{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0001","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfcee31ad69dd7571b1df717ba78cdbc5c840e42fdb2c2741bf427c87cf28e97","family_id":"qf-app20-growth-recurrence-step","seed":202610206001,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-8,1],\"factor\":[1,2]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-8/1, factor=1/2. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-8, 1/2, -4)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-8, 1/2, -4)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0002","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b8839c61c4d3ea70fd702818f2d7d78e9fabe19a378910debf9f7e96d8b2cf","family_id":"qf-app20-growth-recurrence-step","seed":202610206002,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-7,2],\"factor\":[2,3]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-7/2, factor=2/3. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-7/2, 2/3, -7/3)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-7/2, 2/3, -7/3)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0003","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b061c9666f15d8a9b53a24aa38450bd58831ef6a7611ba00c6a34f4d38c1ab50","family_id":"qf-app20-growth-recurrence-step","seed":202610206003,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-6,3],\"factor\":[3,4]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-6/3, factor=3/4. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-2, 3/4, -3/2)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-2, 3/4, -3/2)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0004","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87247c6499f530f5348935d8f3b5fc5e9540d8c6eefd2daee164e523852f02e2","family_id":"qf-app20-growth-recurrence-step","seed":202610206004,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-5,4],\"factor\":[4,5]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-5/4, factor=4/5. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-5/4, 4/5, -1)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-5/4, 4/5, -1)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0005","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4102f1e7c315c8e4d8213c3542a9028d13f897d03ad133883de3581cc869f524","family_id":"qf-app20-growth-recurrence-step","seed":202610206005,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-4,5],\"factor\":[5,6]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-4/5, factor=5/6. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-4/5, 5/6, -2/3)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-4/5, 5/6, -2/3)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0006","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab4831d89a273a1ea5e726e5425638c72a79373282f5de095c24e04051ab8c89","family_id":"qf-app20-growth-recurrence-step","seed":202610206006,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-3,1],\"factor\":[6,7]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-3/1, factor=6/7. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-3, 6/7, -18/7)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-3, 6/7, -18/7)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0007","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39029653384e7cb3fac3b79ae40dbec0d51cdaab961c0ebca1a54bcedf562443","family_id":"qf-app20-growth-recurrence-step","seed":202610206007,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-2,2],\"factor\":[7,2]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-2/2, factor=7/2. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-1, 7/2, -7/2)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1, 7/2, -7/2)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0008","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11c4fc3e4610e88395338ff1bb3838752f732b231677f1f42f6ac08fe2a4726b","family_id":"qf-app20-growth-recurrence-step","seed":202610206008,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[-1,3],\"factor\":[1,3]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=-1/3, factor=1/3. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(-1/3, 1/3, -1/9)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1/3, 1/3, -1/9)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0009","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c0f4b1843111af7bd6670a0d7674dc4e1f76746b042f1ed45c8701bcbff87310","family_id":"qf-app20-growth-recurrence-step","seed":202610206009,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[0,4],\"factor\":[2,4]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=0/4, factor=2/4. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(0, 1/2, 0)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, 1/2, 0)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0010","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13d8fc6da5f037941ee03a43662e4585cab1770bdc4849385f3957d08c76deed","family_id":"qf-app20-growth-recurrence-step","seed":202610206010,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[1,5],\"factor\":[3,5]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=1/5, factor=3/5. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(1/5, 3/5, 3/25)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/5, 3/5, 3/25)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0011","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"319e4a20a62d5962371d1832d2475e29ea9bde80c93fe1461e0199d11d16c4bb","family_id":"qf-app20-growth-recurrence-step","seed":202610206011,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[2,1],\"factor\":[4,6]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=2/1, factor=4/6. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(2, 2/3, 4/3)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 2/3, 4/3)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0012","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22c1e03c52c7ec84ab8c248944efeedb96d418b107b4824f03a8b49b1d5bb298","family_id":"qf-app20-growth-recurrence-step","seed":202610206012,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[3,2],\"factor\":[5,7]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=3/2, factor=5/7. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(3/2, 5/7, 15/14)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, 5/7, 15/14)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0013","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46aa75731c8f35d2a7c0e656a28eae6241a69fbc978844e54709c95fac6e4b5c","family_id":"qf-app20-growth-recurrence-step","seed":202610206013,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[4,3],\"factor\":[6,2]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=4/3, factor=6/2. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(4/3, 3, 4)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/3, 3, 4)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0014","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"166d4141e0c035b22c05d0600fcfcdeba03c0f45acf7a4a43025d2f98e22cc85","family_id":"qf-app20-growth-recurrence-step","seed":202610206014,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[5,4],\"factor\":[7,3]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=5/4, factor=7/3. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(5/4, 7/3, 35/12)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/4, 7/3, 35/12)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0015","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14b89ce7ad524b295ab9888caf078eba229a55254a1cfc06c14371d8166e7b54","family_id":"qf-app20-growth-recurrence-step","seed":202610206015,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[6,5],\"factor\":[1,4]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=6/5, factor=1/4. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(6/5, 1/4, 3/10)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/5, 1/4, 3/10)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0016","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e5ab48eef3064fce019dc1a786dff9b4f51d8d10de4b97b0337cd7c962e9c22","family_id":"qf-app20-growth-recurrence-step","seed":202610206016,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[7,1],\"factor\":[2,5]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=7/1, factor=2/5. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(7, 2/5, 14/5)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 2/5, 14/5)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0017","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b06b0381cc7519f01552e444d554bbe32a41bb4648b3e90dc1505c38935430a","family_id":"qf-app20-growth-recurrence-step","seed":202610206017,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[8,2],\"factor\":[3,6]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=8/2, factor=3/6. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(4, 1/2, 2)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, 1/2, 2)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0018","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9e5b470b8d09e740803a5035b9c01da03917fec7a1df334fc12e9e8343ea94c","family_id":"qf-app20-growth-recurrence-step","seed":202610206018,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[9,3],\"factor\":[4,7]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=9/3, factor=4/7. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(3, 4/7, 12/7)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 4/7, 12/7)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0019","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47a3009cfe4e0aaa6f83aaf4f425592d697ca219883f32710e3204f8b2bfc1a2","family_id":"qf-app20-growth-recurrence-step","seed":202610206019,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[10,4],\"factor\":[5,2]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=10/4, factor=5/2. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(5/2, 5/2, 25/4)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 5/2, 25/4)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0020","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe79ff841717ce87a78323ec75482638866e467eb83f28cd71d9100f72542b82","family_id":"qf-app20-growth-recurrence-step","seed":202610206020,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[11,5],\"factor\":[6,3]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=11/5, factor=6/3. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(11/5, 2, 22/5)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 2, 22/5)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0021","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fadbe7f6c7a46c22f6021c3434bb038e457275c413c62d7758d968dad6f65c6","family_id":"qf-app20-growth-recurrence-step","seed":202610206021,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[12,1],\"factor\":[7,4]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=12/1, factor=7/4. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(12, 7/4, 21)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (12, 7/4, 21)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0022","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fc8a7427e905e4aa35980af4da834196e7cfbda4437d74f96fcf6e639d271a7","family_id":"qf-app20-growth-recurrence-step","seed":202610206022,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[13,2],\"factor\":[1,5]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=13/2, factor=1/5. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(13/2, 1/5, 13/10)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13/2, 1/5, 13/10)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0023","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"583c32d25c1e19092e3778a7b398c542f844a3784f71dcc95227e5b4f6b6b175","family_id":"qf-app20-growth-recurrence-step","seed":202610206023,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[14,3],\"factor\":[2,6]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=14/3, factor=2/6. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(14/3, 1/3, 14/9)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (14/3, 1/3, 14/9)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0024","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4089d360f1174d714ff586b80d2fe7a2e150fc440e6e46a1a988e53094e26b6e","family_id":"qf-app20-growth-recurrence-step","seed":202610206024,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[15,4],\"factor\":[3,7]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=15/4, factor=3/7. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(15/4, 3/7, 45/28)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/4, 3/7, 45/28)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0025","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c90a6466f6741ad345198f53921806fcbd4da0067993d087d496a0b0573b1013","family_id":"qf-app20-growth-recurrence-step","seed":202610206025,"semantic_template":"qf-app20-growth-recurrence-step/{\"current\":[16,5],\"factor\":[4,2]}"},"fields":{"title":"One Exact Fixed-Factor Discrete Update","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. current=16/5, factor=4/2. Give (current, factor, next=current*factor).","answer":{"kind":"expression","value":"(16/5, 2, 32/5)","explanation":"Multiply exact rational values; the sign and zero behavior come from the given initial data, not a prediction. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (16/5, 2, 32/5)."},"skills":["One Exact Fixed-Factor Discrete Update"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0026","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"355621a87d8380db999db61c490c7820ee8905e925723bcc921fcc158bd16b7b","family_id":"qf-app20-growth-nth-term","seed":202610206026,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,2],\"initial\":[-8,1],\"n\":0}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-8/1, factor=1/2, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-8, 1/2, 0, -8)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-8, 1/2, 0, -8)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0027","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d6a47f548980b20dcab1f93161ad02156b7d685da8ff4c04c3168aac1b21af5","family_id":"qf-app20-growth-nth-term","seed":202610206027,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,3],\"initial\":[-7,2],\"n\":1}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-7/2, factor=2/3, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-7/2, 2/3, 1, -7/3)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-7/2, 2/3, 1, -7/3)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0028","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4972e6a89fdd87a8b21660523b48b5396c51da338329dd5acce696856c3984d9","family_id":"qf-app20-growth-nth-term","seed":202610206028,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,4],\"initial\":[-6,3],\"n\":2}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-6/3, factor=3/4, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-2, 3/4, 2, -9/8)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-2, 3/4, 2, -9/8)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0029","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba9d8ecf823a9a2ca84b5f63da11e66bb79a7c9bb437cdd46d46080675404bca","family_id":"qf-app20-growth-nth-term","seed":202610206029,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,5],\"initial\":[-5,4],\"n\":3}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-5/4, factor=4/5, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-5/4, 4/5, 3, -16/25)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-5/4, 4/5, 3, -16/25)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0030","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80aabb936bc0ecbad6c989c69e5267d5eaaf55aeb2c9cbd86478974667d51db3","family_id":"qf-app20-growth-nth-term","seed":202610206030,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,6],\"initial\":[-4,5],\"n\":4}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-4/5, factor=1/6, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-4/5, 1/6, 4, -1/1620)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-4/5, 1/6, 4, -1/1620)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0031","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b89216a96824f76980e2eaef867979008939cc28fc201f39b779daf6895328a","family_id":"qf-app20-growth-nth-term","seed":202610206031,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,7],\"initial\":[-3,1],\"n\":0}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-3/1, factor=2/7, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-3, 2/7, 0, -3)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-3, 2/7, 0, -3)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0032","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf424a1b1e1bf1950df44bf0b67d0e95204ee48d01541c9a80163f2ec49fb3b8","family_id":"qf-app20-growth-nth-term","seed":202610206032,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,2],\"initial\":[-2,2],\"n\":1}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-2/2, factor=3/2, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-1, 3/2, 1, -3/2)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1, 3/2, 1, -3/2)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0033","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f0e41e9a1aa19801631b49ffc4bd00787713192058bf6c236a2a178eaa15e08","family_id":"qf-app20-growth-nth-term","seed":202610206033,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,3],\"initial\":[-1,3],\"n\":2}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/3, factor=4/3, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(-1/3, 4/3, 2, -16/27)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1/3, 4/3, 2, -16/27)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0034","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4ea407a6658a36cc5eadd05ff4909bb4be6dfa38d34addb8237076fae5065c0","family_id":"qf-app20-growth-nth-term","seed":202610206034,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,4],\"initial\":[0,4],\"n\":3}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=0/4, factor=1/4, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(0, 1/4, 3, 0)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, 1/4, 3, 0)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0035","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47f022c0bff9a0f2f48f129fb02a4df83a02b338751e4deee87e506506b715e2","family_id":"qf-app20-growth-nth-term","seed":202610206035,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,5],\"initial\":[1,5],\"n\":4}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=1/5, factor=2/5, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(1/5, 2/5, 4, 16/3125)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/5, 2/5, 4, 16/3125)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0036","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59d860c4b8534eed4ad6d222915cbd4e22af94ab7f3df80bce2411fc8b6a3afa","family_id":"qf-app20-growth-nth-term","seed":202610206036,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,6],\"initial\":[2,1],\"n\":0}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=2/1, factor=3/6, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(2, 1/2, 0, 2)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 1/2, 0, 2)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0037","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1837f426bc204b89862cc9f434dda6713be9dec6843afd43926603762b59edb","family_id":"qf-app20-growth-nth-term","seed":202610206037,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,7],\"initial\":[3,2],\"n\":1}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=3/2, factor=4/7, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(3/2, 4/7, 1, 6/7)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, 4/7, 1, 6/7)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0038","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7551cd24822e12428d0845985cd287038c51ee576f5a075b48643dc9768d3d10","family_id":"qf-app20-growth-nth-term","seed":202610206038,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,2],\"initial\":[4,3],\"n\":2}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=4/3, factor=1/2, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(4/3, 1/2, 2, 1/3)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/3, 1/2, 2, 1/3)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0039","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1f66dbbd3569a4aa2dae49f63a30c2a70efe6949c32b51ae2bf8acfa40677d5","family_id":"qf-app20-growth-nth-term","seed":202610206039,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,3],\"initial\":[5,4],\"n\":3}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=5/4, factor=2/3, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(5/4, 2/3, 3, 10/27)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/4, 2/3, 3, 10/27)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0040","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ed99ddf739adbb916a1b932c25ab637ec4434084324697416a5c2ebc09c9027","family_id":"qf-app20-growth-nth-term","seed":202610206040,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,4],\"initial\":[6,5],\"n\":4}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=6/5, factor=3/4, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(6/5, 3/4, 4, 243/640)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/5, 3/4, 4, 243/640)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0041","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"421debe3c442012d47a582c4398267aaa21eb87a271a485b3f81c9b4aa716bfc","family_id":"qf-app20-growth-nth-term","seed":202610206041,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,5],\"initial\":[7,1],\"n\":0}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=7/1, factor=4/5, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(7, 4/5, 0, 7)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 4/5, 0, 7)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0042","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae77243ab1f6aa86eae2caf6b157af8560ca667a1f3875cca8038f612fd7f88e","family_id":"qf-app20-growth-nth-term","seed":202610206042,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,6],\"initial\":[8,2],\"n\":1}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=8/2, factor=1/6, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(4, 1/6, 1, 2/3)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, 1/6, 1, 2/3)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0043","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"795effafb0ddea656bd182e9a33e5b0ee7c539a4e8f298dfb078c382d5572f75","family_id":"qf-app20-growth-nth-term","seed":202610206043,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,7],\"initial\":[9,3],\"n\":2}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=9/3, factor=2/7, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(3, 2/7, 2, 12/49)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 2/7, 2, 12/49)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0044","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d218c0426734a5d799de685fb74795e23f681d554e502aa49beb57f716a7f2eb","family_id":"qf-app20-growth-nth-term","seed":202610206044,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,2],\"initial\":[10,4],\"n\":3}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=10/4, factor=3/2, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(5/2, 3/2, 3, 135/16)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 3/2, 3, 135/16)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0045","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dbc45a7cdf74b629fd053c57d687616525fd75374fe2e9e0513dcda75c3d1d9","family_id":"qf-app20-growth-nth-term","seed":202610206045,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,3],\"initial\":[11,5],\"n\":4}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=11/5, factor=4/3, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(11/5, 4/3, 4, 2816/405)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 4/3, 4, 2816/405)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0046","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76007063a97e5cbc3e9f2f12dd66f84730c404842d6722b0d799c1b2e285f21d","family_id":"qf-app20-growth-nth-term","seed":202610206046,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,4],\"initial\":[12,1],\"n\":0}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=12/1, factor=1/4, n=0≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(12, 1/4, 0, 12)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (12, 1/4, 0, 12)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0047","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88b8314e0dde31a310ceb761dd8600795f295f927c77c97bd2e824a8af0cc538","family_id":"qf-app20-growth-nth-term","seed":202610206047,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[2,5],\"initial\":[13,2],\"n\":1}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=13/2, factor=2/5, n=1≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(13/2, 2/5, 1, 13/5)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13/2, 2/5, 1, 13/5)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0048","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7df9ba0b8b9421723f7364d5b381ea90f06a2a064c1410bf2b28040faacd649d","family_id":"qf-app20-growth-nth-term","seed":202610206048,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[3,6],\"initial\":[14,3],\"n\":2}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=14/3, factor=3/6, n=2≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(14/3, 1/2, 2, 7/6)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (14/3, 1/2, 2, 7/6)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0049","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f9eb24d5b0de110612297f2a8b47eead43ee8af630d65c1f565d47a764cdc36","family_id":"qf-app20-growth-nth-term","seed":202610206049,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[4,7],\"initial\":[15,4],\"n\":3}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=15/4, factor=4/7, n=3≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(15/4, 4/7, 3, 240/343)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/4, 4/7, 3, 240/343)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0050","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a3430512e53dde1abe886d880d60cdf06288dfe5618ded889a1892de69dd7cc","family_id":"qf-app20-growth-nth-term","seed":202610206050,"semantic_template":"qf-app20-growth-nth-term/{\"factor\":[1,2],\"initial\":[16,5],\"n\":4}"},"fields":{"title":"The Exact Natural-Indexed Term Including Zero Factor and Time","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=16/5, factor=1/2, n=4≥0. Give (initial, factor, n, initial*factor^n). The zeroth power is the empty product 1, including factor=0.","answer":{"kind":"expression","value":"(16/5, 1/2, 4, 1/5)","explanation":"Retain the initial value at n=0; a zero factor gives zero only after a positive number of steps. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (16/5, 1/2, 4, 1/5)."},"skills":["The Exact Natural-Indexed Term Including Zero Factor and Time"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0051","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9fbe16880e6ffd565e4fa014c5bed3631e31beabeec32bb531cc6d0ebd1d476","family_id":"qf-app20-growth-closed-form-audit","seed":202610206051,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-7,1],\"factor\":[1,2],\"initial\":[-8,1],\"n\":0}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-8/1, factor=1/2, n=0, candidate=-7/1. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-8, 1/2, 0, -8, -7, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-8, 1/2, 0, -8, -7, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0052","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a596f641dfff62279c47d93bd1f34c2ba3b55aaad3e2f2b2f76bb4bf905cc83","family_id":"qf-app20-growth-closed-form-audit","seed":202610206052,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-7,3],\"factor\":[2,3],\"initial\":[-7,2],\"n\":1}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-7/2, factor=2/3, n=1, candidate=-7/3. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-7/2, 2/3, 1, -7/3, -7/3, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-7/2, 2/3, 1, -7/3, -7/3, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0053","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18177e6aa16347ff56c206377d94622191e69a2a154644595f7d6abb67e3d03d","family_id":"qf-app20-growth-closed-form-audit","seed":202610206053,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-9,8],\"factor\":[3,4],\"initial\":[-2,1],\"n\":2}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-2/1, factor=3/4, n=2, candidate=-9/8. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-2, 3/4, 2, -9/8, -9/8, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-2, 3/4, 2, -9/8, -9/8, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0054","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6303c8f213780ba9cd481058780b9021211b9743aa4e0434c12da49839f727f8","family_id":"qf-app20-growth-closed-form-audit","seed":202610206054,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-15,25],\"factor\":[4,5],\"initial\":[-5,4],\"n\":3}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-5/4, factor=4/5, n=3, candidate=-15/25. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-5/4, 4/5, 3, -16/25, -3/5, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-5/4, 4/5, 3, -16/25, -3/5, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0055","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fa851ef55b8dcbbd3534776b2706ac21d263b4d8340dfcaea5852cd9b958cb1","family_id":"qf-app20-growth-closed-form-audit","seed":202610206055,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-1,1620],\"factor\":[1,6],\"initial\":[-4,5],\"n\":4}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-4/5, factor=1/6, n=4, candidate=-1/1620. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-4/5, 1/6, 4, -1/1620, -1/1620, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-4/5, 1/6, 4, -1/1620, -1/1620, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0056","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db04e694d9c1a425516aea38ca39c337557c4ec4318789e68285d78bfe82ace9","family_id":"qf-app20-growth-closed-form-audit","seed":202610206056,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-3,1],\"factor\":[2,7],\"initial\":[-3,1],\"n\":0}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-3/1, factor=2/7, n=0, candidate=-3/1. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-3, 2/7, 0, -3, -3, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-3, 2/7, 0, -3, -3, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0057","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6301f8675a4a4890e9c3ea3f2daaba3f09844ea77a69407e83a4f7c0b849a471","family_id":"qf-app20-growth-closed-form-audit","seed":202610206057,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-2,2],\"factor\":[3,2],\"initial\":[-1,1],\"n\":1}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/1, factor=3/2, n=1, candidate=-2/2. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-1, 3/2, 1, -3/2, -1, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1, 3/2, 1, -3/2, -1, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0058","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b27a71643a25b111634b904ea87a05957aeb29f5bd2e87f708b30c34146ca220","family_id":"qf-app20-growth-closed-form-audit","seed":202610206058,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[-16,27],\"factor\":[4,3],\"initial\":[-1,3],\"n\":2}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=-1/3, factor=4/3, n=2, candidate=-16/27. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(-1/3, 4/3, 2, -16/27, -16/27, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (-1/3, 4/3, 2, -16/27, -16/27, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0059","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f18a07e7bc78d6dbd092c784468f08658bb687f53f0baeb20cb8cd4f3ed4356e","family_id":"qf-app20-growth-closed-form-audit","seed":202610206059,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[0,1],\"factor\":[1,4],\"initial\":[0,1],\"n\":3}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=0/1, factor=1/4, n=3, candidate=0/1. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(0, 1/4, 3, 0, 0, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, 1/4, 3, 0, 0, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0060","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65529d26bcb75c5f3938c5dead61254cb96301f026d6d7457b092ce8444f2d69","family_id":"qf-app20-growth-closed-form-audit","seed":202610206060,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[17,3125],\"factor\":[2,5],\"initial\":[1,5],\"n\":4}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=1/5, factor=2/5, n=4, candidate=17/3125. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(1/5, 2/5, 4, 16/3125, 17/3125, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/5, 2/5, 4, 16/3125, 17/3125, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0061","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33015010cd09db1d81d61c27f9b954b0102306db764ba1d9efe46779e34a650f","family_id":"qf-app20-growth-closed-form-audit","seed":202610206061,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,1],\"factor\":[1,2],\"initial\":[2,1],\"n\":0}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=2/1, factor=1/2, n=0, candidate=2/1. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(2, 1/2, 0, 2, 2, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, 1/2, 0, 2, 2, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0062","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a925d017b11562b7f249b136f2862dc18100ae77522879cd17458ca8f101edc","family_id":"qf-app20-growth-closed-form-audit","seed":202610206062,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[6,7],\"factor\":[4,7],\"initial\":[3,2],\"n\":1}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=3/2, factor=4/7, n=1, candidate=6/7. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(3/2, 4/7, 1, 6/7, 6/7, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, 4/7, 1, 6/7, 6/7, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0063","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"087f26f6909742ac4dc6037023ec803c70d51e3ca334c19a6940158767026aa0","family_id":"qf-app20-growth-closed-form-audit","seed":202610206063,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,3],\"factor\":[1,2],\"initial\":[4,3],\"n\":2}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=4/3, factor=1/2, n=2, candidate=2/3. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(4/3, 1/2, 2, 1/3, 2/3, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/3, 1/2, 2, 1/3, 2/3, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0064","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1ac3fe2965168ab1528fd7b310f77e9b776ae5f8a18d64a5c34a5f38ca771d4","family_id":"qf-app20-growth-closed-form-audit","seed":202610206064,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[10,27],\"factor\":[2,3],\"initial\":[5,4],\"n\":3}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=5/4, factor=2/3, n=3, candidate=10/27. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(5/4, 2/3, 3, 10/27, 10/27, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/4, 2/3, 3, 10/27, 10/27, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0065","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"453685681759713802d55698c141d00c38bcd05b367656b9fd51d335ca513205","family_id":"qf-app20-growth-closed-form-audit","seed":202610206065,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[243,640],\"factor\":[3,4],\"initial\":[6,5],\"n\":4}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=6/5, factor=3/4, n=4, candidate=243/640. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(6/5, 3/4, 4, 243/640, 243/640, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/5, 3/4, 4, 243/640, 243/640, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0066","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0af04b43219e9d8b410ac1d782e7888b7bc5cba32a6e98d05b1fb2d8819dcb9d","family_id":"qf-app20-growth-closed-form-audit","seed":202610206066,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[8,1],\"factor\":[4,5],\"initial\":[7,1],\"n\":0}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=7/1, factor=4/5, n=0, candidate=8/1. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(7, 4/5, 0, 7, 8, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, 4/5, 0, 7, 8, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0067","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ee1624d3a1c42bd895e3f22cb6c10e62089deb162951d604c908226a5edfd3e","family_id":"qf-app20-growth-closed-form-audit","seed":202610206067,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,3],\"factor\":[1,6],\"initial\":[4,1],\"n\":1}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=4/1, factor=1/6, n=1, candidate=2/3. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(4, 1/6, 1, 2/3, 2/3, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, 1/6, 1, 2/3, 2/3, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0068","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69e8aaf4d6a137c5cd05155bac1ed88c1c9ea9c1b56e2d6f13e46bd7d2df9186","family_id":"qf-app20-growth-closed-form-audit","seed":202610206068,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[12,49],\"factor\":[2,7],\"initial\":[3,1],\"n\":2}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=3/1, factor=2/7, n=2, candidate=12/49. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(3, 2/7, 2, 12/49, 12/49, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, 2/7, 2, 12/49, 12/49, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0069","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a41e9dc31b14445b69f2173bd2a9b973313fea8a7be8ef0878fcb1edd619cfd8","family_id":"qf-app20-growth-closed-form-audit","seed":202610206069,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[136,16],\"factor\":[3,2],\"initial\":[5,2],\"n\":3}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=5/2, factor=3/2, n=3, candidate=136/16. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(5/2, 3/2, 3, 135/16, 17/2, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/2, 3/2, 3, 135/16, 17/2, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0070","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46b2b0537fb2bcbb61e166f9c07734c2c57dc4a53dc642566b2ed1e84da18259","family_id":"qf-app20-growth-closed-form-audit","seed":202610206070,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2816,405],\"factor\":[4,3],\"initial\":[11,5],\"n\":4}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=11/5, factor=4/3, n=4, candidate=2816/405. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(11/5, 4/3, 4, 2816/405, 2816/405, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (11/5, 4/3, 4, 2816/405, 2816/405, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0071","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19c5100477b251c8e9849ae71ad8e4b4bfd1eedb29244f6780e4d45019adce30","family_id":"qf-app20-growth-closed-form-audit","seed":202610206071,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[12,1],\"factor\":[1,4],\"initial\":[12,1],\"n\":0}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=12/1, factor=1/4, n=0, candidate=12/1. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(12, 1/4, 0, 12, 12, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (12, 1/4, 0, 12, 12, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0072","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b3642135f17c2de470ca02036b089aa23b574a6fea2de3c432e3c67605ad710","family_id":"qf-app20-growth-closed-form-audit","seed":202610206072,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[14,5],\"factor\":[2,5],\"initial\":[13,2],\"n\":1}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=13/2, factor=2/5, n=1, candidate=14/5. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(13/2, 2/5, 1, 13/5, 14/5, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (13/2, 2/5, 1, 13/5, 14/5, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0073","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff7be48aaa3922ec931058598b5a2a540d4d1969a70367e51f0e7adc7aa90582","family_id":"qf-app20-growth-closed-form-audit","seed":202610206073,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[7,6],\"factor\":[1,2],\"initial\":[14,3],\"n\":2}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=14/3, factor=1/2, n=2, candidate=7/6. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(14/3, 1/2, 2, 7/6, 7/6, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (14/3, 1/2, 2, 7/6, 7/6, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0074","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c257813fdd5c88d8986bd275695ddbb284db764b98de9ff48c164440178eac2","family_id":"qf-app20-growth-closed-form-audit","seed":202610206074,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[240,343],\"factor\":[4,7],\"initial\":[15,4],\"n\":3}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=15/4, factor=4/7, n=3, candidate=240/343. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(15/4, 4/7, 3, 240/343, 240/343, Yes)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (15/4, 4/7, 3, 240/343, 240/343, Yes)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0075","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51a905b6566df0605e650af95cd9aa5ee72b34a1cef4897592f5ffb4164f0c69","family_id":"qf-app20-growth-closed-form-audit","seed":202610206075,"semantic_template":"qf-app20-growth-closed-form-audit/{\"candidate\":[2,5],\"factor\":[1,2],\"initial\":[16,5],\"n\":4}"},"fields":{"title":"Normalize a Candidate before a Fixed-Factor Closed-Form Audit","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. Use exact rational inputs. Normalize denominator signs and reduce output fractions; print integers without /1. These specified-value checks do not identify truncation, rounding, measurement or model-error causes. initial=16/5, factor=1/2, n=4, candidate=2/5. Give (initial, factor, n, exact term, normalized candidate, equal?).","answer":{"kind":"expression","value":"(16/5, 1/2, 4, 1/5, 2/5, No)","explanation":"Compute the explicit recurrence-model term and compare rational values, not raw representations. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (16/5, 1/2, 4, 1/5, 2/5, No)."},"skills":["Normalize a Candidate before a Fixed-Factor Closed-Form Audit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0076","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"860e9a9f3582025c34336758450e500fc402db57d657e3321c49ce2b4c2b7bbc","family_id":"qf-app20-growth-factor-regime","seed":202610206076,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[0,1]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=0/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(0, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0077","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0836f04098678a2590daad928e8030da38bd416ce904dfb41f67610775ff8bd7","family_id":"qf-app20-growth-factor-regime","seed":202610206077,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[1,2]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=1/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(1/2, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/2, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0078","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4aa24cc61362c5318ba43597d1815b058dadfe19f509b47f36a5e0044a977204","family_id":"qf-app20-growth-factor-regime","seed":202610206078,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[2,3]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=2/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(2/3, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2/3, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0079","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f9d04e156ff070a3c433d9018852a49fafb13f0d036f49312e7c92059e16178","family_id":"qf-app20-growth-factor-regime","seed":202610206079,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[3,4]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(3/4, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/4, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0080","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2a59ac82d380277fb2f694a2d3f54a320d33ad7f76395007f3a947c23ca9083","family_id":"qf-app20-growth-factor-regime","seed":202610206080,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[4,5]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=4/5. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(4/5, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/5, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0081","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc51663f2e53ff7f454616a3c7245b7ea0948c22c8f71fdd626040aab1e8bf9e","family_id":"qf-app20-growth-factor-regime","seed":202610206081,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[5,6]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=5/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(5/6, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/6, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0082","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3eada156c72134d1eef83dbb120a2cc3533ddad312d3af0378e88e9250ca233","family_id":"qf-app20-growth-factor-regime","seed":202610206082,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[6,7]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=6/7. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(6/7, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (6/7, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0083","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40404cff0a5f5733169d9b72743e049685b1ed772ac4a597b56409bb3662df34","family_id":"qf-app20-growth-factor-regime","seed":202610206083,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[7,1]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=7/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(7, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0084","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18087bee263c84da85ef149ab79d20fadd9bddde21b17423e0a4fcd585c17173","family_id":"qf-app20-growth-factor-regime","seed":202610206084,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[8,2]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=8/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(4, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0085","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eea56e88224ff75d72bc248d46671a409adbb09db8205375e4e82e34fe1f169d","family_id":"qf-app20-growth-factor-regime","seed":202610206085,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[0,3]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=0/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(0, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0086","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed66d030538a3d0bf61e3ac2811c87bad6f9008f3a500713862db5d35bf52b3b","family_id":"qf-app20-growth-factor-regime","seed":202610206086,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[1,4]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=1/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(1/4, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/4, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0087","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"016a30d9399e4f34be6b453bcc77f25744070c942e8542a7d1ce0dbf5c22fd8d","family_id":"qf-app20-growth-factor-regime","seed":202610206087,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[2,5]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=2/5. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(2/5, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2/5, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0088","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73eba36b28572f230334b392ed2339abbdb1b681835dbe082f511ec10781f8eb","family_id":"qf-app20-growth-factor-regime","seed":202610206088,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[3,6]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(1/2, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/2, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0089","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0904448d6ffff1b3ae69ce31a39c64ba3ff4ab649bf867a5bbc6460b99c7f1ac","family_id":"qf-app20-growth-factor-regime","seed":202610206089,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[4,7]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=4/7. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(4/7, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (4/7, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0090","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb343656b81c4f2a13ba81afcb8e99362e797cbe2227f02bfc62aaa964e9a2c1","family_id":"qf-app20-growth-factor-regime","seed":202610206090,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[5,1]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=5/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(5, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0091","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1be924c92102c1f84dd819729133721cd59e5b1e255f237767c19c64bb909009","family_id":"qf-app20-growth-factor-regime","seed":202610206091,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[6,2]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=6/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(3, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0092","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4403157879d2ec6bc774313cfb58de7ee3b15bf33651c677109b7b9dea232f3","family_id":"qf-app20-growth-factor-regime","seed":202610206092,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[7,3]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=7/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(7/3, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (7/3, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0093","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"711f2ae861bc7651e7fa1dabd197b46260a3d37b15e82302988ba6ae185e3e2b","family_id":"qf-app20-growth-factor-regime","seed":202610206093,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[8,4]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=8/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(2, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0094","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1bb52bad857d2244f34673a160cec60dff6d2ef0d1f7b8a583b8c3e39e9aa8a","family_id":"qf-app20-growth-factor-regime","seed":202610206094,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[0,5]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=0/5. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(0, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (0, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0095","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c7880144a98ad6e97627f3e46c2ed4d69de6f623a16b1e3842405732a71ddbb3","family_id":"qf-app20-growth-factor-regime","seed":202610206095,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[1,6]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=1/6. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(1/6, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (1/6, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0096","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6991f08bd8b715c4154bcfbb3960e92b875120cfdab738980a1870aaef88d7fb","family_id":"qf-app20-growth-factor-regime","seed":202610206096,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[2,7]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=2/7. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(2/7, factor at least 0 and below 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2/7, factor at least 0 and below 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0097","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8acb8e3cf8dd95f58b407f7f4ca3ad1dc79659b7d4fb256dd46055acd0343077","family_id":"qf-app20-growth-factor-regime","seed":202610206097,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[3,1]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=3/1. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(3, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0098","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fed226fb84d142e883765d3ad9fb9a8ceb4a49690ffa8c0d4dbf2a3de8814b6","family_id":"qf-app20-growth-factor-regime","seed":202610206098,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[4,2]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=4/2. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(2, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (2, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0099","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24cd02a11c641dfe623f775f822d2a10d3da22e0e417dbc7b4749b5ca6c4a6f0","family_id":"qf-app20-growth-factor-regime","seed":202610206099,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[5,3]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=5/3. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(5/3, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (5/3, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261020-r06-discrete-exponential-growth-0100","batch_id":"batch-20261013-advance-20261020-r06-discrete-exponential-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2d80c3f27614792af66aa10900db30e99425d0c6490795d4da5f2bf49afa22a","family_id":"qf-app20-growth-factor-regime","seed":202610206100,"semantic_template":"qf-app20-growth-factor-regime/{\"factor\":[6,4]}"},"fields":{"title":"Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity","stem":"Use fixed discrete time steps and one exact nonnegative rational factor throughout. This is not a continuous-time model or an empirical population prediction; signed initial quantities and the zero case must retain their specified values. factor=6/4. Give (normalized factor, one label: factor greater than 1 / factor equals 1 / factor at least 0 and below 1). The last case includes factor 0. A zero initial value remains zero, and signed initial data do not all increase when the factor exceeds 1.","answer":{"kind":"expression","value":"(3/2, factor greater than 1)","explanation":"Compare the given factor with 1; this regime label does not certify observed growth, empirical fit or continuous-time behavior. Exact finite model checks are not general theorems or empirical forecasts. The exact result, retaining the requested order, is (3/2, factor greater than 1)."},"skills":["Classify the Nonnegative Factor, Not Every Orbit’s Monotonicity"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}