties encode the reason these specified groups are not isomorphic."
    },
    "P35": {
      "verdict": "wrong",
      "errors": [
        "wrong_type",
        "quantifier"
      ],
      "notes": "One fixed-arity function is applied at incompatible arities instead of specifying a coherent family of products."
    },
    "P36": {
      "verdict": "wrong",
      "errors": [
        "wrong_type",
        "wrong_conclusion"
      ],
      "notes": "The ring carrier replaces the concrete root-of-unity group, and neither its order nor the specified generator is stated."
    },
    "P37": {
      "verdict": "wrong",
      "errors": [
        "degenerate",
        "wrong_conclusion"
      ],
      "notes": "The declaration has no defining expression for the requested subgroup."
    },
    "P38": {
      "verdict": "wrong",
      "errors": [
        "missing_hypothesis"
      ],
      "notes": "Nonemptiness is missing, so the empty subset satisfies both closure assumptions."
    },
    "P39": {
      "verdict": "wrong",
      "errors": [
        "wrong_type",
        "wrong_conclusion"
      ],
      "notes": "Laurent series replace finite Laurent polynomials, and the requested units are not characterized."
    },
    "P40": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The supplied structure body fixes the function to the canonical integer residue map."
    },
    "P41": {
      "verdict": "wrong",
      "errors": [
        "extra_hypothesis"
      ],
      "notes": "The cardinality assertion is restricted to the range where noncommutativity holds, dropping the smaller symmetric groups."
    },
    "P42": {
      "verdict": "wrong",
      "errors": [
        "missing_hypothesis",
        "wrong_conclusion"
      ],
      "notes": "A necessary closure condition is promoted to sufficiency without inverse closure."
    },
    "P43": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Any two groups of the specified size are related by a multiplicative equivalence."
    },
    "P44": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The defining field is exactly commutativity of the group operation."
    },
    "P45": {
      "verdict": "wrong",
      "errors": [
        "quantifier",
        "wrong_conclusion"
      ],
      "notes": "Literal uniqueness of a carrier/structure replaces uniqueness up to group isomorphism."
    },
    "P46": {
      "verdict": "wrong",
      "errors": [
        "wrong_type",
        "wrong_conclusion"
      ],
      "notes": "Univariate polynomials over a product ring replace three-variable real polynomials; the units question is omitted."
    },
    "P47": {
      "verdict": "minor_drift",
      "errors": [
        "wrong_conclusion"
      ],
      "notes": "The integer-span equality captures the generated set, but the equality for opposite generators is not explicitly stated."
    },
    "P48": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The one-sided inverse hypothesis gives the specified inverse element."
    },
    "P49": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The theorem statement is exactly involutivity of group inversion."
    },
    "P50": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The custom structure supplies additive commutative group, associative multiplication and both distributive laws without imposing a unit."
    },
    "P51": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The carrier, multiplication and identity encode the specified finite group; existence of the chosen API is not audited."
    },
    "P52": {
      "verdict": "minor_drift",
      "errors": [
        "wrong_conclusion",
        "degenerate"
      ],
      "notes": "Ring structure and unit-to-constant direction are present, but identity characterization is replaced by reflexivity."
    },
    "P53": {
      "verdict": "minor_drift",
      "errors": [
        "extra_hypothesis"
      ],
      "notes": "Cardinality is represented only for finite groups although the definition is not restricted to them."
    },
    "P54": {
      "verdict": "wrong",
      "errors": [
        "wrong_type"
      ],
      "notes": "Multiplicative subgroup closure of a residue is used in place of the additive cyclic subgroup generated by it."
    },
    "P55": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The abelian-property equivalence is stated under an isomorphism."
    },
    "P56": {
      "verdict": "not_formalizable",
      "errors": [
        "other"
      ],
      "notes": "The design question does not specify an authenticity predicate or formal adversarial model."
    },
    "P57": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion"
      ],
      "notes": "The homomorphism criterion is retained but the separate isomorphism definition is absent."
    },
    "P58": {
      "verdict": "wrong",
      "errors": [
        "wrong_type"
      ],
      "notes": "A multiplicative equivalence on the full residue ring does not represent the additive cyclic group in the comparison."
    },
    "P59": {
      "verdict": "minor_drift",
      "errors": [
        "wrong_type"
      ],
      "notes": "The inherited-group criterion is recognizable, but operations on the subset are not specified as restricted operations."
    },
    "P60": {
      "verdict": "wrong",
      "errors": [
        "wrong_type",
        "missing_hypothesis"
      ],
      "notes": "Global bijections with local differentiability conditions do not specify self-diffeomorphisms of the open subset."
    },
    "P61": {
      "verdict": "faithful",
      "errors": [],
      "notes": "The homeomorphism type supplies the intended carrier and group structure under composition; notation validity is not assessed."
    },
    "P62": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Both least-positive exponent and the bounded-power description are retained, ignoring set-minimum API syntax."
    },
    "P63": {
      "verdict": "faithful",
      "errors": [],
      "notes": "Both two-sided identity hypotheses and their equality are stated."
    },
    "P64": {
      "verdict": "wrong",
      "errors": [
        "missing_hypothesis",
        "extra_hypothesis"
      ],
      "notes": "The exponent relation uses the ciphertext modulus where the RSA exponent modulus is required."
    },
    "P65": {
      "verdict": "wrong",
      "errors": [
        "wrong_conclusion",
        "degenerate"
      ],
      "notes": "The arbitrary finite group is unrelated to the requested geometric symmetries, with the classification left in a comment."
    }
  }
}