{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0001","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e27ce8e51cb43be49bcf5778ce15105c94586f023dfa19a160733511e9a9e170","family_id":"qf-mod30-cardinality-injection","seed":202610305001,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[100,101],\"source\":[0,1],\"target\":[100,101,102]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders, with mapping=[100, 101] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([0, 1], [100, 101, 102], [100, 101], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([0, 1], [100, 101, 102], [100, 101], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0002","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"494bf3eb852996066024d49886915036a6d8894e0bf90ca8f33a264a52f2244b","family_id":"qf-mod30-cardinality-injection","seed":202610305002,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[110,111,112],\"source\":[10,11,12],\"target\":[110,111,112,113]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders, with mapping=[110, 111, 112] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0003","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ff9406008c4287ba28bf3e7e45750434de5c61fa8a8c542e9f512b665ae443f","family_id":"qf-mod30-cardinality-injection","seed":202610305003,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[120,121,122,123],\"source\":[20,21,22,23],\"target\":[120,121,122,123,124]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders, with mapping=[120, 121, 122, 123] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0004","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d668465acc69d8fd112dc48cf7686fe585d7be649d82472d46296617b546845c","family_id":"qf-mod30-cardinality-injection","seed":202610305004,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[130,131,130,131,130],\"source\":[30,31,32,33,34],\"target\":[130,131]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders, with mapping=[130, 131, 130, 131, 130] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, no_comparison_certificate)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, no_comparison_certificate)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0005","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3ce28aec88763a9a2833973a2c35a2f519c9aa8f7ed4d7f3ed23434a53c800f","family_id":"qf-mod30-cardinality-injection","seed":202610305005,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[140,141],\"source\":[40,41],\"target\":[140,141,142]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([40, 41], [140, 141, 142], [140, 141], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([40, 41], [140, 141, 142], [140, 141], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0006","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b56cf6c5633891e4c663c12f6781529e7ef066c185c52d0521402fab451621da","family_id":"qf-mod30-cardinality-injection","seed":202610305006,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[150,151,152],\"source\":[50,51,52],\"target\":[150,151,152,153]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders, with mapping=[150, 151, 152] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0007","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dee3905022ac8dc1cb9c7c7be165800fd73e81ebc1839fe965e6d15c61b1fb93","family_id":"qf-mod30-cardinality-injection","seed":202610305007,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[160,161,162,163],\"source\":[60,61,62,63],\"target\":[160,161,162,163,164]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders, with mapping=[160, 161, 162, 163] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0008","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc539d4a2a509c9c1e9ebff27f1e2f4fbfc7b7dbb9a6cf6500221639f3c5dcbd","family_id":"qf-mod30-cardinality-injection","seed":202610305008,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[170,171,170,171,170],\"source\":[70,71,72,73,74],\"target\":[170,171]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders, with mapping=[170, 171, 170, 171, 170] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, no_comparison_certificate)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, no_comparison_certificate)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0009","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c767b61c7e9bd71728df4532b3b167d69c9dc599fb6f4267afd5c4390990fda","family_id":"qf-mod30-cardinality-injection","seed":202610305009,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[180,181],\"source\":[80,81],\"target\":[180,181,182]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders, with mapping=[180, 181] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([80, 81], [180, 181, 182], [180, 181], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([80, 81], [180, 181, 182], [180, 181], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0010","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a744834160b8809435cdb76f7942fbe2c7cab4f95171188b4a59af84e2c1418","family_id":"qf-mod30-cardinality-injection","seed":202610305010,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[190,191,192],\"source\":[90,91,92],\"target\":[190,191,192,193]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders, with mapping=[190, 191, 192] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0011","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a23439c6d6a7f67bfd8a619b953cac73608c4d550547cd257509e033b4c667f","family_id":"qf-mod30-cardinality-injection","seed":202610305011,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[200,201,202,203],\"source\":[100,101,102,103],\"target\":[200,201,202,203,204]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders, with mapping=[200, 201, 202, 203] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0012","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0020ab9aa3c6dc6d2d244c8ba18e5f3ad8972e61dffc7af2eca2d751ee5e769d","family_id":"qf-mod30-cardinality-injection","seed":202610305012,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[210,211,210,211,210],\"source\":[110,111,112,113,114],\"target\":[210,211]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders, with mapping=[210, 211, 210, 211, 210] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, no_comparison_certificate)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, no_comparison_certificate)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0013","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45acf2fbc145253eaed6a19fb5cf3e1aa756b94ce0edf81426bbf4fda18dd463","family_id":"qf-mod30-cardinality-injection","seed":202610305013,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[220,221],\"source\":[120,121],\"target\":[220,221,222]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders, with mapping=[220, 221] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([120, 121], [220, 221, 222], [220, 221], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([120, 121], [220, 221, 222], [220, 221], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0014","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bd9ed577b4c9ad9ea74873c450b2fcca54b680f9001afdac91b155ec2c67b39","family_id":"qf-mod30-cardinality-injection","seed":202610305014,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[230,231,232],\"source\":[130,131,132],\"target\":[230,231,232,233]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders, with mapping=[230, 231, 232] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0015","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60e8e257670eef53095af3de02c1e6b303d15c369c98e23afa5ac8820cd74ce3","family_id":"qf-mod30-cardinality-injection","seed":202610305015,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[240,241,242,243],\"source\":[140,141,142,143],\"target\":[240,241,242,243,244]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders, with mapping=[240, 241, 242, 243] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0016","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2fa581da0f57005d2252820f390dfe7a8132cc4e5ab8048b92a5e4ee707e115","family_id":"qf-mod30-cardinality-injection","seed":202610305016,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[250,251,250,251,250],\"source\":[150,151,152,153,154],\"target\":[250,251]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, no_comparison_certificate)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, no_comparison_certificate)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0017","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27679af723bd5aa95cd2563e0bed34633f2d32d757548affa66aeea43fb2afeb","family_id":"qf-mod30-cardinality-injection","seed":202610305017,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[260,261],\"source\":[160,161],\"target\":[260,261,262]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders, with mapping=[260, 261] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([160, 161], [260, 261, 262], [260, 261], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([160, 161], [260, 261, 262], [260, 261], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0018","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa1473e612163caa7b425b5f5b69dcb754ad39cb85d7fe798798d64227c479e3","family_id":"qf-mod30-cardinality-injection","seed":202610305018,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[270,271,272],\"source\":[170,171,172],\"target\":[270,271,272,273]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders, with mapping=[270, 271, 272] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0019","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5384851caa1656c0497157a213834806359b7a3c730f8e4d3e04879f472a98be","family_id":"qf-mod30-cardinality-injection","seed":202610305019,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[280,281,282,283],\"source\":[180,181,182,183],\"target\":[280,281,282,283,284]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders, with mapping=[280, 281, 282, 283] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0020","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfe3e63dadd55d4ac3601b49433edb82037d9803a732a163a21bdf26ebe5f7cd","family_id":"qf-mod30-cardinality-injection","seed":202610305020,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[290,291,290,291,290],\"source\":[190,191,192,193,194],\"target\":[290,291]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders, with mapping=[290, 291, 290, 291, 290] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, no_comparison_certificate)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, no_comparison_certificate)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0021","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"815631b0a156bd1cb73b5fa57f2bf9b88ad49e2f574dad4f28f758d28306dadd","family_id":"qf-mod30-cardinality-injection","seed":202610305021,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[300,301],\"source\":[200,201],\"target\":[300,301,302]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders, with mapping=[300, 301] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([200, 201], [300, 301, 302], [300, 301], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([200, 201], [300, 301, 302], [300, 301], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0022","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c39f2e356b791778afd7a89a0dcd80d1fd481f91cc50334509337c033992ac86","family_id":"qf-mod30-cardinality-injection","seed":202610305022,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[310,311,312],\"source\":[210,211,212],\"target\":[310,311,312,313]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0023","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b5a538fa3fc51144e20cd066cfba9790f73f7d44eb741a69b9b7893182c870c","family_id":"qf-mod30-cardinality-injection","seed":202610305023,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[320,321,322,323],\"source\":[220,221,222,223],\"target\":[320,321,322,323,324]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders, with mapping=[320, 321, 322, 323] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0024","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17c7a477c8acdf4365e5050e706c6264a7a4a934c8b031453942390814a61e87","family_id":"qf-mod30-cardinality-injection","seed":202610305024,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[330,331,330,331,330],\"source\":[230,231,232,233,234],\"target\":[330,331]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders, with mapping=[330, 331, 330, 331, 330] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, no_comparison_certificate)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, no_comparison_certificate)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0025","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"589c45b11b82051bb64f6d3b74b714bbb9d16d84075b3316b37ce11797cb67ce","family_id":"qf-mod30-cardinality-injection","seed":202610305025,"semantic_template":"qf-mod30-cardinality-injection/{\"mapping\":[340,341],\"source\":[240,241],\"target\":[340,341,342]}"},"fields":{"title":"An Injection Certifies One Finite Cardinality Inequality","stem":"Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders, with mapping=[340, 341] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,comparison). comparison is |source|≤|target| when this mapping is injective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([240, 241], [340, 341, 342], [340, 341], Yes, Yes)","explanation":"Distinct source elements must have distinct images. Retain all list orders and the exact target. Failure of this mapping does not prove that no other injection exists. The exact result, retaining the requested order, is ([240, 241], [340, 341, 342], [340, 341], Yes, Yes)."},"skills":["An Injection Certifies One Finite Cardinality Inequality"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0026","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"267213a53c502253d926b8ae3ccf3454881781b511e615191bd6a429f3448ee5","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305026,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[100,101],\"source\":[0,1],\"target\":[100,101,102]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders, with mapping=[100, 101] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([0, 1], [100, 101, 102], [100, 101], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([0, 1], [100, 101, 102], [100, 101], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0027","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca0ecdf8aa5f863f518c4845654f7fe433d0a7ab426137dd74fe7bbc30bdb4c4","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305027,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[110,111,112],\"source\":[10,11,12],\"target\":[110,111,112,113]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders, with mapping=[110, 111, 112] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0028","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"384bb840c3725e39653f997d3d569a89deb55660532cdcbcec39afe59f160be2","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305028,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[120,121,122,123],\"source\":[20,21,22,23],\"target\":[120,121,122,123,124]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders, with mapping=[120, 121, 122, 123] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0029","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72936542047bf2bcdf00a23c20fdde1dc04c938a584287fae91d27b37886ecb6","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305029,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[130,131,130,131,130],\"source\":[30,31,32,33,34],\"target\":[130,131]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders, with mapping=[130, 131, 130, 131, 130] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], Yes, Yes, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], Yes, Yes, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0030","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5dc03060e7322041ae7a897575c2540b20de0c30ad1af3d27052697c5b3bbbb","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305030,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[140,141],\"source\":[40,41],\"target\":[140,141,142]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([40, 41], [140, 141, 142], [140, 141], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([40, 41], [140, 141, 142], [140, 141], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0031","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"581a45aa454771724578d0bbd5f94af88cfbdba01ca5aa3e0f852ba642da2534","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305031,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[150,151,152],\"source\":[50,51,52],\"target\":[150,151,152,153]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders, with mapping=[150, 151, 152] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0032","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3c9e572b22fa65ba2c02e7fa274c88ff67ece07a1a6ef1b0911a396727cac3e","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305032,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[160,161,162,163],\"source\":[60,61,62,63],\"target\":[160,161,162,163,164]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders, with mapping=[160, 161, 162, 163] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0033","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7caabf34d02a4989cb0efa8537a6b5cc462dceffa7ef5e5535cb612e3d711b38","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305033,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[170,171,170,171,170],\"source\":[70,71,72,73,74],\"target\":[170,171]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders, with mapping=[170, 171, 170, 171, 170] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], Yes, Yes, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], Yes, Yes, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0034","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d2d7d0550db4b681744bff72fdf803cecc01a9d4847b730823708824f3106fe","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305034,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[180,181],\"source\":[80,81],\"target\":[180,181,182]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders, with mapping=[180, 181] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([80, 81], [180, 181, 182], [180, 181], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([80, 81], [180, 181, 182], [180, 181], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0035","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63f3fd9c00a53d41d3f22e342026122ca4169253eb417a316a490dd3b1ad85e4","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305035,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[190,191,192],\"source\":[90,91,92],\"target\":[190,191,192,193]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders, with mapping=[190, 191, 192] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0036","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6b469029594dc435f5b564785272568a309ab06b6f2a3cbd69bd6917f467bd8","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305036,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[200,201,202,203],\"source\":[100,101,102,103],\"target\":[200,201,202,203,204]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders, with mapping=[200, 201, 202, 203] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0037","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"321ca2959322fa8ea0ef2be66e252fd93e8db32193030181b74e6d94aa96269c","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305037,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[210,211,210,211,210],\"source\":[110,111,112,113,114],\"target\":[210,211]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders, with mapping=[210, 211, 210, 211, 210] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], Yes, Yes, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], Yes, Yes, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0038","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d9f89e040e0e2595f371108ff2fac4e097e453c444d761986e6d27f58f28b81","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305038,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[220,221],\"source\":[120,121],\"target\":[220,221,222]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders, with mapping=[220, 221] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([120, 121], [220, 221, 222], [220, 221], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([120, 121], [220, 221, 222], [220, 221], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0039","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff60ce8f884c295b691623e78e7a9fe66dfe75c57510c72af8e2b3ba85c70496","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305039,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[230,231,232],\"source\":[130,131,132],\"target\":[230,231,232,233]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders, with mapping=[230, 231, 232] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0040","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5654e2bff376c7f1ba243904607d16f4a328242d3950a55125aad492f1b1c2b4","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305040,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[240,241,242,243],\"source\":[140,141,142,143],\"target\":[240,241,242,243,244]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders, with mapping=[240, 241, 242, 243] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0041","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8c809d366c0494ba4aef5c108ec8a5a56e149d3b417b4da9e9c23a225b66e5a","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305041,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[250,251,250,251,250],\"source\":[150,151,152,153,154],\"target\":[250,251]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], Yes, Yes, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], Yes, Yes, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0042","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21a6428c43b8ec40c7967c066a2f1571bbbfa472729dd2989b01fa0a00e7d0a7","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305042,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[260,261],\"source\":[160,161],\"target\":[260,261,262]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders, with mapping=[260, 261] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([160, 161], [260, 261, 262], [260, 261], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([160, 161], [260, 261, 262], [260, 261], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0043","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fe4684b49159e63774835129c41f48feaa8027b77260cdca4739535f43f5edc","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305043,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[270,271,272],\"source\":[170,171,172],\"target\":[270,271,272,273]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders, with mapping=[270, 271, 272] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0044","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"402343bf4f9b8b8276bfb09475dc274395c1ef9afbc64187d77c01ab2b4a82a9","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305044,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[280,281,282,283],\"source\":[180,181,182,183],\"target\":[280,281,282,283,284]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders, with mapping=[280, 281, 282, 283] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0045","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d9bc4eb51f2ee6043b63031df17c87f40e62193b7dc1f82376030f1b2b4c331","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305045,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[290,291,290,291,290],\"source\":[190,191,192,193,194],\"target\":[290,291]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders, with mapping=[290, 291, 290, 291, 290] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], Yes, Yes, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], Yes, Yes, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0046","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff5faa59588b874272b20e9fe1298da4e5d3d5f70fe4de7cc9a12345eb77548","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305046,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[300,301],\"source\":[200,201],\"target\":[300,301,302]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders, with mapping=[300, 301] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([200, 201], [300, 301, 302], [300, 301], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([200, 201], [300, 301, 302], [300, 301], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0047","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f92896ed41e36a8ae58288ea05950592a249669e9db2eb858fd036fa315cce50","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305047,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[310,311,312],\"source\":[210,211,212],\"target\":[310,311,312,313]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0048","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"688a23a6deb108083fb62a58b2736d614ab439e84c66b71e40fa0661a8669bc8","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305048,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[320,321,322,323],\"source\":[220,221,222,223],\"target\":[320,321,322,323,324]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders, with mapping=[320, 321, 322, 323] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0049","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b11ac93445c5bd030ca844910652437a79585a2ca9e211598bcc2b4d6a2f5d55","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305049,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[330,331,330,331,330],\"source\":[230,231,232,233,234],\"target\":[330,331]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders, with mapping=[330, 331, 330, 331, 330] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], Yes, Yes, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], Yes, Yes, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0050","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6b142cf64803aff06f3e8b5561ca0e0fa94a84cbda5502a52e0545959b3e0f8","family_id":"qf-mod30-cardinality-surjection-finite","seed":202610305050,"semantic_template":"qf-mod30-cardinality-surjection-finite/{\"mapping\":[340,341],\"source\":[240,241],\"target\":[340,341,342]}"},"fields":{"title":"A Surjection Supplies the Reverse Comparison Only in This Finite Scope","stem":"Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders, with mapping=[340, 341] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,surjective?,comparison,finite_only). comparison is |source|≥|target| when this mapping is surjective, otherwise no_comparison_certificate.","answer":{"kind":"expression","value":"([240, 241], [340, 341, 342], [340, 341], No, no_comparison_certificate, finite_only)","explanation":"Test coverage of every target element by the specified mapping. The reverse finite-cardinality comparison is a certificate from this mapping, not an unrestricted assertion about infinite sets or choice. The exact result, retaining the requested order, is ([240, 241], [340, 341, 342], [340, 341], No, no_comparison_certificate, finite_only)."},"skills":["A Surjection Supplies the Reverse Comparison Only in This Finite Scope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0051","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2add5b08557a3424d2cd61e01a39fbab8ba9fe38a4ce6b15b23db1d0a9cfe2e","family_id":"qf-mod30-cardinality-bijection","seed":202610305051,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[100,101],\"source\":[0,1],\"target\":[100,101,102]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders, with mapping=[100, 101] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([0, 1], [100, 101, 102], [100, 101], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([0, 1], [100, 101, 102], [100, 101], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0052","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"238db1599f93ffbcf048516d479bff37adff40228e584de3ccfe2f06a25e55ba","family_id":"qf-mod30-cardinality-bijection","seed":202610305052,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[110,111,112],\"source\":[10,11,12],\"target\":[110,111,112,113]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders, with mapping=[110, 111, 112] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([10, 11, 12], [110, 111, 112, 113], [110, 111, 112], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0053","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86d26dd00dbd2f2224295fea56c6baf039b5230deddf6298a6bb59ab0d464639","family_id":"qf-mod30-cardinality-bijection","seed":202610305053,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[120,121,122,123],\"source\":[20,21,22,23],\"target\":[120,121,122,123,124]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders, with mapping=[120, 121, 122, 123] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([20, 21, 22, 23], [120, 121, 122, 123, 124], [120, 121, 122, 123], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0054","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5b82960ee657a68fa5e3f1cdbf6407de210a1c2859f819bc0a3868dd915d2ab","family_id":"qf-mod30-cardinality-bijection","seed":202610305054,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[130,131,130,131,130],\"source\":[30,31,32,33,34],\"target\":[130,131]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders, with mapping=[130, 131, 130, 131, 130] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, Yes, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([30, 31, 32, 33, 34], [130, 131], [130, 131, 130, 131, 130], No, Yes, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0055","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0621e2b2a62ab996a41adbef74925f7ecd9fcd4176e2dccb194e8a01df282a2","family_id":"qf-mod30-cardinality-bijection","seed":202610305055,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[140,141],\"source\":[40,41],\"target\":[140,141,142]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders, with mapping=[140, 141] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([40, 41], [140, 141, 142], [140, 141], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([40, 41], [140, 141, 142], [140, 141], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0056","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"996e66d8be674cec58dd59d0822f61a92b660d2d1c19a9f21cc734d3cb51a92a","family_id":"qf-mod30-cardinality-bijection","seed":202610305056,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[150,151,152],\"source\":[50,51,52],\"target\":[150,151,152,153]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders, with mapping=[150, 151, 152] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([50, 51, 52], [150, 151, 152, 153], [150, 151, 152], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0057","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d6d9094bd20a48fe99cc8568758231b2b8066de989928af15d32db9d1a354ab","family_id":"qf-mod30-cardinality-bijection","seed":202610305057,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[160,161,162,163],\"source\":[60,61,62,63],\"target\":[160,161,162,163,164]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders, with mapping=[160, 161, 162, 163] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([60, 61, 62, 63], [160, 161, 162, 163, 164], [160, 161, 162, 163], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0058","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dde330393a8659c08e88439b02af7a3d6f7a9ad59be9005aac16e6f0e5b3496","family_id":"qf-mod30-cardinality-bijection","seed":202610305058,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[170,171,170,171,170],\"source\":[70,71,72,73,74],\"target\":[170,171]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders, with mapping=[170, 171, 170, 171, 170] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, Yes, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([70, 71, 72, 73, 74], [170, 171], [170, 171, 170, 171, 170], No, Yes, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0059","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19c3575cb6a06d9a60c6a7fdefab06e981ebf679ea41d854e90a4c834f17afc4","family_id":"qf-mod30-cardinality-bijection","seed":202610305059,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[180,181],\"source\":[80,81],\"target\":[180,181,182]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders, with mapping=[180, 181] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([80, 81], [180, 181, 182], [180, 181], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([80, 81], [180, 181, 182], [180, 181], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0060","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f7c00e7fa4a711c3dd0b8b2014654931aa9ffd712aad1e3ff1879275b451d77","family_id":"qf-mod30-cardinality-bijection","seed":202610305060,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[190,191,192],\"source\":[90,91,92],\"target\":[190,191,192,193]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders, with mapping=[190, 191, 192] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([90, 91, 92], [190, 191, 192, 193], [190, 191, 192], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0061","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"390ae5be9284af6339918d40d02ad87115022946d955c2054e5e00b508aef428","family_id":"qf-mod30-cardinality-bijection","seed":202610305061,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[200,201,202,203],\"source\":[100,101,102,103],\"target\":[200,201,202,203,204]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders, with mapping=[200, 201, 202, 203] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([100, 101, 102, 103], [200, 201, 202, 203, 204], [200, 201, 202, 203], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0062","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bac3adaad2aa61f3a87adcb30322e09230a0ee0026b4c9f6fc2e3345627ae4db","family_id":"qf-mod30-cardinality-bijection","seed":202610305062,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[210,211,210,211,210],\"source\":[110,111,112,113,114],\"target\":[210,211]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders, with mapping=[210, 211, 210, 211, 210] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, Yes, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([110, 111, 112, 113, 114], [210, 211], [210, 211, 210, 211, 210], No, Yes, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0063","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f54eb73ce5d78e3637f2996db539d58503d118fd95066badcde28f65bd45ab16","family_id":"qf-mod30-cardinality-bijection","seed":202610305063,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[220,221],\"source\":[120,121],\"target\":[220,221,222]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders, with mapping=[220, 221] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([120, 121], [220, 221, 222], [220, 221], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([120, 121], [220, 221, 222], [220, 221], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0064","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc3685342f1595fa88c85dd1625e576f29c126d37cb9306d8a983ac23fa5e1a6","family_id":"qf-mod30-cardinality-bijection","seed":202610305064,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[230,231,232],\"source\":[130,131,132],\"target\":[230,231,232,233]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders, with mapping=[230, 231, 232] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([130, 131, 132], [230, 231, 232, 233], [230, 231, 232], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0065","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f6c30396a9fafaaf21dda750bbba7ced67fc68a6b18c88d55a4c0008a684eb4","family_id":"qf-mod30-cardinality-bijection","seed":202610305065,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[240,241,242,243],\"source\":[140,141,142,143],\"target\":[240,241,242,243,244]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders, with mapping=[240, 241, 242, 243] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([140, 141, 142, 143], [240, 241, 242, 243, 244], [240, 241, 242, 243], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0066","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff8029db21bc2e29bf6d2f0073a431dffd73bc140383e5a442ace44592f466a6","family_id":"qf-mod30-cardinality-bijection","seed":202610305066,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[250,251,250,251,250],\"source\":[150,151,152,153,154],\"target\":[250,251]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders, with mapping=[250, 251, 250, 251, 250] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, Yes, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([150, 151, 152, 153, 154], [250, 251], [250, 251, 250, 251, 250], No, Yes, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0067","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cb75cbf3589e12790264cb693905b181c7bd4de6248bf1eae127ab882c7ecdb","family_id":"qf-mod30-cardinality-bijection","seed":202610305067,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[260,261],\"source\":[160,161],\"target\":[260,261,262]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders, with mapping=[260, 261] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([160, 161], [260, 261, 262], [260, 261], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([160, 161], [260, 261, 262], [260, 261], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0068","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9ba10f7913114dd0bdafbfe93a08158060d4d2dd8aaf52d583f082320e020ff","family_id":"qf-mod30-cardinality-bijection","seed":202610305068,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[270,271,272],\"source\":[170,171,172],\"target\":[270,271,272,273]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders, with mapping=[270, 271, 272] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([170, 171, 172], [270, 271, 272, 273], [270, 271, 272], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0069","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2db82ea92caa26a3d4cc6163898f36567b83c38f52d4b2c6d0920a52287cbe6","family_id":"qf-mod30-cardinality-bijection","seed":202610305069,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[280,281,282,283],\"source\":[180,181,182,183],\"target\":[280,281,282,283,284]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders, with mapping=[280, 281, 282, 283] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([180, 181, 182, 183], [280, 281, 282, 283, 284], [280, 281, 282, 283], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0070","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1cdaf6734a7c645d7b2e0f791f9e4366bacaf77199ac4b9e86904a5e098b655","family_id":"qf-mod30-cardinality-bijection","seed":202610305070,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[290,291,290,291,290],\"source\":[190,191,192,193,194],\"target\":[290,291]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders, with mapping=[290, 291, 290, 291, 290] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, Yes, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([190, 191, 192, 193, 194], [290, 291], [290, 291, 290, 291, 290], No, Yes, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0071","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43bc54b698922c133d5f3d442ff721feade10a521e115ae5673c6a7799e50295","family_id":"qf-mod30-cardinality-bijection","seed":202610305071,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[300,301],\"source\":[200,201],\"target\":[300,301,302]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders, with mapping=[300, 301] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([200, 201], [300, 301, 302], [300, 301], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([200, 201], [300, 301, 302], [300, 301], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0072","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce4a1d52884eceb19be789818654436b2b6537219ca9cacdcb8f22864440192d","family_id":"qf-mod30-cardinality-bijection","seed":202610305072,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[310,311,312],\"source\":[210,211,212],\"target\":[310,311,312,313]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders, with mapping=[310, 311, 312] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([210, 211, 212], [310, 311, 312, 313], [310, 311, 312], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0073","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6b55213b13a50cc5b1df6271ed75c433c1bc32126b8775891efee518c2e4275","family_id":"qf-mod30-cardinality-bijection","seed":202610305073,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[320,321,322,323],\"source\":[220,221,222,223],\"target\":[320,321,322,323,324]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders, with mapping=[320, 321, 322, 323] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([220, 221, 222, 223], [320, 321, 322, 323, 324], [320, 321, 322, 323], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0074","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8565ea8978a283740ffe2dcaf52285b852b51ef63fca46d03e1b77a52fb9f0a7","family_id":"qf-mod30-cardinality-bijection","seed":202610305074,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[330,331,330,331,330],\"source\":[230,231,232,233,234],\"target\":[330,331]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders, with mapping=[330, 331, 330, 331, 330] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, Yes, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([230, 231, 232, 233, 234], [330, 331], [330, 331, 330, 331, 330], No, Yes, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0075","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cc8d84b1252c9bf653c4577324c1aac9579e314aedcec8d74e44612015652b7","family_id":"qf-mod30-cardinality-bijection","seed":202610305075,"semantic_template":"qf-mod30-cardinality-bijection/{\"mapping\":[340,341],\"source\":[240,241],\"target\":[340,341,342]}"},"fields":{"title":"The Last Equality Flag Is a Bijection-Supported Certificate","stem":"Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders, with mapping=[340, 341] aligned to source order. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. Give (source,target,mapping,injective?,surjective?,bijective?,equal_cardinality). In this protocol the last flag is conjoined with the bijective flag: No means this mapping supplied no equality certificate, not that the two finite cardinalities are unequal.","answer":{"kind":"expression","value":"([240, 241], [340, 341, 342], [340, 341], Yes, No, No, No)","explanation":"Check injectivity and target coverage separately. Only their conjunction supplies a bijection-based equality certificate. A nonbijective mapping between equal-sized sets can therefore have a No last flag without refuting equality of their sizes. The exact result, retaining the requested order, is ([240, 241], [340, 341, 342], [340, 341], Yes, No, No, No)."},"skills":["The Last Equality Flag Is a Bijection-Supported Certificate"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0076","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bc8afcf81ccd1b06f7550212aea22e0e5d1d61abba86c7df94f6dae9bf3f622","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305076,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[0,1],\"target\":[100,101,102]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[0, 1] and target=[100, 101, 102] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(2, 3, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0077","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a77b81ce7b7dbdcaddda0ef5371f4a455bc360b570258bca5809ed6a04747cf7","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305077,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[10,11,12],\"target\":[110,111,112,113]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[10, 11, 12] and target=[110, 111, 112, 113] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(3, 4, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0078","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c54b4f2940e5059ae96ab6ff4f58288c7c8da8e672d8e39f4846d1d40d8295e","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305078,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[20,21,22,23],\"target\":[120,121,122,123,124]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[20, 21, 22, 23] and target=[120, 121, 122, 123, 124] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(4, 5, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0079","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96cbae4f2c5e47fc862d03e0854e92defb478621739ddacd0b76d0ea19777742","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305079,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[30,31,32,33,34],\"target\":[130,131]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[30, 31, 32, 33, 34] and target=[130, 131] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(5, 2, Yes, no_injection_finite_pigeonhole)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0080","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65c5017ee9304d6f1b678e1cd22d4da5f8a61155e5dbe411b4d9402e4d59450a","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305080,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[40,41],\"target\":[140,141,142]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[40, 41] and target=[140, 141, 142] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(2, 3, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0081","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d4201f9f2fd8cf8884a6bd06314515b99b332adc42cfbd724cbd59b4a07b21a","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305081,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[50,51,52],\"target\":[150,151,152,153]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[50, 51, 52] and target=[150, 151, 152, 153] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(3, 4, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0082","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e77719342676ca7fd351998dc98fc1889b000b7889b9bc960faac2a87791978","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305082,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[60,61,62,63],\"target\":[160,161,162,163,164]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[60, 61, 62, 63] and target=[160, 161, 162, 163, 164] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(4, 5, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0083","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88d9cc3299db25034c0099b247b81297dd4af53c97a7bd79b5d2a5468ac75ebb","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305083,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[70,71,72,73,74],\"target\":[170,171]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[70, 71, 72, 73, 74] and target=[170, 171] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(5, 2, Yes, no_injection_finite_pigeonhole)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0084","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"254824e9d35c09362cbc8edcb88e4dbc15e12bf51669d9acbd7d69b0c9a84e45","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305084,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[80,81],\"target\":[180,181,182]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[80, 81] and target=[180, 181, 182] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(2, 3, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0085","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8428063ffbd3d708c07988b77357a1f0a80af576be6aa8db9b2acf8912db0496","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305085,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[90,91,92],\"target\":[190,191,192,193]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[90, 91, 92] and target=[190, 191, 192, 193] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(3, 4, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0086","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c63427b6bb5531bfb7a5968dfdeb2a78063178502676c7f6d009a3f982d2ad1","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305086,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[100,101,102,103],\"target\":[200,201,202,203,204]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[100, 101, 102, 103] and target=[200, 201, 202, 203, 204] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(4, 5, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0087","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"751ae8653c7b253568937c8ff3db53c2ef9b46c6562916859bf78cdfa26d1e00","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305087,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[110,111,112,113,114],\"target\":[210,211]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[110, 111, 112, 113, 114] and target=[210, 211] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(5, 2, Yes, no_injection_finite_pigeonhole)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0088","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"251ab8cbe92b69bafafc80cd3f7b24a5999bc3c43d7563d6b09778c247d78d78","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305088,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[120,121],\"target\":[220,221,222]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[120, 121] and target=[220, 221, 222] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(2, 3, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0089","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1063343aa5e799af66cd49238fe1c61a6246cb42a775dbaddfa9b9034d784c9","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305089,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[130,131,132],\"target\":[230,231,232,233]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[130, 131, 132] and target=[230, 231, 232, 233] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(3, 4, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0090","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54bd1221f10aa9ddd8e87a262ff00376d77ba0270f0ae956d592bbfa6e4c91e4","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305090,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[140,141,142,143],\"target\":[240,241,242,243,244]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[140, 141, 142, 143] and target=[240, 241, 242, 243, 244] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(4, 5, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0091","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab15dfdc93172d243187782628a3a2f27b6f5b731200677764dc19032607fb48","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305091,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[150,151,152,153,154],\"target\":[250,251]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[150, 151, 152, 153, 154] and target=[250, 251] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(5, 2, Yes, no_injection_finite_pigeonhole)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0092","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6756eed2a27199acdfb375bce81a12ced22c66c57ed05ac38277b5dd917951bf","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305092,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[160,161],\"target\":[260,261,262]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[160, 161] and target=[260, 261, 262] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(2, 3, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0093","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecf677df105bd6cb4f4201da0dbfda6043e3217d3ab5c260af73b85c410420ab","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305093,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[170,171,172],\"target\":[270,271,272,273]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[170, 171, 172] and target=[270, 271, 272, 273] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(3, 4, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0094","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eca1ee7d44a5a9848b4718b9b11dbd7a6a4dd7ac15d74932e1e8aac912a286d","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305094,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[180,181,182,183],\"target\":[280,281,282,283,284]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[180, 181, 182, 183] and target=[280, 281, 282, 283, 284] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(4, 5, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0095","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa3f70f60bf8bcd9caface161b50037d914d96445f75f33a959efabb528c4087","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305095,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[190,191,192,193,194],\"target\":[290,291]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[190, 191, 192, 193, 194] and target=[290, 291] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(5, 2, Yes, no_injection_finite_pigeonhole)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0096","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"005a1c6dc60e7efe5989a92853191d4a918c16f83c9a1eafa35f811981ecaef4","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305096,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[200,201],\"target\":[300,301,302]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[200, 201] and target=[300, 301, 302] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(2, 3, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0097","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0a67d44948189abace65affa25b515933062f3fd113c975e5623b6c05cb911c","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305097,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[210,211,212],\"target\":[310,311,312,313]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[210, 211, 212] and target=[310, 311, 312, 313] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(3, 4, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (3, 4, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0098","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"958e0adfded9d73aaa47822d55176b101343d9d42bcfc7140d32b4c24c8a1e32","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305098,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[220,221,222,223],\"target\":[320,321,322,323,324]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[220, 221, 222, 223] and target=[320, 321, 322, 323, 324] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(4, 5, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (4, 5, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0099","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55839216aa9a606e37c7131bcf6a0dc29237d19f413013ba9aa7beda62005b99","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305099,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[230,231,232,233,234],\"target\":[330,331]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[230, 231, 232, 233, 234] and target=[330, 331] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(5, 2, Yes, no_injection_finite_pigeonhole)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (5, 2, Yes, no_injection_finite_pigeonhole)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r05-cardinality-comparison-0100","batch_id":"batch-20261014-advance-20261030-r05-cardinality-comparison-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba57a0b372894364b5bf70a55ef6e70d29f67537284d90e507bc349610c4b672","family_id":"qf-mod30-cardinality-pigeonhole","seed":202610305100,"semantic_template":"qf-mod30-cardinality-pigeonhole/{\"source\":[240,241],\"target\":[340,341,342]}"},"fields":{"title":"A Larger Finite Source Rules Out Every Injection","stem":"Use the complete finite source=[240, 241] and target=[340, 341, 342] as distinct-element lists in their original orders. A mapping must be total on that exact source and every image must belong to that exact target. Injectivity and surjectivity are independent mapping properties, not guesses from equal list lengths. The comparison certificates here are finite-only; no unrestricted infinite-set or choice conclusion is inferred. No mapping is supplied in this family. Give (source_size,target_size,larger?,conclusion). conclusion is no_injection_finite_pigeonhole or no_impossibility_certificate.","answer":{"kind":"expression","value":"(2, 3, No, no_impossibility_certificate)","explanation":"Compare the finite set sizes. A strictly larger source rules out an injection by the finite pigeonhole principle. A No larger flag supplies no such impossibility certificate; it is not the failure of a particular mapping. The exact result, retaining the requested order, is (2, 3, No, no_impossibility_certificate)."},"skills":["A Larger Finite Source Rules Out Every Injection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}