{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261016-r07-surjective-0001","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e151efa8e8a896b37904c23f7260fe87469fb31defafd9e3bf493b60dfcbd339","family_id":"qf-density16-surj-fiber-inventory","seed":202610167001,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[],\"m\":0,\"n\":0}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([], [], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0002","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ed70e25e295e1dc06497cf156c8c72616a41c4bf3c55372786670236f02c069","family_id":"qf-density16-surj-fiber-inventory","seed":202610167002,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[],\"m\":2,\"n\":0}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(0)=∅→Fin(2), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([], [[], []], [0, 1], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[], []], [0, 1], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0003","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e63e2445054c7099560cc2c34831b48e16c3bcf810fe128ce10f44206a1a6453","family_id":"qf-density16-surj-fiber-inventory","seed":202610167003,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,1,1],\"m\":2,\"n\":3}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 1, 1], [[0], [1, 2]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [[0], [1, 2]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0004","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7d7e724d06f67f0b005f33f61cd6bb5d9232147e189582fdfcf286bd23c4afa","family_id":"qf-density16-surj-fiber-inventory","seed":202610167004,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 0, 0, 0, 1], [[0, 1, 2, 3], [4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [[0, 1, 2, 3], [4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0005","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0425276a20cc310d9d0ebf606ffb2abefb9fd174cbc919d583c7ca983f8ce922","family_id":"qf-density16-surj-fiber-inventory","seed":202610167005,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[1,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([1, 0, 0, 0, 1], [[1, 2, 3], [0, 4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [[1, 2, 3], [0, 4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0006","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cea3aedc5889dfd55f6d87c9a001e07a757c866c950675920fef47cea54be84","family_id":"qf-density16-surj-fiber-inventory","seed":202610167006,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[2,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([2, 0, 0, 0, 1], [[1, 2, 3], [4], [0]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [[1, 2, 3], [4], [0]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0007","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6938d3c7e6b45351b7c026be752263bed3823dc45c778d77bdf1dc729fe1288c","family_id":"qf-density16-surj-fiber-inventory","seed":202610167007,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 1, 0, 0, 1], [[0, 2, 3], [1, 4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [[0, 2, 3], [1, 4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0008","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bf9b901ed5ed179089490ab9f282585e3db0676ceab749368da735287bc4e92","family_id":"qf-density16-surj-fiber-inventory","seed":202610167008,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[1,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([1, 1, 0, 0, 1], [[2, 3], [0, 1, 4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [[2, 3], [0, 1, 4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0009","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2cdcd8a02d0ade9e9f33b5661b9157c3bc35ab3e9b66e0002af09e183522556","family_id":"qf-density16-surj-fiber-inventory","seed":202610167009,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[2,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([2, 1, 0, 0, 1], [[2, 3], [1, 4], [0]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [[2, 3], [1, 4], [0]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0010","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d82075c2ebc87d4fff00010dfa7e63ef5fa42bf9063b7f9591e6a84e9032efad","family_id":"qf-density16-surj-fiber-inventory","seed":202610167010,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 2, 0, 0, 1], [[0, 2, 3], [4], [1]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [[0, 2, 3], [4], [1]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0011","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc7478d5addac85e3ef99d3dc631c3c80c59b78cdb5504c4f27dc63671284083","family_id":"qf-density16-surj-fiber-inventory","seed":202610167011,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[1,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([1, 2, 0, 0, 1], [[2, 3], [0, 4], [1]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [[2, 3], [0, 4], [1]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0012","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb32ac58a9a5985a8dcd681a8e8deac2948895872e6f52e82d31ba216f341eaa","family_id":"qf-density16-surj-fiber-inventory","seed":202610167012,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[2,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([2, 2, 0, 0, 1], [[2, 3], [4], [0, 1]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [[2, 3], [4], [0, 1]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0013","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24c06e6fef9a027c55e2631daa3f87e3fc71ff05ac4e0de1905d12b8ea455c54","family_id":"qf-density16-surj-fiber-inventory","seed":202610167013,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 0, 1, 0, 1], [[0, 1, 3], [2, 4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [[0, 1, 3], [2, 4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0014","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15b67c61dff83c959182355699a823870d7b09e2090fd66126d8ec18fc9aa5df","family_id":"qf-density16-surj-fiber-inventory","seed":202610167014,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[1,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([1, 0, 1, 0, 1], [[1, 3], [0, 2, 4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [[1, 3], [0, 2, 4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0015","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94839dddf3b9243698fce01ea77e3bc0dbfebc282d21c66e2ab79697a35be357","family_id":"qf-density16-surj-fiber-inventory","seed":202610167015,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[2,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([2, 0, 1, 0, 1], [[1, 3], [2, 4], [0]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [[1, 3], [2, 4], [0]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0016","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"daeb2d2183735f1fa73507964e247dbad87146b7ce4fa6c0c99f4fc0f248f4b0","family_id":"qf-density16-surj-fiber-inventory","seed":202610167016,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 1, 1, 0, 1], [[0, 3], [1, 2, 4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [[0, 3], [1, 2, 4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0017","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"589c76f07e85c72f688e759e0c3e6e09672964c078d859ee68a4b800c64f5e56","family_id":"qf-density16-surj-fiber-inventory","seed":202610167017,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[1,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([1, 1, 1, 0, 1], [[3], [0, 1, 2, 4], []], [2], No)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [[3], [0, 1, 2, 4], []], [2], No)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0018","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2de0593a0481e9aa9da5ca4d6dfca941946b4a626d3e965c7f3e8c0c184e4df4","family_id":"qf-density16-surj-fiber-inventory","seed":202610167018,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[2,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([2, 1, 1, 0, 1], [[3], [1, 2, 4], [0]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [[3], [1, 2, 4], [0]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0019","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e65f5a369b0f9f4e8aea52646c3d9abb7efad9f422f46fa4075db6a80cdf3a6","family_id":"qf-density16-surj-fiber-inventory","seed":202610167019,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 2, 1, 0, 1], [[0, 3], [2, 4], [1]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [[0, 3], [2, 4], [1]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0020","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30f11e3c5fcef801075755f2786df54621d6ac228b2b96a25572c14ea646c713","family_id":"qf-density16-surj-fiber-inventory","seed":202610167020,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[1,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([1, 2, 1, 0, 1], [[3], [0, 2, 4], [1]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [[3], [0, 2, 4], [1]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0021","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"390fa0b0e59121706f46ac609fadfdc0535b564cde41013b299e5c608d920b7e","family_id":"qf-density16-surj-fiber-inventory","seed":202610167021,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[2,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([2, 2, 1, 0, 1], [[3], [2, 4], [0, 1]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [[3], [2, 4], [0, 1]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0022","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e11e532fbdbcda9c1b0cc18547766390e84bdb1a3ce6369a6e2bb5cbfc3f50f","family_id":"qf-density16-surj-fiber-inventory","seed":202610167022,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 0, 2, 0, 1], [[0, 1, 3], [4], [2]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [[0, 1, 3], [4], [2]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0023","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a55e7bd93bd6c783bef877091d4b529a8761676c0ee2f8c9877335340de18703","family_id":"qf-density16-surj-fiber-inventory","seed":202610167023,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[1,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([1, 0, 2, 0, 1], [[1, 3], [0, 4], [2]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [[1, 3], [0, 4], [2]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0024","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84d61e5ebcd3441e7ffc3fd4812ed79ba65cfe78f4c1b3797e5c879f22da2285","family_id":"qf-density16-surj-fiber-inventory","seed":202610167024,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[2,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([2, 0, 2, 0, 1], [[1, 3], [4], [0, 2]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [[1, 3], [4], [0, 2]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0025","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7357cea09d37632084d2dd407e8149d2d081efd85bfdbb66101884cb1b12f3d0","family_id":"qf-density16-surj-fiber-inventory","seed":202610167025,"semantic_template":"qf-density16-surj-fiber-inventory/{\"images\":[0,1,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Every Declared Codomain Fiber and Missing Target","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.","answer":{"kind":"expression","value":"([0, 1, 2, 0, 1], [[0, 3], [1, 4], [2]], [], Yes)","explanation":"List every declared target’s source-index fiber; any empty fiber prevents surjectivity. The empty-to-empty map is surjective, unlike an empty source with nonempty codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [[0, 3], [1, 4], [2]], [], Yes)."},"skills":["Every Declared Codomain Fiber and Missing Target"],"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-20261012-advance-20261016-r07-surjective-0026","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bb18792624db4fcc30c8b07fa50c7b4d643d7961ff313a77dfc01ec83d91385","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167026,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[],\"images\":[],\"m\":0,\"n\":0}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([], [], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0027","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdc4f4aa4923549bad50d28126d0379d9b393e8a7986db5ccfde06e5c6b50f8d","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167027,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[-1],\"images\":[],\"m\":1,\"n\":0}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(0)=∅→Fin(1), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[-1] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([], [-1], Yes, [-1], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [-1], Yes, [-1], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0028","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57de0752faac2e5aa89f7a6c80253fa04c75ff74c601d35306eda28421780be3","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167028,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0],\"images\":[0,1],\"m\":3,\"n\":2}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 1], [0], No, outside-codomain, not-evaluated)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0], No, outside-codomain, not-evaluated)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0029","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cfb08b8bd1a36edbfc05b3bf45648a8067ff523b4d24d9716672fcda6f3da4b","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167029,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1],\"images\":[0,1],\"m\":3,\"n\":2}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 1], [0, 1], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0, 1], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0030","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3618b1d2cad8a8197457bab8c16e8005fb1b13030a1da789837100264849084f","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167030,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,1],\"m\":3,\"n\":2}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0031","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7ce9e0c5bd6548974bcff7ca3906606d3a3f5c7ded59e234e1dba471b7ae3c3","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167031,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0032","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2840f4cfa081ae2a83478bff4e9d6966980f2115e7f3bc8f1de941a6551fea41","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167032,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[1,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([1, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0033","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd0bbdb0c20ed4ae773b424090a2e42521c2bd89ee030903261a5117a203dd7c","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167033,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[2,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([2, 0, 0, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0034","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3910c295ba7b4aaf6cff00cc042a38f1fbc4187a9f56a96fc92ee8be051b2394","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167034,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 1, 0, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0035","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c816d2244a1071adb551d26e3bad2f188319ba70327d5d21763891df2dbb52b8","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167035,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[1,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([1, 1, 0, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0036","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23569f795b49e81c0bf9fdcc1f6c156fa4cd501c0de583e565b938a4f507e4ee","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167036,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[2,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0037","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d5826fb31a62d46c7b462fb1391bfa154c8fc1511e11bc2e3eb7bba95bd5d16","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167037,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0038","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca003279268e26de1df3acca3c308839fb3d961edb8e76f41667cc5d1977cf5f","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167038,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[1,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([1, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0039","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a97e79eec678c14169fbec706c75da17812eb2d705c8636dc17b0597897afe1","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167039,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[2,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([2, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0040","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f3fe70a15e5f2b4aca0909c5b0b9fef8c6d7fd48d4e75dd15e92d0145b3b4af","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167040,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 0, 1, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0041","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6a2aef65dd60caf296882145556f1ba9c32ee0f6cf1c5f82e9b86ac9429516d","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167041,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[1,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([1, 0, 1, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0042","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9932c1af278853ecee3b2fe86f79680de65edba88ebb4578656a68c99131745b","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167042,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[2,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([2, 0, 1, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0043","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4591a9d2888c381dbd616b17f1bed1b1841bb7a4cc92632ff94ab653a8d104be","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167043,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 1, 1, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0044","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c047b808294d6d5c0544df8be75b3cdc5de211226340fd9f647e5e3b0ccddf7","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167044,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[1,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([1, 1, 1, 0, 1], [0, 1, 2], Yes, [2], No)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [0, 1, 2], Yes, [2], No)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0045","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e49019d0a600f4c54c96691f2336e4b444b3e26d8796ea3da8eca1fb28d6dac","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167045,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[2,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([2, 1, 1, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0046","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19d706acbfd5a1802d3f9b0bbd87cae7a58276d3eae2c024957d29caf3084ae1","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167046,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0047","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b105914f8b5338bcc1242229e083f3f9123cfc1d684305882f30eae38e7ab280","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167047,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[1,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([1, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0048","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddb750e77d0fe581aa5b8eebf6f6ac958a04b32051376df64161aa5e64541086","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167048,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[2,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([2, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0049","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3612b9ce4298f0dd34c41ae51c5c587dccef01d943423407c09513137371685a","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167049,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[0,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([0, 0, 2, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0050","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6886501938507ffb907cc20072a05602dd7a5269a7871f90073ed0cb92bd442c","family_id":"qf-density16-surj-changed-codomain-qualification","seed":202610167050,"semantic_template":"qf-density16-surj-changed-codomain-qualification/{\"codomain\":[0,1,2],\"images\":[1,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Changed Codomain Must Contain All Original Outputs","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).","answer":{"kind":"expression","value":"([1, 0, 2, 0, 1], [0, 1, 2], Yes, [], Yes)","explanation":"Retain and qualify every existing output before computing missing targets; dropping offending outputs cannot legalize the new codomain. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [0, 1, 2], Yes, [], Yes)."},"skills":["Changed Codomain Must Contain All Original Outputs"],"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-20261012-advance-20261016-r07-surjective-0051","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcf7df98a4c03deb2c7dafa7239111c04a2ba6a8aa613986fb29bc2c85c697c2","family_id":"qf-density16-surj-delete-input-missing","seed":202610167051,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":0,\"images\":[0],\"m\":1,\"n\":1}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0], [], [], [], [0], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], [], [], [], [0], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0052","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"216c0597c1a1ab9fb8211dc2c5cdf0f9f0802bf167b0cbc903494c78961c2155","family_id":"qf-density16-surj-delete-input-missing","seed":202610167052,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":1,\"images\":[0,1,1],\"m\":2,\"n\":3}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 1, 1], [], [0, 2], [0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [], [0, 2], [0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0053","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b71589c650a45606e9f143b04ae55105afccd45faa392aa58051aafb0cf949f","family_id":"qf-density16-surj-delete-input-missing","seed":202610167053,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":2,\"images\":[0,1,1],\"m\":3,\"n\":3}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 1, 1], [2], [0, 1], [0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1], [2], [0, 1], [0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0054","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fe192ef131a11944c1b9fe81eb15187c63a93f539f285a85cf02418aa9b6f18","family_id":"qf-density16-surj-delete-input-missing","seed":202610167054,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":0,\"images\":[0,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 0, 0, 0, 1], [2], [1, 2, 3, 4], [0, 0, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [2], [1, 2, 3, 4], [0, 0, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0055","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6297abaadb188a4213dda6e561a24ccd2a5e38350edccc60a62ef0c828ea6fba","family_id":"qf-density16-surj-delete-input-missing","seed":202610167055,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":1,\"images\":[1,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([1, 0, 0, 0, 1], [2], [0, 2, 3, 4], [1, 0, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [2], [0, 2, 3, 4], [1, 0, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0056","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4621f0eab4329d3bb4232eedbfa0ea83dc30089afafe72e395002253f2bd23a6","family_id":"qf-density16-surj-delete-input-missing","seed":202610167056,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":2,\"images\":[2,0,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([2, 0, 0, 0, 1], [], [0, 1, 3, 4], [2, 0, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [], [0, 1, 3, 4], [2, 0, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0057","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d90cc14e9359e4d9b17a9d52505883481e154d9c927add006b64217760e781a8","family_id":"qf-density16-surj-delete-input-missing","seed":202610167057,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":3,\"images\":[0,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 1, 0, 0, 1], [2], [0, 1, 2, 4], [0, 1, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [2], [0, 1, 2, 4], [0, 1, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0058","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa2d9f9ebe2819426773dab083958dacfe7b681671c22c5660a5d74cf1dee591","family_id":"qf-density16-surj-delete-input-missing","seed":202610167058,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":4,\"images\":[1,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([1, 1, 0, 0, 1], [2], [0, 1, 2, 3], [1, 1, 0, 0], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [2], [0, 1, 2, 3], [1, 1, 0, 0], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0059","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54d20687770645b02d99625841d7dab9c71952d23466669ea97cbe8bc5068688","family_id":"qf-density16-surj-delete-input-missing","seed":202610167059,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":0,\"images\":[2,1,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([2, 1, 0, 0, 1], [], [1, 2, 3, 4], [1, 0, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [], [1, 2, 3, 4], [1, 0, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0060","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5964d84caf3d1419e34ae386f236cd5e304ec7eb4bdeb771ceda8cabb09c94c5","family_id":"qf-density16-surj-delete-input-missing","seed":202610167060,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":1,\"images\":[0,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 2, 0, 0, 1], [], [0, 2, 3, 4], [0, 0, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [], [0, 2, 3, 4], [0, 0, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0061","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6424b5f4c6f357daec2d03f8cd97cc842303ea6a275ac511b99ed4838afa3c2","family_id":"qf-density16-surj-delete-input-missing","seed":202610167061,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":2,\"images\":[1,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([1, 2, 0, 0, 1], [], [0, 1, 3, 4], [1, 2, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [], [0, 1, 3, 4], [1, 2, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0062","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd8827bca4fc2978d1a692b9c566c43b86c2e89dae33095ffb852b7be9fae267","family_id":"qf-density16-surj-delete-input-missing","seed":202610167062,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":3,\"images\":[2,2,0,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([2, 2, 0, 0, 1], [], [0, 1, 2, 4], [2, 2, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [], [0, 1, 2, 4], [2, 2, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0063","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89f3d80efcca47f95f881a639f988518da22e1590307fb6fd94dca50440771f1","family_id":"qf-density16-surj-delete-input-missing","seed":202610167063,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":4,\"images\":[0,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 0, 1, 0, 1], [2], [0, 1, 2, 3], [0, 0, 1, 0], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [2], [0, 1, 2, 3], [0, 0, 1, 0], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0064","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15779c169373a34ffb52a3182b86a34cecf073136029eaa45dea2df2918a93cb","family_id":"qf-density16-surj-delete-input-missing","seed":202610167064,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":0,\"images\":[1,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([1, 0, 1, 0, 1], [2], [1, 2, 3, 4], [0, 1, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [2], [1, 2, 3, 4], [0, 1, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0065","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2dfa60ecd22fda806ad1a60a37cb03a3e76b9aea07cf630cc26416e136842057","family_id":"qf-density16-surj-delete-input-missing","seed":202610167065,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":1,\"images\":[2,0,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([2, 0, 1, 0, 1], [], [0, 2, 3, 4], [2, 1, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [], [0, 2, 3, 4], [2, 1, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0066","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"645277c3b1edad2d57d61202a904120887248d44d8e0d3bf8af9c74c78a30c44","family_id":"qf-density16-surj-delete-input-missing","seed":202610167066,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":2,\"images\":[0,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 1, 1, 0, 1], [2], [0, 1, 3, 4], [0, 1, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [2], [0, 1, 3, 4], [0, 1, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0067","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edef61d0e7b209bfb28c8e584d011b35285fe15e85f69e10f899decdd5e20598","family_id":"qf-density16-surj-delete-input-missing","seed":202610167067,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":3,\"images\":[1,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([1, 1, 1, 0, 1], [2], [0, 1, 2, 4], [1, 1, 1, 1], [0, 2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [2], [0, 1, 2, 4], [1, 1, 1, 1], [0, 2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0068","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc73c9d83bd1eced5ab363845398cc53a18bd2b112f6b06965e8ff143c1212ce","family_id":"qf-density16-surj-delete-input-missing","seed":202610167068,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":4,\"images\":[2,1,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([2, 1, 1, 0, 1], [], [0, 1, 2, 3], [2, 1, 1, 0], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [], [0, 1, 2, 3], [2, 1, 1, 0], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0069","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbc64eac6670f722bf7eb72b5d5cd437e89195dfdb483cfe452d944ca71a2d94","family_id":"qf-density16-surj-delete-input-missing","seed":202610167069,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":0,\"images\":[0,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 2, 1, 0, 1], [], [1, 2, 3, 4], [2, 1, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [], [1, 2, 3, 4], [2, 1, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0070","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eef66f50d47c6ff9cc290eb7a76b6c2ff3532912a7b2b3ddbce46d1a57825858","family_id":"qf-density16-surj-delete-input-missing","seed":202610167070,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":1,\"images\":[1,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([1, 2, 1, 0, 1], [], [0, 2, 3, 4], [1, 1, 0, 1], [2], No)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [], [0, 2, 3, 4], [1, 1, 0, 1], [2], No)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0071","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9d9aceab475cdb89a8e04e7fdfe0d878b3a55f74977353179f953620054a4cf","family_id":"qf-density16-surj-delete-input-missing","seed":202610167071,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":2,\"images\":[2,2,1,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([2, 2, 1, 0, 1], [], [0, 1, 3, 4], [2, 2, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [], [0, 1, 3, 4], [2, 2, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0072","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44f9c6ba9ae362f669d71d2c902100ac8ee0a5d5ed8959f8db250e2304197f47","family_id":"qf-density16-surj-delete-input-missing","seed":202610167072,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":3,\"images\":[0,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 0, 2, 0, 1], [], [0, 1, 2, 4], [0, 0, 2, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [], [0, 1, 2, 4], [0, 0, 2, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0073","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ec1431b784205dd8bb7493f119f3182bffe9858b937f8ea67eba2e96be85d4f","family_id":"qf-density16-surj-delete-input-missing","seed":202610167073,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":4,\"images\":[1,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([1, 0, 2, 0, 1], [], [0, 1, 2, 3], [1, 0, 2, 0], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [], [0, 1, 2, 3], [1, 0, 2, 0], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0074","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aa6ed03894cae3e0a475f7ccebf99f1abf4597ac8529c23d11862d6ac5c9f80","family_id":"qf-density16-surj-delete-input-missing","seed":202610167074,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":0,\"images\":[2,0,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([2, 0, 2, 0, 1], [], [1, 2, 3, 4], [0, 2, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [], [1, 2, 3, 4], [0, 2, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0075","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52a27a4423aeac4c5ffc3661131f4f235cca36b0658334b03e61b9ee30eed8e4","family_id":"qf-density16-surj-delete-input-missing","seed":202610167075,"semantic_template":"qf-density16-surj-delete-input-missing/{\"at\":1,\"images\":[0,1,2,0,1],\"m\":3,\"n\":5}"},"fields":{"title":"Source Deletion with Declared Codomain Kept","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).","answer":{"kind":"expression","value":"([0, 1, 2, 0, 1], [], [0, 2, 3, 4], [0, 2, 0, 1], [], Yes)","explanation":"Remove the source entry while preserving its other original labels and the declared target set; deleting a unique preimage can create a newly missing target. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [], [0, 2, 3, 4], [0, 2, 0, 1], [], Yes)."},"skills":["Source Deletion with Declared Codomain Kept"],"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-20261012-advance-20261016-r07-surjective-0076","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c708da7858561bcead512a4f2510d8e8d23b9648f6c57dd59dfd09d4d2f0155b","family_id":"qf-density16-surj-two-target-preimages","seed":202610167076,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[],\"m\":0,\"n\":0,\"queries\":[0,-1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, -1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([], [No, No], [outside-codomain, outside-codomain], [outside-codomain, outside-codomain])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [No, No], [outside-codomain, outside-codomain], [outside-codomain, outside-codomain])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0077","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cb34271a03000be5a0e9e26173d723e56227c1f8df4df8eb0d3062180880084","family_id":"qf-density16-surj-two-target-preimages","seed":202610167077,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[],\"m\":2,\"n\":0,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(0)=∅→Fin(2), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([], [Yes, Yes], [[], []], [does-not-exist, does-not-exist])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [Yes, Yes], [[], []], [does-not-exist, does-not-exist])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0078","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4eaa8a7958a693bef7fe39b4ef2e7c438a280fe01bbe7c4e66894ac51b0da3c5","family_id":"qf-density16-surj-two-target-preimages","seed":202610167078,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,1],\"m\":3,\"n\":3,\"queries\":[2,3]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[2, 3]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 1], [Yes, No], [[], outside-codomain], [does-not-exist, outside-codomain])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [Yes, No], [[], outside-codomain], [does-not-exist, outside-codomain])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0079","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f5625ef3088f4afb5480ee6fa0c152116709ef0724655d326b0e875688bb491","family_id":"qf-density16-surj-two-target-preimages","seed":202610167079,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,1],\"m\":2,\"n\":3,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 1], [Yes, Yes], [[0, 1], [2]], [0, 2])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [Yes, Yes], [[0, 1], [2]], [0, 2])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0080","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ba10d9a008e48df8c6eb04c6a12015244d6d3980f980bbb685292f3227669c4","family_id":"qf-density16-surj-two-target-preimages","seed":202610167080,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,0,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 0, 0, 0], [Yes, Yes], [[0, 1, 2, 3, 4], []], [0, does-not-exist])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [Yes, Yes], [[0, 1, 2, 3, 4], []], [0, does-not-exist])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0081","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d210bfeb89bb176b8275ea0d058c6eed21201788c7747a62043f92a695e07e0","family_id":"qf-density16-surj-two-target-preimages","seed":202610167081,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,0,0,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 0, 0, 0, 0], [Yes, Yes], [[1, 2, 3, 4], [0]], [1, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], [Yes, Yes], [[1, 2, 3, 4], [0]], [1, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0082","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ee50bcd0468e23003a13c6a9bf34d41dcc406d89ff0b59ef49b0de9e8153873","family_id":"qf-density16-surj-two-target-preimages","seed":202610167082,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,1,0,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 1, 0, 0, 0], [Yes, Yes], [[0, 2, 3, 4], [1]], [0, 1])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], [Yes, Yes], [[0, 2, 3, 4], [1]], [0, 1])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0083","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36b0cfb9f10206b8770fd43757d21e237be94f14b8ba13ba2df471c0a842298e","family_id":"qf-density16-surj-two-target-preimages","seed":202610167083,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,1,0,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 1, 0, 0, 0], [Yes, Yes], [[2, 3, 4], [0, 1]], [2, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], [Yes, Yes], [[2, 3, 4], [0, 1]], [2, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0084","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"083f24d5f7e95ace803f94a1f24f51c12a3728618d25407f3d3469fcf045d574","family_id":"qf-density16-surj-two-target-preimages","seed":202610167084,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,1,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 1, 0, 0], [Yes, Yes], [[0, 1, 3, 4], [2]], [0, 2])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [Yes, Yes], [[0, 1, 3, 4], [2]], [0, 2])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0085","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"945d9fad31e61fc4559bad85dc40784df6198ec7d63596beee77bf986c852a67","family_id":"qf-density16-surj-two-target-preimages","seed":202610167085,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,0,1,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 0, 1, 0, 0], [Yes, Yes], [[1, 3, 4], [0, 2]], [1, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], [Yes, Yes], [[1, 3, 4], [0, 2]], [1, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0086","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7cf912edcdd5b5ca1fa04e850ac1cf4a9e5e28d274040c19f4af18ede47b0a8","family_id":"qf-density16-surj-two-target-preimages","seed":202610167086,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,1,1,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 1, 1, 0, 0], [Yes, Yes], [[0, 3, 4], [1, 2]], [0, 1])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], [Yes, Yes], [[0, 3, 4], [1, 2]], [0, 1])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0087","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"217f25c8719554d7471eeae5c77b63326b428576573301c5b19195f2175a8924","family_id":"qf-density16-surj-two-target-preimages","seed":202610167087,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,1,1,0,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 1, 1, 0, 0], [Yes, Yes], [[3, 4], [0, 1, 2]], [3, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], [Yes, Yes], [[3, 4], [0, 1, 2]], [3, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0088","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b27ef4055e0f51865d9c0b4f7e6619a60f39c762941e340be71eb742d4e8d450","family_id":"qf-density16-surj-two-target-preimages","seed":202610167088,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,0,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 0, 1, 0], [Yes, Yes], [[0, 1, 2, 4], [3]], [0, 3])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 1, 0], [Yes, Yes], [[0, 1, 2, 4], [3]], [0, 3])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0089","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a16aea6fc5a454a6a4927207ce04afa4635db301a6a1d3ef119e67cd54df3e6","family_id":"qf-density16-surj-two-target-preimages","seed":202610167089,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,0,0,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 0, 0, 1, 0], [Yes, Yes], [[1, 2, 4], [0, 3]], [1, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 1, 0], [Yes, Yes], [[1, 2, 4], [0, 3]], [1, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0090","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"318bacfbf5c1747275a0af5089d690813928eea3f75e2d6e10a2daef839e525b","family_id":"qf-density16-surj-two-target-preimages","seed":202610167090,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,1,0,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 1, 0, 1, 0], [Yes, Yes], [[0, 2, 4], [1, 3]], [0, 1])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0], [Yes, Yes], [[0, 2, 4], [1, 3]], [0, 1])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0091","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6b3066c88b04016fd3845df905e00cc4469389a7724390eb9f1c7e202120500","family_id":"qf-density16-surj-two-target-preimages","seed":202610167091,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,1,0,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 1, 0, 1, 0], [Yes, Yes], [[2, 4], [0, 1, 3]], [2, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 1, 0], [Yes, Yes], [[2, 4], [0, 1, 3]], [2, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0092","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74932cfd730c5afcfcf44d630392933716606c5ada671fdb0667f0abc76adbf9","family_id":"qf-density16-surj-two-target-preimages","seed":202610167092,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,1,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 1, 1, 0], [Yes, Yes], [[0, 1, 4], [2, 3]], [0, 2])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 1, 0], [Yes, Yes], [[0, 1, 4], [2, 3]], [0, 2])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0093","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"adf3a2d30a0327812af44c531d9698862e419ceeec7db33b6c9a88b1fcfe57c6","family_id":"qf-density16-surj-two-target-preimages","seed":202610167093,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,0,1,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 0, 1, 1, 0], [Yes, Yes], [[1, 4], [0, 2, 3]], [1, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 1, 0], [Yes, Yes], [[1, 4], [0, 2, 3]], [1, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0094","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b20adce36044fb7ae67421da53597735d660ffe4b545e57653788407ab867748","family_id":"qf-density16-surj-two-target-preimages","seed":202610167094,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,1,1,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 1, 1, 1, 0], [Yes, Yes], [[0, 4], [1, 2, 3]], [0, 1])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 1, 0], [Yes, Yes], [[0, 4], [1, 2, 3]], [0, 1])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0095","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a40df9f5fef935c41eaf46c0351afc980226edaf18ec074ab39d15a7c0dd29a5","family_id":"qf-density16-surj-two-target-preimages","seed":202610167095,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,1,1,1,0],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 1, 1, 1, 0], [Yes, Yes], [[4], [0, 1, 2, 3]], [4, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 1, 0], [Yes, Yes], [[4], [0, 1, 2, 3]], [4, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0096","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5317249d83c6c5dfa212b3c6bdbba08a2178a0e1098a8ffbe2e63d168879a949","family_id":"qf-density16-surj-two-target-preimages","seed":202610167096,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,0,0,1],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 0, 0, 1], [Yes, Yes], [[0, 1, 2, 3], [4]], [0, 4])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [Yes, Yes], [[0, 1, 2, 3], [4]], [0, 4])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0097","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a728b394e7242a763636de3124243ab876379f8cccea8992f6e35d90dd435b2","family_id":"qf-density16-surj-two-target-preimages","seed":202610167097,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,0,0,0,1],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 0, 0, 0, 1], [Yes, Yes], [[1, 2, 3], [0, 4]], [1, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [Yes, Yes], [[1, 2, 3], [0, 4]], [1, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0098","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b107b812beea94ccabdf97103fda192419284c6cc0c1fb16f7b7444e02605d28","family_id":"qf-density16-surj-two-target-preimages","seed":202610167098,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,1,0,0,1],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 1, 0, 0, 1], [Yes, Yes], [[0, 2, 3], [1, 4]], [0, 1])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [Yes, Yes], [[0, 2, 3], [1, 4]], [0, 1])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0099","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92126d715eebdf5eeeee5ed9f692b40543260037822bc06e22d7509a069ade17","family_id":"qf-density16-surj-two-target-preimages","seed":202610167099,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[1,1,0,0,1],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([1, 1, 0, 0, 1], [Yes, Yes], [[2, 3], [0, 1, 4]], [2, 0])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [Yes, Yes], [[2, 3], [0, 1, 4]], [2, 0])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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-20261012-advance-20261016-r07-surjective-0100","batch_id":"batch-20261012-advance-20261016-r07-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa0c6638c2973c12ab1bd19160298171940b730a4672515dd33f006ca633458e","family_id":"qf-density16-surj-two-target-preimages","seed":202610167100,"semantic_template":"qf-density16-surj-two-target-preimages/{\"images\":[0,0,1,0,1],\"m\":3,\"n\":5,\"queries\":[0,1]}"},"fields":{"title":"Two Target-Type Guards, Complete Fibers, and Least Preimages","stem":"Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).","answer":{"kind":"expression","value":"([0, 0, 1, 0, 1], [Yes, Yes], [[0, 1, 3], [2, 4]], [0, 2])","explanation":"Scan the complete table only for a typed target; a legal empty fiber has no preimage and is not the same as an outside-codomain candidate. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [Yes, Yes], [[0, 1, 3], [2, 4]], [0, 2])."},"skills":["Two Target-Type Guards, Complete Fibers, and Least Preimages"],"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."}}