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