{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261118-r05-p-adic-valuation-integers-0001","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7017e5919cb79e28383954b5f65ae2ec93e32cc56c3134900b82e92cfdac97fa","family_id":"qf-mod49-pval-factor-extraction","seed":202611185001,"semantic_template":"pval-factor-extraction|{a:2,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=2. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, 2, 2, 1, 1, [2], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 2, 2, 1, 1, [2], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0002","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1003d248580fb0f67150772ee34bb8105b1c73e41429b6aebb1690703ab858a","family_id":"qf-mod49-pval-factor-extraction","seed":202611185002,"semantic_template":"pval-factor-extraction|{a:-18,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=-18. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, -18, 18, 2, 2, [3, 9], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -18, 18, 2, 2, [3, 9], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0003","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"478a3ab75e462ceae81352419ab70643bbe68b7f76fe384e0c71efe9599dd18d","family_id":"qf-mod49-pval-factor-extraction","seed":202611185003,"semantic_template":"pval-factor-extraction|{a:375,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=375. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, 375, 375, 3, 3, [5, 25, 125], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 375, 375, 3, 3, [5, 25, 125], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0004","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a73a794cf6f39b7388b64d32fdfd42e27f9e486a3c2708908ce752c36c9ea1bb","family_id":"qf-mod49-pval-factor-extraction","seed":202611185004,"semantic_template":"pval-factor-extraction|{a:-8,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=-8. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, -8, 8, 3, 1, [2, 4, 8], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -8, 8, 3, 1, [2, 4, 8], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0005","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8abad70668e1f5792b27a3644bb6cf0069ffac070883517f8335395a91e51161","family_id":"qf-mod49-pval-factor-extraction","seed":202611185005,"semantic_template":"pval-factor-extraction|{a:45,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=45. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, 45, 45, 2, 5, [3, 9], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 45, 45, 2, 5, [3, 9], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0006","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"478f559c1b577166e42d477c1d5779eb95c62618528e19b92ea20bb2fa5d1867","family_id":"qf-mod49-pval-factor-extraction","seed":202611185006,"semantic_template":"pval-factor-extraction|{a:-750,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=-750. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, -750, 750, 3, 6, [5, 25, 125], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -750, 750, 3, 6, [5, 25, 125], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0007","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6658b54e2c9090c550e96ff166deac32dfb52619b7fb6b181d6ee9cf857358a7","family_id":"qf-mod49-pval-factor-extraction","seed":202611185007,"semantic_template":"pval-factor-extraction|{a:14,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=14. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, 14, 14, 1, 7, [2], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 14, 14, 1, 7, [2], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0008","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2a2d7d1322e31c08633ce61768df38fb94192a8ccdee9fa984bd1d4e4175c80","family_id":"qf-mod49-pval-factor-extraction","seed":202611185008,"semantic_template":"pval-factor-extraction|{a:-9,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=-9. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, -9, 9, 2, 1, [3, 9], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -9, 9, 2, 1, [3, 9], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0009","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1b82ced65f420ea59b019f9ba223a7191ad48a7fa9f2cb1aecc41b7e956812d","family_id":"qf-mod49-pval-factor-extraction","seed":202611185009,"semantic_template":"pval-factor-extraction|{a:250,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=250. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, 250, 250, 3, 2, [5, 25, 125], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 250, 250, 3, 2, [5, 25, 125], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0010","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98f329f9e17e4af4e32fe6913dc2d0c737d33ead79093aafc3599dcc319c4236","family_id":"qf-mod49-pval-factor-extraction","seed":202611185010,"semantic_template":"pval-factor-extraction|{a:-6,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=-6. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, -6, 6, 1, 3, [2], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -6, 6, 1, 3, [2], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0011","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53b94cc08f2730ed845ae64f33b0054706a2b409c4a0fbeb16c8fb5a2ec52196","family_id":"qf-mod49-pval-factor-extraction","seed":202611185011,"semantic_template":"pval-factor-extraction|{a:36,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=36. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, 36, 36, 2, 4, [3, 9], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 36, 36, 2, 4, [3, 9], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0012","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06d3b1201fe1b159ab4d92c0764b339bdafe3dd0c166d5b8ed43a9ad6405646f","family_id":"qf-mod49-pval-factor-extraction","seed":202611185012,"semantic_template":"pval-factor-extraction|{a:-625,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=-625. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, -625, 625, 4, 1, [5, 25, 125, 625], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -625, 625, 4, 1, [5, 25, 125, 625], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0013","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d0a74478cec502e7a16eaba0575e5a36f146076933c16071d30174d75a602ba","family_id":"qf-mod49-pval-factor-extraction","seed":202611185013,"semantic_template":"pval-factor-extraction|{a:12,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=12. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, 12, 12, 2, 3, [2, 4], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 12, 12, 2, 3, [2, 4], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0014","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d381a4b3e9e1ddb1496d97be1b3521437bad8fe72184c71d7064caf468436f38","family_id":"qf-mod49-pval-factor-extraction","seed":202611185014,"semantic_template":"pval-factor-extraction|{a:-63,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=-63. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, -63, 63, 2, 7, [3, 9], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -63, 63, 2, 7, [3, 9], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0015","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2ebfdeb4fea4fa902747b3659f9e79865947ec30433c6984b85ed819367f114","family_id":"qf-mod49-pval-factor-extraction","seed":202611185015,"semantic_template":"pval-factor-extraction|{a:125,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=125. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, 125, 125, 3, 1, [5, 25, 125], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 125, 125, 3, 1, [5, 25, 125], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0016","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65479fc9e05cd02c6353823b1afbde41f08dbe0db7342707e4c6c3f7bff3e44c","family_id":"qf-mod49-pval-factor-extraction","seed":202611185016,"semantic_template":"pval-factor-extraction|{a:-4,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=-4. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, -4, 4, 2, 1, [2, 4], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -4, 4, 2, 1, [2, 4], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0017","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ca9eddb10e839376938a0c1c409769dde5a29e983d3e9f397491adc91486018","family_id":"qf-mod49-pval-factor-extraction","seed":202611185017,"semantic_template":"pval-factor-extraction|{a:27,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=27. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, 27, 27, 3, 1, [3, 9, 27], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 27, 27, 3, 1, [3, 9, 27], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0018","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9fb01838366abdf09185407c5469dd14d057417925d937d03edfc2884fad68d","family_id":"qf-mod49-pval-factor-extraction","seed":202611185018,"semantic_template":"pval-factor-extraction|{a:-500,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=-500. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, -500, 500, 3, 4, [5, 25, 125], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -500, 500, 3, 4, [5, 25, 125], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0019","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44f182e8c698932baf53d941541e10e50f7ad90254ce93de6f8e8df183f57003","family_id":"qf-mod49-pval-factor-extraction","seed":202611185019,"semantic_template":"pval-factor-extraction|{a:10,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=10. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, 10, 10, 1, 5, [2], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 10, 10, 1, 5, [2], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0020","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17e6c7674498c682595a803b434ec2980143f46eeba67a431eb93d88ae186e89","family_id":"qf-mod49-pval-factor-extraction","seed":202611185020,"semantic_template":"pval-factor-extraction|{a:-54,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=-54. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, -54, 54, 3, 2, [3, 9, 27], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, -54, 54, 3, 2, [3, 9, 27], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0021","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01745035d85e8c3a9e45c6f0eff6b8593ed59a0e13f19c01ca4e0529e1a46935","family_id":"qf-mod49-pval-factor-extraction","seed":202611185021,"semantic_template":"pval-factor-extraction|{a:875,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=875. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, 875, 875, 3, 7, [5, 25, 125], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, 875, 875, 3, 7, [5, 25, 125], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0022","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1adefcc411c1d6db4dd79839bf6d26f28885faaf720742d9970ab6b3d876e8cb","family_id":"qf-mod49-pval-factor-extraction","seed":202611185022,"semantic_template":"pval-factor-extraction|{a:-2,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=-2. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, -2, 2, 1, 1, [2], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, -2, 2, 1, 1, [2], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0023","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a88ff40bbe75de68334521c5d3fcb553287632464487dc278ea788c21107d542","family_id":"qf-mod49-pval-factor-extraction","seed":202611185023,"semantic_template":"pval-factor-extraction|{a:18,p:3}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=3, nonzero integer a=18. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(3, 18, 18, 2, 2, [3, 9], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (3, 18, 18, 2, 2, [3, 9], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0024","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60d7b4e64dfb582f8470e3c58213921d5e68e564f9f9ffd24ed553039bb8014b","family_id":"qf-mod49-pval-factor-extraction","seed":202611185024,"semantic_template":"pval-factor-extraction|{a:-375,p:5}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=5, nonzero integer a=-375. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(5, -375, 375, 3, 3, [5, 25, 125], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (5, -375, 375, 3, 3, [5, 25, 125], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0025","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0eedc941a279e6c398d27588c2ee79d1301e5a960048b62316aca803dbf722","family_id":"qf-mod49-pval-factor-extraction","seed":202611185025,"semantic_template":"pval-factor-extraction|{a:8,p:2}"},"fields":{"title":"The Highest Dividing Prime Power Is Extracted from the Absolute Integer","stem":"Use prime p=2, nonzero integer a=8. Give (p, integer_a, absolute_a, valuation, unit_factor, dividing_prime_powers, factorization_identity, unit_not_divisible_by_p).","answer":{"kind":"expression","value":"(2, 8, 8, 3, 1, [2, 4, 8], Yes, Yes)","explanation":"Normalize by |a| and factor it as p^k times a unit not divisible by p. Retain every successive dividing prime power and both exact factorization checks. A negative integer is valid, not a failed-domain witness. The exact result, retaining the requested order, is (2, 8, 8, 3, 1, [2, 4, 8], Yes, Yes)."},"skills":["The Highest Dividing Prime Power Is Extracted from the Absolute Integer"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0026","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb5bfebf0c9aa66116495737aed57c99bf6e97839a97c56a84738c2e9503d8d6","family_id":"qf-mod49-pval-product-additivity","seed":202611185026,"semantic_template":"pval-product-additivity|{a:1,b:-4,p:2}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=2, integers a=1, b=-4, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(2, 1, -4, 0, 2, -4, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 1, -4, 0, 2, -4, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0027","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"558ed1c992594ae6fd8941cefa852b9fd4e1f3640ee02e52bc3acf062756f657","family_id":"qf-mod49-pval-product-additivity","seed":202611185027,"semantic_template":"pval-product-additivity|{a:6,b:-27,p:3}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=3, integers a=6, b=-27, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(3, 6, -27, 1, 3, -162, 4, 4, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 6, -27, 1, 3, -162, 4, 4, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0028","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf8c0188f1451c301191e216e7a66270198b3f9a8d9673c6a7c0a07bca342054","family_id":"qf-mod49-pval-product-additivity","seed":202611185028,"semantic_template":"pval-product-additivity|{a:75,b:-4,p:5}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=5, integers a=75, b=-4, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(5, 75, -4, 2, 0, -300, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 75, -4, 2, 0, -300, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0029","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f508f789db7cf31d324ef87721414e0b743948d1cc78eefefb3ac7ea9585b2dc","family_id":"qf-mod49-pval-product-additivity","seed":202611185029,"semantic_template":"pval-product-additivity|{a:4,b:-35,p:7}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=7, integers a=4, b=-35, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(7, 4, -35, 0, 1, -140, 1, 1, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 4, -35, 0, 1, -140, 1, 1, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0030","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"045657f339cdeeb0e8f1fb14087e4254ff68704708ef0810c836e3b32c2e9886","family_id":"qf-mod49-pval-product-additivity","seed":202611185030,"semantic_template":"pval-product-additivity|{a:55,b:-726,p:11}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=11, integers a=55, b=-726, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(11, 55, -726, 1, 2, -39930, 3, 3, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 55, -726, 1, 2, -39930, 3, 3, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0031","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e5f132a03cf38d0ffa91fff564357578e15166b2c47ea7fca09bc7f1a05156d","family_id":"qf-mod49-pval-product-additivity","seed":202611185031,"semantic_template":"pval-product-additivity|{a:1014,b:-7,p:13}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=13, integers a=1014, b=-7, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(13, 1014, -7, 2, 0, -7098, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 1014, -7, 2, 0, -7098, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0032","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"446ca3ae25b5ede816f0a2fccd2a39ebaacfc30edb64fcb28badae175e509319","family_id":"qf-mod49-pval-product-additivity","seed":202611185032,"semantic_template":"pval-product-additivity|{a:7,b:-16,p:2}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=2, integers a=7, b=-16, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(2, 7, -16, 0, 4, -112, 4, 4, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 7, -16, 0, 4, -112, 4, 4, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0033","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf46553e01feeafbf3209f4464187bd2fcfdc1efba614647b6e43b84f6c1f382","family_id":"qf-mod49-pval-product-additivity","seed":202611185033,"semantic_template":"pval-product-additivity|{a:24,b:-81,p:3}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=3, integers a=24, b=-81, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(3, 24, -81, 1, 4, -1944, 5, 5, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 24, -81, 1, 4, -1944, 5, 5, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0034","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d828378e496bd6fd7f019a8760c4a8892ce767e66a038513d9f5e0767fef539","family_id":"qf-mod49-pval-product-additivity","seed":202611185034,"semantic_template":"pval-product-additivity|{a:225,b:-10,p:5}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=5, integers a=225, b=-10, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(5, 225, -10, 2, 1, -2250, 3, 3, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 225, -10, 2, 1, -2250, 3, 3, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0035","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7deb68833543dee6d3e950d4fb1980e4178151341b8316ec523fd4979f4fbe01","family_id":"qf-mod49-pval-product-additivity","seed":202611185035,"semantic_template":"pval-product-additivity|{a:10,b:-77,p:7}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=7, integers a=10, b=-77, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(7, 10, -77, 0, 1, -770, 1, 1, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 10, -77, 0, 1, -770, 1, 1, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0036","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b372082675550264126dd7179cbe33f56c2541df0f245a02b40cbae41c100ce5","family_id":"qf-mod49-pval-product-additivity","seed":202611185036,"semantic_template":"pval-product-additivity|{a:121,b:-1452,p:11}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=11, integers a=121, b=-1452, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(11, 121, -1452, 2, 2, -175692, 4, 4, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 121, -1452, 2, 2, -175692, 4, 4, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0037","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb245e27ab1822014157a717f5c144e7633a335f85158aa7d869a6958b699aad","family_id":"qf-mod49-pval-product-additivity","seed":202611185037,"semantic_template":"pval-product-additivity|{a:2028,b:-13,p:13}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=13, integers a=2028, b=-13, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(13, 2028, -13, 2, 1, -26364, 3, 3, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 2028, -13, 2, 1, -26364, 3, 3, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0038","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41e838f3bb584c7275bc305dbd5298d952a6903f1255ba630cd48ff5320142cc","family_id":"qf-mod49-pval-product-additivity","seed":202611185038,"semantic_template":"pval-product-additivity|{a:13,b:-28,p:2}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=2, integers a=13, b=-28, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(2, 13, -28, 0, 2, -364, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 13, -28, 0, 2, -364, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0039","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cbdfd4990e28be437250cca436ce3397c5911041df9d481d7da3dd57d6c5b1c","family_id":"qf-mod49-pval-product-additivity","seed":202611185039,"semantic_template":"pval-product-additivity|{a:42,b:-135,p:3}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=3, integers a=42, b=-135, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 42, -135, 1, 3, -5670, 4, 4, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0040","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b298f4bdc7888b4797994c2e28243073f71a259d65a4c75761bc58d076a85507","family_id":"qf-mod49-pval-product-additivity","seed":202611185040,"semantic_template":"pval-product-additivity|{a:375,b:-16,p:5}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=5, integers a=375, b=-16, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(5, 375, -16, 3, 0, -6000, 3, 3, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 375, -16, 3, 0, -6000, 3, 3, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0041","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9329ba9cc7d9873b26a0d037193b588b8972af7ec3696b422e044df6f8210269","family_id":"qf-mod49-pval-product-additivity","seed":202611185041,"semantic_template":"pval-product-additivity|{a:16,b:-119,p:7}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=7, integers a=16, b=-119, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(7, 16, -119, 0, 1, -1904, 1, 1, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 16, -119, 0, 1, -1904, 1, 1, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0042","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13b3233ac358e125008cc47b2ca6c50e380dc2b7ee41c124fd83ed705dea65a5","family_id":"qf-mod49-pval-product-additivity","seed":202611185042,"semantic_template":"pval-product-additivity|{a:187,b:-2178,p:11}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=11, integers a=187, b=-2178, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(11, 187, -2178, 1, 2, -407286, 3, 3, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 187, -2178, 1, 2, -407286, 3, 3, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0043","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b40589c1bae3417b4ee2ac11e20a1400fbca6ba768c01152f52f4ca83b66c36","family_id":"qf-mod49-pval-product-additivity","seed":202611185043,"semantic_template":"pval-product-additivity|{a:3042,b:-19,p:13}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=13, integers a=3042, b=-19, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(13, 3042, -19, 2, 0, -57798, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 3042, -19, 2, 0, -57798, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0044","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a75882985d54f8a0b9b866f167ad532cfaedae445ff6cc4a5c0fca1f1777b98c","family_id":"qf-mod49-pval-product-additivity","seed":202611185044,"semantic_template":"pval-product-additivity|{a:19,b:-40,p:2}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=2, integers a=19, b=-40, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(2, 19, -40, 0, 3, -760, 3, 3, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 19, -40, 0, 3, -760, 3, 3, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0045","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2df5cc003d1d6ffcb6a73bdd55a117a71e9fcdf8debcd4a2b3bc435a1c4482e6","family_id":"qf-mod49-pval-product-additivity","seed":202611185045,"semantic_template":"pval-product-additivity|{a:60,b:-189,p:3}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=3, integers a=60, b=-189, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(3, 60, -189, 1, 3, -11340, 4, 4, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (3, 60, -189, 1, 3, -11340, 4, 4, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0046","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ab07534e18c07c311da23fe9e4f74c58f22dbae208b601026ba5d966bcc0cf9","family_id":"qf-mod49-pval-product-additivity","seed":202611185046,"semantic_template":"pval-product-additivity|{a:525,b:-22,p:5}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=5, integers a=525, b=-22, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(5, 525, -22, 2, 0, -11550, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (5, 525, -22, 2, 0, -11550, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0047","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0731994ce347bcadde0cb403ac2cef5398cc27436e754eecd845a37bc692719b","family_id":"qf-mod49-pval-product-additivity","seed":202611185047,"semantic_template":"pval-product-additivity|{a:22,b:-161,p:7}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=7, integers a=22, b=-161, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(7, 22, -161, 0, 1, -3542, 1, 1, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (7, 22, -161, 0, 1, -3542, 1, 1, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0048","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83b8515bff8be18010894e429c7e8efd01a0871d9e1b15cf6c92922c096f35d1","family_id":"qf-mod49-pval-product-additivity","seed":202611185048,"semantic_template":"pval-product-additivity|{a:253,b:-2904,p:11}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=11, integers a=253, b=-2904, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(11, 253, -2904, 1, 2, -734712, 3, 3, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (11, 253, -2904, 1, 2, -734712, 3, 3, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0049","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f4cead457b0be3bcd9dff24c4d0a666a0837f85ff76de6f0eb9211d3729236a","family_id":"qf-mod49-pval-product-additivity","seed":202611185049,"semantic_template":"pval-product-additivity|{a:4056,b:-25,p:13}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=13, integers a=4056, b=-25, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(13, 4056, -25, 2, 0, -101400, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (13, 4056, -25, 2, 0, -101400, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0050","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e062ab935d67c030786201f72ebb27507b324fcb8a58983825178f754e7001bc","family_id":"qf-mod49-pval-product-additivity","seed":202611185050,"semantic_template":"pval-product-additivity|{a:25,b:-52,p:2}"},"fields":{"title":"Integer Valuations Add under Products, Including the Fixed Zero Convention","stem":"Use prime p=2, integers a=25, b=-52, with v_p(0)=infinity. Give (p, a, b, valuation_a, valuation_b, product, valuation_product, sum_of_valuations, additivity, infinity_convention).","answer":{"kind":"expression","value":"(2, 25, -52, 0, 2, -1300, 2, 2, Yes, infinity_plus_k_equals_infinity)","explanation":"Compute both valuations and the product valuation. If either factor is zero, the valuation sum is the single infinity sentinel. Otherwise add the finite exponents. This integer instance is not a construction of a p-adic completion. The exact result, retaining the requested order, is (2, 25, -52, 0, 2, -1300, 2, 2, Yes, infinity_plus_k_equals_infinity)."},"skills":["Integer Valuations Add under Products, Including the Fixed Zero Convention"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0051","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"340787ffd6b7df944e504071cfdd638fe0493dda15ec5d63a5dc31b2b737fc34","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185051,"semantic_template":"pval-sum-ultrametric|{a:2,b:4,p:2}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=2, nonzero summands a=2, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(2, 2, 4, 1, 2, 6, 1, 1, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 2, 4, 1, 2, 6, 1, 1, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0052","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"769159a3eec7a6e2989c89753c65819383f58e829555de64740491109914f929","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185052,"semantic_template":"pval-sum-ultrametric|{a:18,b:27,p:3}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=3, nonzero summands a=18, b=27; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(3, 18, 27, 2, 3, 45, 2, 2, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 18, 27, 2, 3, 45, 2, 2, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0053","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6183ab64e43310f942cba126e0dcb0e8fa67a53a30b5aed9def2db7d2d95ef11","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185053,"semantic_template":"pval-sum-ultrametric|{a:375,b:4,p:5}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=5, nonzero summands a=375, b=4; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(5, 375, 4, 3, 0, 379, 0, 0, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 375, 4, 3, 0, 379, 0, 0, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0054","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38e758ff6933b50f1b4033422e6f6fd4f46331664e2debdf1f5d4e309c5bdf66","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185054,"semantic_template":"pval-sum-ultrametric|{a:28,b:35,p:7}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=7, nonzero summands a=28, b=35; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(7, 28, 35, 1, 1, 63, 1, 1, Yes, No, not_guaranteed)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 28, 35, 1, 1, 63, 1, 1, Yes, No, not_guaranteed)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0055","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"815efda37b1366828d2a0dd4111b41346dc253c5c97c6671328ce6437eb4c7ec","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185055,"semantic_template":"pval-sum-ultrametric|{a:605,b:726,p:11}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=11, nonzero summands a=605, b=726; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(11, 605, 726, 2, 2, 1331, 3, 2, Yes, No, not_guaranteed)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 605, 726, 2, 2, 1331, 3, 2, Yes, No, not_guaranteed)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0056","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5baed20d9649af973400b6c8be12dd1192a8d6fc654405c9056bf26a1e55bfa5","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185056,"semantic_template":"pval-sum-ultrametric|{a:13182,b:7,p:13}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=13, nonzero summands a=13182, b=7; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(13, 13182, 7, 3, 0, 13189, 0, 0, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 13182, 7, 3, 0, 13189, 0, 0, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0057","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94eee37ef090c4ee03a7271c3a088d533275c12c265b7b83aed8c93f8f08ea8c","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185057,"semantic_template":"pval-sum-ultrametric|{a:14,b:16,p:2}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=2, nonzero summands a=14, b=16; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(2, 14, 16, 1, 4, 30, 1, 1, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 14, 16, 1, 4, 30, 1, 1, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0058","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab7993b1a8c5832d0fb93b1725ad81ec765fe8302ea962442be004d8917aec98","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185058,"semantic_template":"pval-sum-ultrametric|{a:72,b:81,p:3}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=3, nonzero summands a=72, b=81; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(3, 72, 81, 2, 4, 153, 2, 2, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 72, 81, 2, 4, 153, 2, 2, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0059","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"902a847d6fc4ff58fa49a29234e5bf97c4aae5e0120e432f3d08253475a3a204","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185059,"semantic_template":"pval-sum-ultrametric|{a:1125,b:10,p:5}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=5, nonzero summands a=1125, b=10; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(5, 1125, 10, 3, 1, 1135, 1, 1, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 1125, 10, 3, 1, 1135, 1, 1, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0060","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92b889da41ca54dfbcf25236f23a0dc5c212e624633c1c365baeb48544106d90","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185060,"semantic_template":"pval-sum-ultrametric|{a:70,b:77,p:7}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=7, nonzero summands a=70, b=77; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(7, 70, 77, 1, 1, 147, 2, 1, Yes, No, not_guaranteed)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 70, 77, 1, 1, 147, 2, 1, Yes, No, not_guaranteed)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0061","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37242408b2231824762853c2db274213f63f644958147dea2e3e137a3890968f","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185061,"semantic_template":"pval-sum-ultrametric|{a:1331,b:1452,p:11}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=11, nonzero summands a=1331, b=1452; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(11, 1331, 1452, 3, 2, 2783, 2, 2, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 1331, 1452, 3, 2, 2783, 2, 2, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0062","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c20450223b4c5f67578cab9fff5569077a8af06f0acc5635243071aed577bf6f","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185062,"semantic_template":"pval-sum-ultrametric|{a:26364,b:13,p:13}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=13, nonzero summands a=26364, b=13; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(13, 26364, 13, 3, 1, 26377, 1, 1, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 26364, 13, 3, 1, 26377, 1, 1, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0063","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"261b8bd6e760467e43ac1662471b336b9e473105794f0ba84a66120169e4d108","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185063,"semantic_template":"pval-sum-ultrametric|{a:26,b:28,p:2}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=2, nonzero summands a=26, b=28; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(2, 26, 28, 1, 2, 54, 1, 1, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 26, 28, 1, 2, 54, 1, 1, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0064","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eb5ab8b269ce1c7c677ecad18ec5474bac59d16d96d9be9a621b5658538bc31","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185064,"semantic_template":"pval-sum-ultrametric|{a:126,b:135,p:3}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=3, nonzero summands a=126, b=135; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(3, 126, 135, 2, 3, 261, 2, 2, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 126, 135, 2, 3, 261, 2, 2, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0065","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50c69694481ad72158122fea92d2d8207ea4d8d69c6670b25c977a39f378eb25","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185065,"semantic_template":"pval-sum-ultrametric|{a:1875,b:16,p:5}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=5, nonzero summands a=1875, b=16; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(5, 1875, 16, 4, 0, 1891, 0, 0, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 1875, 16, 4, 0, 1891, 0, 0, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0066","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23d67f3b3a9b497d02dd83bce664091e65bff43a8300d823b9111e7b1ebc87e4","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185066,"semantic_template":"pval-sum-ultrametric|{a:112,b:119,p:7}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=7, nonzero summands a=112, b=119; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(7, 112, 119, 1, 1, 231, 1, 1, Yes, No, not_guaranteed)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 112, 119, 1, 1, 231, 1, 1, Yes, No, not_guaranteed)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0067","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dff69d86bfb15960825ab41f88e6bf538fcd955ce492f02878f5135caa58f41","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185067,"semantic_template":"pval-sum-ultrametric|{a:2057,b:2178,p:11}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=11, nonzero summands a=2057, b=2178; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(11, 2057, 2178, 2, 2, 4235, 2, 2, Yes, No, not_guaranteed)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 2057, 2178, 2, 2, 4235, 2, 2, Yes, No, not_guaranteed)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0068","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6529087e99dd8f43f13bf225a088e8f697db5d9ce8649fef67a2b167abb1512","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185068,"semantic_template":"pval-sum-ultrametric|{a:39546,b:19,p:13}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=13, nonzero summands a=39546, b=19; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(13, 39546, 19, 3, 0, 39565, 0, 0, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 39546, 19, 3, 0, 39565, 0, 0, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0069","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb4ff0d29050b7e74c664402eb08b8e1419ca1781ffe34a3178f48887da6d87e","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185069,"semantic_template":"pval-sum-ultrametric|{a:38,b:40,p:2}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=2, nonzero summands a=38, b=40; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(2, 38, 40, 1, 3, 78, 1, 1, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 38, 40, 1, 3, 78, 1, 1, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0070","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4289737047df5d842e3d7f551a50574fe7f7ba205cebe46e19562f979947d807","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185070,"semantic_template":"pval-sum-ultrametric|{a:180,b:189,p:3}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=3, nonzero summands a=180, b=189; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(3, 180, 189, 2, 3, 369, 2, 2, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (3, 180, 189, 2, 3, 369, 2, 2, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0071","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2549d3afb83d32aae58571ed351d1c9db216f6c9b6402b44473c133b9d46b96b","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185071,"semantic_template":"pval-sum-ultrametric|{a:2625,b:22,p:5}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=5, nonzero summands a=2625, b=22; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(5, 2625, 22, 3, 0, 2647, 0, 0, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (5, 2625, 22, 3, 0, 2647, 0, 0, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0072","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a56c18c8f4a70e16e82f410b2b9fb1210e4f1d1d81afaf43d410cf2641afd8cb","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185072,"semantic_template":"pval-sum-ultrametric|{a:154,b:161,p:7}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=7, nonzero summands a=154, b=161; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(7, 154, 161, 1, 1, 315, 1, 1, Yes, No, not_guaranteed)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (7, 154, 161, 1, 1, 315, 1, 1, Yes, No, not_guaranteed)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0073","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3217e2d9348a6c2584c39a9539e55492f62f4dd86d5d190d0a35e0f7d390229c","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185073,"semantic_template":"pval-sum-ultrametric|{a:2783,b:2904,p:11}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=11, nonzero summands a=2783, b=2904; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (11, 2783, 2904, 2, 2, 5687, 2, 2, Yes, No, not_guaranteed)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0074","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6b31505bc96bc480a5223d853d011c79873e9d95461632bce93d55cf81b75bb","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185074,"semantic_template":"pval-sum-ultrametric|{a:52728,b:25,p:13}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=13, nonzero summands a=52728, b=25; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(13, 52728, 25, 3, 0, 52753, 0, 0, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (13, 52728, 25, 3, 0, 52753, 0, 0, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0075","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"361c722d346ef464054481433c4794e5c663d63ad06e990009d879e4a8f3a375","family_id":"qf-mod49-pval-sum-ultrametric","seed":202611185075,"semantic_template":"pval-sum-ultrametric|{a:50,b:52,p:2}"},"fields":{"title":"Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations","stem":"Use prime p=2, nonzero summands a=50, b=52; their sum may be zero. Give (p, a, b, valuation_a, valuation_b, sum, valuation_sum, minimum_valuation, ultrametric_inequality, valuations_unequal, equality_when_unequal_or_not_guaranteed).","answer":{"kind":"expression","value":"(2, 50, 52, 1, 2, 102, 1, 1, Yes, Yes, Yes)","explanation":"Compute the two valuations and v_p(a+b), retaining infinity for a zero sum. Compare with the minimum. If valuations differ the sum valuation equals that minimum; when they agree, not_guaranteed withholds a universal equality claim rather than asserting this instance must have strict inequality. The exact result, retaining the requested order, is (2, 50, 52, 1, 2, 102, 1, 1, Yes, Yes, Yes)."},"skills":["Sum Valuation Is at Least the Minimum, with Equality Guaranteed Only at Unequal Valuations"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0076","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58b51b86a8505d97b0857655a75fd2cfbf83bd0ab78b1eb72507b7b5820ba1f1","family_id":"qf-mod49-pval-boundary","seed":202611185076,"semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:2}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=zero_convention, base p=2, integer a=0. Give (mode, p, integer_a, valuation, convention).","answer":{"kind":"expression","value":"(zero_convention, 2, 0, infinity, single_serializable_zero_convention)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 2, 0, infinity, single_serializable_zero_convention)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0077","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11f3a8e37a5ff6983080bf2a39aea35c7d5d5a194d434d1df6a0d06d48132fe2","family_id":"qf-mod49-pval-boundary","seed":202611185077,"semantic_template":"pval-boundary|{a:2,mode:\"nonprime_base\",p:6}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=nonprime_base, base p=6, integer a=2. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).","answer":{"kind":"expression","value":"(nonprime_base, 6, 2, No, not_called_p_adic_valuation)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 6, 2, No, not_called_p_adic_valuation)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0078","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8faf8f48626f992fc427156438b9f3294767d96acc8bffc5d765f4f683bcc4bf","family_id":"qf-mod49-pval-boundary","seed":202611185078,"semantic_template":"pval-boundary|{a:-12,mode:\"negative_normalization\",p:2}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=negative_normalization, base p=2, integer a=-12. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).","answer":{"kind":"expression","value":"(negative_normalization, 2, -12, 12, 2, Yes, negative_integer_is_valid)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 2, -12, 12, 2, Yes, negative_integer_is_valid)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0079","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"995ac5ab3c5c888638ec44d84b6f400048ef30908da19fb27c11379d9ea1a949","family_id":"qf-mod49-pval-boundary","seed":202611185079,"semantic_template":"pval-boundary|{a:8,mode:\"equal_valuations_cancellation\",p:2}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=equal_valuations_cancellation, base p=2, integer a=8. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 2, 8, -8, 3, 3, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0080","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36a3fa3db85864243d6799bbfb2f72a04d0460bdd702e2dcabf1b00147117427","family_id":"qf-mod49-pval-boundary","seed":202611185080,"semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:3}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=zero_convention, base p=3, integer a=0. Give (mode, p, integer_a, valuation, convention).","answer":{"kind":"expression","value":"(zero_convention, 3, 0, infinity, single_serializable_zero_convention)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 3, 0, infinity, single_serializable_zero_convention)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0081","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbf5dadac8db39ad9b5b801b1b8dcac1d07d97e71bec5dce6e0a0cbb521517fe","family_id":"qf-mod49-pval-boundary","seed":202611185081,"semantic_template":"pval-boundary|{a:6,mode:\"nonprime_base\",p:12}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=nonprime_base, base p=12, integer a=6. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).","answer":{"kind":"expression","value":"(nonprime_base, 12, 6, No, not_called_p_adic_valuation)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 12, 6, No, not_called_p_adic_valuation)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0082","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd08d58f2384aa099ce99c1de24a181692c85e867c07a83e488751f5c66f7560","family_id":"qf-mod49-pval-boundary","seed":202611185082,"semantic_template":"pval-boundary|{a:-7,mode:\"negative_normalization\",p:3}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=negative_normalization, base p=3, integer a=-7. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).","answer":{"kind":"expression","value":"(negative_normalization, 3, -7, 7, 0, Yes, negative_integer_is_valid)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 3, -7, 7, 0, Yes, negative_integer_is_valid)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0083","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"94951d657c5ab31d71befb403046ff9acaffb6e4024aae875c9dd86ce1fbf6bf","family_id":"qf-mod49-pval-boundary","seed":202611185083,"semantic_template":"pval-boundary|{a:24,mode:\"equal_valuations_cancellation\",p:3}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=equal_valuations_cancellation, base p=3, integer a=24. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 3, 24, -24, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0084","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1b413e227d1d198b1d9f693153164326719907d1aa99c4c211dcb6761a12604","family_id":"qf-mod49-pval-boundary","seed":202611185084,"semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:5}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=zero_convention, base p=5, integer a=0. Give (mode, p, integer_a, valuation, convention).","answer":{"kind":"expression","value":"(zero_convention, 5, 0, infinity, single_serializable_zero_convention)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 5, 0, infinity, single_serializable_zero_convention)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0085","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"888c881d9cdd08ca711a2a0d5ebecca46fef4d3ce6457243c42cd62c3c332df7","family_id":"qf-mod49-pval-boundary","seed":202611185085,"semantic_template":"pval-boundary|{a:10,mode:\"nonprime_base\",p:18}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=nonprime_base, base p=18, integer a=10. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).","answer":{"kind":"expression","value":"(nonprime_base, 18, 10, No, not_called_p_adic_valuation)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 18, 10, No, not_called_p_adic_valuation)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0086","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81018a7901441d21c4bf35970cc1c9aa6f544459b771f00d43766407782efffc","family_id":"qf-mod49-pval-boundary","seed":202611185086,"semantic_template":"pval-boundary|{a:-55,mode:\"negative_normalization\",p:5}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=negative_normalization, base p=5, integer a=-55. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).","answer":{"kind":"expression","value":"(negative_normalization, 5, -55, 55, 1, Yes, negative_integer_is_valid)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 5, -55, 55, 1, Yes, negative_integer_is_valid)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0087","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f435e294c3b250f0c33b08fa2b83ebc7ad8a4d01aac8bcfb2c8b66d456f7f68","family_id":"qf-mod49-pval-boundary","seed":202611185087,"semantic_template":"pval-boundary|{a:60,mode:\"equal_valuations_cancellation\",p:5}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=equal_valuations_cancellation, base p=5, integer a=60. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 5, 60, -60, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0088","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c2fd26fd43d9c3484c6e86edfccefa97548fa19568b7d0e547ccd1e3ac99f83","family_id":"qf-mod49-pval-boundary","seed":202611185088,"semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:7}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=zero_convention, base p=7, integer a=0. Give (mode, p, integer_a, valuation, convention).","answer":{"kind":"expression","value":"(zero_convention, 7, 0, infinity, single_serializable_zero_convention)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 7, 0, infinity, single_serializable_zero_convention)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0089","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97cd5d95d2137041236871ec224ebf4aab8ecc77e424d9f2203912415e3cca1e","family_id":"qf-mod49-pval-boundary","seed":202611185089,"semantic_template":"pval-boundary|{a:14,mode:\"nonprime_base\",p:24}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=nonprime_base, base p=24, integer a=14. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).","answer":{"kind":"expression","value":"(nonprime_base, 24, 14, No, not_called_p_adic_valuation)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 24, 14, No, not_called_p_adic_valuation)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0090","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a482a20a3fbce0de12b55fb310770aec86d3ae92a77fa5168ef553887d0e38ab","family_id":"qf-mod49-pval-boundary","seed":202611185090,"semantic_template":"pval-boundary|{a:-735,mode:\"negative_normalization\",p:7}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=negative_normalization, base p=7, integer a=-735. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).","answer":{"kind":"expression","value":"(negative_normalization, 7, -735, 735, 2, Yes, negative_integer_is_valid)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 7, -735, 735, 2, Yes, negative_integer_is_valid)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0091","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dac714ea461ba34a134b4bcd2be272c2c0e428810bdf15c7bab4a01395be526","family_id":"qf-mod49-pval-boundary","seed":202611185091,"semantic_template":"pval-boundary|{a:112,mode:\"equal_valuations_cancellation\",p:7}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=equal_valuations_cancellation, base p=7, integer a=112. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 7, 112, -112, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0092","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2e9c43f46ac688109490114aa2fbad671be8a9652c1d47fb23d0ae8be8f3a6b","family_id":"qf-mod49-pval-boundary","seed":202611185092,"semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:11}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=zero_convention, base p=11, integer a=0. Give (mode, p, integer_a, valuation, convention).","answer":{"kind":"expression","value":"(zero_convention, 11, 0, infinity, single_serializable_zero_convention)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 11, 0, infinity, single_serializable_zero_convention)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0093","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6be2aa8286ae25a09ae6552aac2b5b8bd96252fcf6995114aaac0ffa1a0a37c3","family_id":"qf-mod49-pval-boundary","seed":202611185093,"semantic_template":"pval-boundary|{a:18,mode:\"nonprime_base\",p:28}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=nonprime_base, base p=28, integer a=18. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).","answer":{"kind":"expression","value":"(nonprime_base, 28, 18, No, not_called_p_adic_valuation)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 28, 18, No, not_called_p_adic_valuation)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0094","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c291b2c57ade972dd0ef31a67d95f62d9f034a064ed4e4e1ff2b72614af3e17","family_id":"qf-mod49-pval-boundary","seed":202611185094,"semantic_template":"pval-boundary|{a:-19,mode:\"negative_normalization\",p:11}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=negative_normalization, base p=11, integer a=-19. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).","answer":{"kind":"expression","value":"(negative_normalization, 11, -19, 19, 0, Yes, negative_integer_is_valid)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 11, -19, 19, 0, Yes, negative_integer_is_valid)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0095","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"816397e34c58b5d04b6513d86a25b2cac04360efcb1c4c5d29dccac3e98c76ed","family_id":"qf-mod49-pval-boundary","seed":202611185095,"semantic_template":"pval-boundary|{a:220,mode:\"equal_valuations_cancellation\",p:11}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=equal_valuations_cancellation, base p=11, integer a=220. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 11, 220, -220, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0096","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ab44a9bd624d41495233955f2506596c640187093a195194a90a91ec4330fe0","family_id":"qf-mod49-pval-boundary","seed":202611185096,"semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:13}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=zero_convention, base p=13, integer a=0. Give (mode, p, integer_a, valuation, convention).","answer":{"kind":"expression","value":"(zero_convention, 13, 0, infinity, single_serializable_zero_convention)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 13, 0, infinity, single_serializable_zero_convention)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0097","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b219802233e4bc06e3231886b70e5900375cc3dd5e7addd4aa146f66578703ba","family_id":"qf-mod49-pval-boundary","seed":202611185097,"semantic_template":"pval-boundary|{a:22,mode:\"nonprime_base\",p:34}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=nonprime_base, base p=34, integer a=22. Give (mode, base, integer_a, p_adic_valuation_qualification, boundary).","answer":{"kind":"expression","value":"(nonprime_base, 34, 22, No, not_called_p_adic_valuation)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (nonprime_base, 34, 22, No, not_called_p_adic_valuation)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0098","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf5035d3d0d0c7c16ee14cc4f74161c1cca4d3dc72d12ca0967e7f1d98f30188","family_id":"qf-mod49-pval-boundary","seed":202611185098,"semantic_template":"pval-boundary|{a:-299,mode:\"negative_normalization\",p:13}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=negative_normalization, base p=13, integer a=-299. Give (mode, p, integer_a, absolute_a, valuation, negative_integer_valid, scope).","answer":{"kind":"expression","value":"(negative_normalization, 13, -299, 299, 1, Yes, negative_integer_is_valid)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (negative_normalization, 13, -299, 299, 1, Yes, negative_integer_is_valid)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0099","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1b68c10ac1316f1c093fec390d12b850bc2677f8a8283df6740b3ed9d289786","family_id":"qf-mod49-pval-boundary","seed":202611185099,"semantic_template":"pval-boundary|{a:312,mode:\"equal_valuations_cancellation\",p:13}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=equal_valuations_cancellation, base p=13, integer a=312. Give (mode, p, a, opposite_a, valuation_a, valuation_opposite_a, valuation_sum, cancellation_scope, universal_sum_equality_claim).","answer":{"kind":"expression","value":"(equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (equal_valuations_cancellation, 13, 312, -312, 1, 1, infinity, strict_inequality_or_infinity_can_occur_when_valuations_equal, No)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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-20261015-advance-20261118-r05-p-adic-valuation-integers-0100","batch_id":"batch-20261015-advance-20261118-r05-p-adic-valuation-integers-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e0395c8f53ab7a2dc66e8c2b92d448cbd02332f3cb95fca686d8f8d7670519e","family_id":"qf-mod49-pval-boundary","seed":202611185100,"semantic_template":"pval-boundary|{a:0,mode:\"zero_convention\",p:17}"},"fields":{"title":"Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions","stem":"Use mode=zero_convention, base p=17, integer a=0. Give (mode, p, integer_a, valuation, convention).","answer":{"kind":"expression","value":"(zero_convention, 17, 0, infinity, single_serializable_zero_convention)","explanation":"Keep v_p(0)=infinity for prime p. A composite base is not called a p-adic valuation here. Negative integers are normalized by absolute value. Opposite nonzero summands have equal valuations and cancel to0, refuting a universal sum-equality claim. The exact result, retaining the requested order, is (zero_convention, 17, 0, infinity, single_serializable_zero_convention)."},"skills":["Zero, Composite Bases, Negative Inputs and Cancellation Need Distinct Conventions"],"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."}}