{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261021-r07-discrete-logistic-growth-0001","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c0efdca22f10f25c77f8e2c20b711687b7efb1e024a0b3b846f60f9f9bf9c2","family_id":"qf-app21-logistic-single-step","seed":202610217001,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[0,5],\"rate\":[0,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=0/5, rate=0/1, capacity=5/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(0, 0, 5, 0)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 0, 5, 0)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0002","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"374e078241dbd656d28116a540035a2dfa142613e6b657c8ffbd9e96ff1e4ed8","family_id":"qf-app21-logistic-single-step","seed":202610217002,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[1,5],\"rate\":[1,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=1/5, rate=1/4, capacity=6/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(1/5, 1/4, 6, 149/600)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1/5, 1/4, 6, 149/600)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0003","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83f367167137ed134cbc195a4eb4b89a8c4ff428c950eaf3b4a3eb531c0dbf3b","family_id":"qf-app21-logistic-single-step","seed":202610217003,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[2,5],\"rate\":[1,2]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=2/5, rate=1/2, capacity=7/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(2/5, 1/2, 7, 103/175)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2/5, 1/2, 7, 103/175)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0004","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c18133ee52176bf67a8f9384f8a13734cc79fc39f22a15d794d7e715f7cbb93","family_id":"qf-app21-logistic-single-step","seed":202610217004,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[3,5],\"rate\":[3,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=3/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(3/5, 3/4, 8, 813/800)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3/5, 3/4, 8, 813/800)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0005","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b185a89acb45b6b89a5a02bb24989362e8835b9af7f32636586c56a617f752c5","family_id":"qf-app21-logistic-single-step","seed":202610217005,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[4,5],\"rate\":[1,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=4/5, rate=1/1, capacity=9/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(4/5, 1, 9, 344/225)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4/5, 1, 9, 344/225)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0006","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a57bc399839eaf7a2fdc1d293e402e714c62be4bd62582a0568d3fd80282914","family_id":"qf-app21-logistic-single-step","seed":202610217006,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[5,5],\"rate\":[0,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=5/5, rate=0/1, capacity=5/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(1, 0, 5, 1)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 0, 5, 1)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0007","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4242f984d6cb59f0ba6f621e69cf9a72d12368cbf31ab05f420dd57558fca010","family_id":"qf-app21-logistic-single-step","seed":202610217007,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[6,5],\"rate\":[1,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=6/5, rate=1/4, capacity=6/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(6/5, 1/4, 6, 36/25)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6/5, 1/4, 6, 36/25)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0008","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83b3f15474c9ad4afa21789823e21c0be2c497ac6021cd9da3496df0f896aeb4","family_id":"qf-app21-logistic-single-step","seed":202610217008,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[7,5],\"rate\":[1,2]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=7/5, rate=1/2, capacity=7/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(7/5, 1/2, 7, 49/25)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7/5, 1/2, 7, 49/25)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0009","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ad5ec37952730610fff52ef6a46ce16eef1ceb82324ca0fb9e060524645e7eb","family_id":"qf-app21-logistic-single-step","seed":202610217009,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[8,5],\"rate\":[3,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=8/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(8/5, 3/4, 8, 64/25)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8/5, 3/4, 8, 64/25)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0010","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"799f323e755f5ba8d39be6363f8bc3756274043e5d6b3ad385563fffa5b7ef73","family_id":"qf-app21-logistic-single-step","seed":202610217010,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[9,5],\"rate\":[1,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=9/5, rate=1/1, capacity=9/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(9/5, 1, 9, 81/25)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9/5, 1, 9, 81/25)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0011","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96ed4dc09764b44e39f987a45638e5751391e9195c0c7a40c80b5ad56111fc1b","family_id":"qf-app21-logistic-single-step","seed":202610217011,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[10,5],\"rate\":[0,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=10/5, rate=0/1, capacity=5/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(2, 0, 5, 2)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 0, 5, 2)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0012","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b32586736883985c1b7002df8f73b3d97256c5d8ec42e2e7b57e5bd5c628898d","family_id":"qf-app21-logistic-single-step","seed":202610217012,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[11,5],\"rate\":[1,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=11/5, rate=1/4, capacity=6/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(11/5, 1/4, 6, 1529/600)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11/5, 1/4, 6, 1529/600)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0013","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cfd05bb070905c30a57f104ab949eb3b078eb8efcd0c44ba3075a988d642f50","family_id":"qf-app21-logistic-single-step","seed":202610217013,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[12,5],\"rate\":[1,2]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=12/5, rate=1/2, capacity=7/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(12/5, 1/2, 7, 558/175)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12/5, 1/2, 7, 558/175)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0014","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da7227506c626552563068a095186609f4393dfafd9cae2630024b244465f8a","family_id":"qf-app21-logistic-single-step","seed":202610217014,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[13,5],\"rate\":[3,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=13/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(13/5, 3/4, 8, 3133/800)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13/5, 3/4, 8, 3133/800)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0015","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a759380d501f1067368a26e33667acb0a3f61aaa03fadcb6ddd37202fa234f0d","family_id":"qf-app21-logistic-single-step","seed":202610217015,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[14,5],\"rate\":[1,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=14/5, rate=1/1, capacity=9/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(14/5, 1, 9, 1064/225)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14/5, 1, 9, 1064/225)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0016","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d626437f6c5ea1948ae0c9ea052584820613afe5ce76b9c36a2e7f29698b3f48","family_id":"qf-app21-logistic-single-step","seed":202610217016,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[15,5],\"rate\":[0,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=15/5, rate=0/1, capacity=5/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(3, 0, 5, 3)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 0, 5, 3)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0017","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bc1fc8c56743fada50cf420d108d145d3bdaa79afa17ba690dba2f8d2d73076","family_id":"qf-app21-logistic-single-step","seed":202610217017,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[16,5],\"rate\":[1,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=16/5, rate=1/4, capacity=6/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(16/5, 1/4, 6, 268/75)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16/5, 1/4, 6, 268/75)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0018","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fcf5a2ad9d894573256fc42e3513c1a4cfdd765f7171d7d2bea3b1a1e412baf","family_id":"qf-app21-logistic-single-step","seed":202610217018,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[17,5],\"rate\":[1,2]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=17/5, rate=1/2, capacity=7/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(17/5, 1/2, 7, 748/175)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17/5, 1/2, 7, 748/175)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0019","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86a17c3220d86a6a5404018d07e65a5f38fe3b44a260baae66fea95a5f857536","family_id":"qf-app21-logistic-single-step","seed":202610217019,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[18,5],\"rate\":[3,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=18/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(18/5, 3/4, 8, 1017/200)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18/5, 3/4, 8, 1017/200)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0020","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ff90a481b4ccd6d020724a8ead6d8921c1cd47f1239f2b7c47c77b3f9ace131","family_id":"qf-app21-logistic-single-step","seed":202610217020,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[19,5],\"rate\":[1,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=19/5, rate=1/1, capacity=9/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(19/5, 1, 9, 1349/225)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19/5, 1, 9, 1349/225)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0021","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90a80b2a0b861dfaf03df3c98d29569359042ea61968890452d5f3cac51ff6b8","family_id":"qf-app21-logistic-single-step","seed":202610217021,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[5,1],\"current\":[20,5],\"rate\":[0,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=20/5, rate=0/1, capacity=5/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(4, 0, 5, 4)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 0, 5, 4)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0022","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad92b8e19a0b3d9a4e91103391012a15bbfed3cc4b3ada96212bcf714e15f68d","family_id":"qf-app21-logistic-single-step","seed":202610217022,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[6,1],\"current\":[21,5],\"rate\":[1,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=21/5, rate=1/4, capacity=6/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(21/5, 1/4, 6, 903/200)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21/5, 1/4, 6, 903/200)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0023","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e00e181416e9ccf4e5d55b7697d6a2d1418ff88d7b0b973d54a1ca9130268fc","family_id":"qf-app21-logistic-single-step","seed":202610217023,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[7,1],\"current\":[22,5],\"rate\":[1,2]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=22/5, rate=1/2, capacity=7/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(22/5, 1/2, 7, 913/175)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22/5, 1/2, 7, 913/175)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0024","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cfdee41117d50714570efb6b562836ce482606c3ab2ca42963eb63a16ff52f3","family_id":"qf-app21-logistic-single-step","seed":202610217024,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[8,1],\"current\":[23,5],\"rate\":[3,4]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=23/5, rate=3/4, capacity=8/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(23/5, 3/4, 8, 4853/800)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23/5, 3/4, 8, 4853/800)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0025","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fef7b339c191cbc11a4052afdd6853a27718b3b44e59b814d16a5ba1896a509","family_id":"qf-app21-logistic-single-step","seed":202610217025,"semantic_template":"qf-app21-logistic-single-step/{\"capacity\":[9,1],\"current\":[24,5],\"rate\":[1,1]}"},"fields":{"title":"One Exact Capacity-Limited Discrete Increment","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=24/5, rate=1/1, capacity=9/1. Give (current,rate,capacity,next).","answer":{"kind":"expression","value":"(24/5, 1, 9, 176/25)","explanation":"Compute the increment and add it to the current state; do not replace this model with r*x*(1−x/K). This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24/5, 1, 9, 176/25)."},"skills":["One Exact Capacity-Limited Discrete Increment"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0026","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c560cb92c0913edc8b50d2ccd7d3226af4178bf24f819ccf8156a0ae6afd6272","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217026,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[0,25],\"rate\":[0,1],\"steps\":2}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=0/25, rate=0/1, capacity=5/1, updates=2. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(0, 0, 5, 2, [0, 0, 0])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 0, 5, 2, [0, 0, 0])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0027","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eba1a70568958298f77cf04e903b955f6bb0f9d7df3df6881f6de25630149dcf","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217027,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[1,25],\"rate\":[1,4],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=1/25, rate=1/4, capacity=6/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(1/25, 1/4, 6, 1, [1/25, 749/15000])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1/25, 1/4, 6, 1, [1/25, 749/15000])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0028","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a449884eee319e0e23ef45df7d63ffdd0913a2a5dea8b4403f55274c90d24c0e","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217028,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[2,25],\"rate\":[1,2],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=2/25, rate=1/2, capacity=7/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(2/25, 1/2, 7, 0, [2/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2/25, 1/2, 7, 0, [2/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0029","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95876828a7dc7e8d9eeb63c4acd3e6805b913f302494f2232602a6f2a987f647","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217029,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[3,25],\"rate\":[3,4],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=3/25, rate=3/4, capacity=8/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(3/25, 3/4, 8, 1, [3/25, 4173/20000])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3/25, 3/4, 8, 1, [3/25, 4173/20000])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0030","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38058d194898eac46e323f8c3bf1ed724954192d46ddee6afaf01bd2b8fd4db4","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217030,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[4,25],\"rate\":[1,1],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=4/25, rate=1/1, capacity=9/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(4/25, 1, 9, 0, [4/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4/25, 1, 9, 0, [4/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0031","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57c3bda3ab5fbbc9ef4dee505775595fa607fae2b44c64ff6ef4790a3924dd48","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217031,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[5,25],\"rate\":[0,1],\"steps\":2}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=5/25, rate=0/1, capacity=5/1, updates=2. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(1/5, 0, 5, 2, [1/5, 1/5, 1/5])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1/5, 0, 5, 2, [1/5, 1/5, 1/5])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0032","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a59009ac32f87cba54a77e759a4d61f543a04187463f905329f78161b2f1597b","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217032,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[6,25],\"rate\":[1,4],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=6/25, rate=1/4, capacity=6/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(6/25, 1/4, 6, 0, [6/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6/25, 1/4, 6, 0, [6/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0033","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4830894b01fec58195ff0aa31a290185fdbe4a330425784d4c21365601dada5","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217033,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[7,25],\"rate\":[1,2],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=7/25, rate=1/2, capacity=7/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(7/25, 1/2, 7, 1, [7/25, 259/625])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7/25, 1/2, 7, 1, [7/25, 259/625])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0034","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4afed28be1bff40abf7706329d69cd3a7667a315833507b38cb38f994dceb017","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217034,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[8,25],\"rate\":[3,4],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=8/25, rate=3/4, capacity=8/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(8/25, 3/4, 8, 0, [8/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8/25, 3/4, 8, 0, [8/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0035","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39717df847e00fdcaf02d7f462e0c4959494b8778495f05a42de17188755313a","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217035,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[9,25],\"rate\":[1,1],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=9/25, rate=1/1, capacity=9/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(9/25, 1, 9, 1, [9/25, 441/625])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9/25, 1, 9, 1, [9/25, 441/625])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0036","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae6d162a2975b0a1092b94356d62d31b7bc84f90d78f9b27630085a4a27f4bf4","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217036,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[10,25],\"rate\":[0,1],\"steps\":2}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=10/25, rate=0/1, capacity=5/1, updates=2. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(2/5, 0, 5, 2, [2/5, 2/5, 2/5])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2/5, 0, 5, 2, [2/5, 2/5, 2/5])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0037","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a859b9da945b8db569340b5b7ab78718ec3d537e2e35545db86f35900ea96ced","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217037,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[11,25],\"rate\":[1,4],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=11/25, rate=1/4, capacity=6/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(11/25, 1/4, 6, 1, [11/25, 8129/15000])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11/25, 1/4, 6, 1, [11/25, 8129/15000])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0038","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"734e914760feec3022c948e31c87489ed457e95b3d6c79df7731a8e076598f72","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217038,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[12,25],\"rate\":[1,2],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=12/25, rate=1/2, capacity=7/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(12/25, 1/2, 7, 0, [12/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12/25, 1/2, 7, 0, [12/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0039","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d482260fbeb83fec60398f7c07b09b218b160034d916d3ad2a17e6bc1c499433","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217039,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[13,25],\"rate\":[3,4],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=13/25, rate=3/4, capacity=8/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(13/25, 3/4, 8, 1, [13/25, 17693/20000])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13/25, 3/4, 8, 1, [13/25, 17693/20000])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0040","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"245969877e4fd40bc5dba13cee0f007033a33cb7ee6dfd5eb925a8f688dfd048","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217040,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[14,25],\"rate\":[1,1],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=14/25, rate=1/1, capacity=9/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(14/25, 1, 9, 0, [14/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14/25, 1, 9, 0, [14/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0041","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f423f08e6bf1b93080aa04da32e2b25ecae01096215bb2c38503d9c9e4ce1723","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217041,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[15,25],\"rate\":[0,1],\"steps\":2}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=15/25, rate=0/1, capacity=5/1, updates=2. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(3/5, 0, 5, 2, [3/5, 3/5, 3/5])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3/5, 0, 5, 2, [3/5, 3/5, 3/5])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0042","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e827e64de01638bcbf547645e9c365a15fcc2faa172784a30d1c41bc418643f","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217042,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[16,25],\"rate\":[1,4],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=16/25, rate=1/4, capacity=6/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(16/25, 1/4, 6, 0, [16/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16/25, 1/4, 6, 0, [16/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0043","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a9eab473d225525cc20d50c5eb0adb8cffdb70d3163b93f92cafa0870517962","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217043,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[17,25],\"rate\":[1,2],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=17/25, rate=1/2, capacity=7/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(17/25, 1/2, 7, 1, [17/25, 4318/4375])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17/25, 1/2, 7, 1, [17/25, 4318/4375])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0044","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c9a0125e99c7d37ac4839ca997d3cda1680017127f035c43e6dfc0bf04d4747","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217044,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[18,25],\"rate\":[3,4],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=18/25, rate=3/4, capacity=8/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(18/25, 3/4, 8, 0, [18/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18/25, 3/4, 8, 0, [18/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0045","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbe315829f6ebd7d7a8dc8d0ac9a64383788a61ae65af3ae372beda03d34ab12","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217045,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[19,25],\"rate\":[1,1],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=19/25, rate=1/1, capacity=9/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(19/25, 1, 9, 1, [19/25, 8189/5625])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19/25, 1, 9, 1, [19/25, 8189/5625])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0046","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d1ea4b02230416536f610eb9c9295dd34d69f01f848fe50d90b4b69c35938ca","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217046,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[5,1],\"initial\":[20,25],\"rate\":[0,1],\"steps\":2}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=20/25, rate=0/1, capacity=5/1, updates=2. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(4/5, 0, 5, 2, [4/5, 4/5, 4/5])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4/5, 0, 5, 2, [4/5, 4/5, 4/5])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0047","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0ad9a5061528993ed30d5c91ae9161c30e9e5e4abf8eefe3511510f92758cb5","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217047,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[6,1],\"initial\":[21,25],\"rate\":[1,4],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=21/25, rate=1/4, capacity=6/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(21/25, 1/4, 6, 1, [21/25, 5103/5000])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21/25, 1/4, 6, 1, [21/25, 5103/5000])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0048","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d810e686a3f090c971a3f72762ae6f6c72ec504e8f966fef0d0b7096384178f","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217048,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[7,1],\"initial\":[22,25],\"rate\":[1,2],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=22/25, rate=1/2, capacity=7/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(22/25, 1/2, 7, 0, [22/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22/25, 1/2, 7, 0, [22/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0049","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b06bf75750da05d8ee2eb2c6be82c1871f62e80659cc6812983b57f8d074838","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217049,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[8,1],\"initial\":[23,25],\"rate\":[3,4],\"steps\":1}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=23/25, rate=3/4, capacity=8/1, updates=1. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(23/25, 3/4, 8, 1, [23/25, 30613/20000])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23/25, 3/4, 8, 1, [23/25, 30613/20000])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0050","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc7f8bc902bb801e5382acdb9745df6a5cc4320778785ebb8a144c28b9442df8","family_id":"qf-app21-logistic-orbit-prefix","seed":202610217050,"semantic_template":"qf-app21-logistic-orbit-prefix/{\"capacity\":[9,1],\"initial\":[24,25],\"rate\":[1,1],\"steps\":0}"},"fields":{"title":"A Short Exact Orbit of the Specified Logistic Increment Model","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. initial=24/25, rate=1/1, capacity=9/1, updates=0. Give (initial,rate,capacity,update count,orbit including initial). With zero updates retain just the initial value.","answer":{"kind":"expression","value":"(24/25, 1, 9, 0, [24/25])","explanation":"Reapply the same exact update, retaining all rational values without rounding. A short orbit is not a long-term stability or convergence proof. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24/25, 1, 9, 0, [24/25])."},"skills":["A Short Exact Orbit of the Specified Logistic Increment Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0051","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f56a2942efb91a8a6efe29182f83ce24228f44abb760a09ffce2c744c9ffb8","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217051,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[0,5],\"rate\":[1,2]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=0/5, rate=1/2, capacity=5/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(0, 1/2, 5, 0, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 1/2, 5, 0, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0052","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb0af59214d7642fed0fe5e1460912b262b1cdc73308d24182cabb176c2a97bb","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217052,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[1,5],\"rate\":[3,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=1/5, rate=3/4, capacity=6/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(1/5, 3/4, 6, 69/200, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1/5, 3/4, 6, 69/200, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0053","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dfc91dc667e41bff556f1b8b8c957783a123f74f54d270e839b3695bbcf3c8f","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217053,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[2,5],\"rate\":[1,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=2/5, rate=1/1, capacity=7/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(2/5, 1, 7, 136/175, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2/5, 1, 7, 136/175, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0054","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"124d94276aed92caaa10fc5f0c719d427f5d8f1a4be5b5a4e51d9885344ab599","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217054,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[3,5],\"rate\":[0,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=3/5, rate=0/1, capacity=8/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(3/5, 0, 8, 3/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3/5, 0, 8, 3/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0055","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b22af41fbd64b8d81c18375860fa70128d64c474b7cb373149c37893a4b087a7","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217055,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[4,5],\"rate\":[1,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=4/5, rate=1/4, capacity=9/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(4/5, 1/4, 9, 221/225, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4/5, 1/4, 9, 221/225, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0056","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23b6dc3d00d0ca8745774b522e1141db7a3f532421f38aa8a9d1c10bcd55eb8c","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217056,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[5,5],\"rate\":[1,2]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=5/5, rate=1/2, capacity=5/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(1, 1/2, 5, 7/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 1/2, 5, 7/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0057","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dc5db7c29a899eec95cc754814e82d649f5f1bed3999b61e71cb727259cbb08","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217057,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[6,5],\"rate\":[3,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=6/5, rate=3/4, capacity=6/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(6/5, 3/4, 6, 48/25, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6/5, 3/4, 6, 48/25, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0058","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db5011f22822ca83f502051fd16b711786c53950628961275547b45b652adcad","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217058,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[7,5],\"rate\":[1,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=7/5, rate=1/1, capacity=7/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(7/5, 1, 7, 63/25, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (7/5, 1, 7, 63/25, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0059","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eb3a42f0622ae121e2c5c59e2c4b3dd62b44ae4835d2b414b7ab28c54cb03b9","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217059,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[8,5],\"rate\":[0,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=8/5, rate=0/1, capacity=8/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(8/5, 0, 8, 8/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (8/5, 0, 8, 8/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0060","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74912a6ae1bcb0d606a78f0a309d75914f40155f245b24b7e324334d1a86d4b9","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217060,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[9,5],\"rate\":[1,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=9/5, rate=1/4, capacity=9/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(9/5, 1/4, 9, 54/25, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9/5, 1/4, 9, 54/25, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0061","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"123d09190f0b0156b3893030e832121f616e5e37fcec87763d2e4b3859a96b00","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217061,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[10,5],\"rate\":[1,2]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=10/5, rate=1/2, capacity=5/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(2, 1/2, 5, 13/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (2, 1/2, 5, 13/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0062","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c7d5319aa9d426ef2f06f292eb43368ea61539a1ab6e954fff1556b05386f46","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217062,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[11,5],\"rate\":[3,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=11/5, rate=3/4, capacity=6/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(11/5, 3/4, 6, 649/200, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (11/5, 3/4, 6, 649/200, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0063","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"343e6aa7e01c471b925dcfcdd10afb2ad2e71e40ad145ceed24798950c1f7492","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217063,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[12,5],\"rate\":[1,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=12/5, rate=1/1, capacity=7/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(12/5, 1, 7, 696/175, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12/5, 1, 7, 696/175, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0064","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9de3da1d1fe013d468dcde41edf53d00ce5c5e4dff22eb52268d016383991b1c","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217064,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[13,5],\"rate\":[0,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=13/5, rate=0/1, capacity=8/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(13/5, 0, 8, 13/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (13/5, 0, 8, 13/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0065","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d129ceb18919fddddb2e647eae67b54277c6d86caa103dfa13b6ae222229e54","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217065,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[14,5],\"rate\":[1,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=14/5, rate=1/4, capacity=9/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(14/5, 1/4, 9, 1477/450, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (14/5, 1/4, 9, 1477/450, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0066","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ed81c3bc28345ce47657327212b38f18479d56f0aafdaa18fede520cd210dc1","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217066,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[15,5],\"rate\":[1,2]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=15/5, rate=1/2, capacity=5/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(3, 1/2, 5, 18/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (3, 1/2, 5, 18/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0067","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a270946a99f9276fdcdad219189afa6dcbf746844477337a9873a4eefac3b830","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217067,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[16,5],\"rate\":[3,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=16/5, rate=3/4, capacity=6/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(16/5, 3/4, 6, 108/25, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (16/5, 3/4, 6, 108/25, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0068","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f784511992a20d155fa4bf3f98999b8f066da0ccb2f1f98485bda6d51b60a226","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217068,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[17,5],\"rate\":[1,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=17/5, rate=1/1, capacity=7/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(17/5, 1, 7, 901/175, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (17/5, 1, 7, 901/175, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0069","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48453093d0dead3c50b15bafbc31da7fa62c02bc841adc12dc726c3944f97e72","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217069,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[18,5],\"rate\":[0,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=18/5, rate=0/1, capacity=8/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(18/5, 0, 8, 18/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18/5, 0, 8, 18/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0070","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06d4eba81e8f33956462fe20e22bd34d90517a445b9ff4182e77167f36bfb772","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217070,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[19,5],\"rate\":[1,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=19/5, rate=1/4, capacity=9/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(19/5, 1/4, 9, 1957/450, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (19/5, 1/4, 9, 1957/450, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0071","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93bbdae5c05f4b15dde13f6aa975c1118c2292b2decf539ac80da20e16d35d05","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217071,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[5,1],\"current\":[20,5],\"rate\":[1,2]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=20/5, rate=1/2, capacity=5/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(4, 1/2, 5, 22/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (4, 1/2, 5, 22/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0072","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e357bb625a8755776e42e8c6e259b8d485331db78c50471a487753e2e50462c5","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217072,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[6,1],\"current\":[21,5],\"rate\":[3,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=21/5, rate=3/4, capacity=6/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(21/5, 3/4, 6, 1029/200, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21/5, 3/4, 6, 1029/200, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0073","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a99d801a6501c06bd0c10f046bd00bc1a0a547ee32eca36a9ea337e1b1a32be","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217073,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[7,1],\"current\":[22,5],\"rate\":[1,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=22/5, rate=1/1, capacity=7/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(22/5, 1, 7, 1056/175, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (22/5, 1, 7, 1056/175, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0074","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47e1abfbae4024b3e63e29fe7565941651ded7d7b1769acdc0bab87ac6c1306c","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217074,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[8,1],\"current\":[23,5],\"rate\":[0,1]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=23/5, rate=0/1, capacity=8/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(23/5, 0, 8, 23/5, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (23/5, 0, 8, 23/5, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0075","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9dc1228d7081703894fa16e7420426343bd06ac99ac4ed263a6d12bc3680dcc7","family_id":"qf-app21-logistic-capacity-interval-audit","seed":202610217075,"semantic_template":"qf-app21-logistic-capacity-interval-audit/{\"capacity\":[9,1],\"current\":[24,5],\"rate\":[1,4]}"},"fields":{"title":"One-Step Retention of the Declared Capacity Interval","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. current=24/5, rate=1/4, capacity=9/1. Give (current,rate,capacity,next,0≤next≤capacity?).","answer":{"kind":"expression","value":"(24/5, 1/4, 9, 134/25, Yes)","explanation":"Evaluate the exact next state and both endpoint inequalities only in the qualified parameter domain; do not extrapolate to other logistic definitions or rates. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24/5, 1/4, 9, 134/25, Yes)."},"skills":["One-Step Retention of the Declared Capacity Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0076","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cca91b192289caf0b4668df08d87be6651dc3a0e19cd00f89866bbc52f8a74aa","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217076,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[5,1],\"rate\":[1,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=1/4, capacity=5/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 1/4, 5, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 1/4, 5, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0077","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f2dbcd4a2ececd5c03b2855350a6c1cafac8d49e2de90287fb925c3aa90ca38","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217077,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[6,1],\"capacity\":[6,1],\"rate\":[1,2]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=6/1, rate=1/2, capacity=6/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(6, 1/2, 6, 6, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (6, 1/2, 6, 6, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0078","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e9587ff9e9880289944aef51df0b01e9ffcdf931f4373e344ac0496cbc494ce","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217078,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[7,1],\"rate\":[3,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=3/4, capacity=7/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 3/4, 7, 23/14, No)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 3/4, 7, 23/14, No)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0079","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff89602982ad785dcb0389819e59c5dc580e0cc49df192a5b33438a4f059510b","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217079,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[8,1],\"rate\":[1,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=1/1, capacity=8/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 1, 8, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 1, 8, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0080","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2842b571a60e17e95293b9d3ac1e39ca7076b1bc6189b2f1285635e2b51ccab5","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217080,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[9,1],\"capacity\":[9,1],\"rate\":[0,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=9/1, rate=0/1, capacity=9/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(9, 0, 9, 9, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (9, 0, 9, 9, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0081","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"096288a57ed85ef7f5bd25f87560b3ab335d94042cac36da6b47e7c69aa40a89","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217081,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[10,1],\"rate\":[1,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=1/4, capacity=10/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 1/4, 10, 49/40, No)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 1/4, 10, 49/40, No)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0082","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"284ba22fed7e3b843a474cf43295596af0f9fb6f2fa081e73a77f3df928ad49a","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217082,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[11,1],\"rate\":[1,2]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=1/2, capacity=11/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 1/2, 11, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 1/2, 11, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0083","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8255a9dc597202d2f2f7ccf92f8960b1e4b8f3ce93e6b61ca5a5f9357ccacb7","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217083,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[12,1],\"capacity\":[12,1],\"rate\":[3,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=12/1, rate=3/4, capacity=12/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(12, 3/4, 12, 12, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (12, 3/4, 12, 12, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0084","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afa8a0e96404f3972d01ed3c9c25462ce8d463162f55122c59b85f89101cd4bf","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217084,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[13,1],\"rate\":[1,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=1/1, capacity=13/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 1, 13, 25/13, No)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 1, 13, 25/13, No)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0085","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c8f3758b09087bffb0bff42eb2a5063205233cfc5e6eb5642001a71989432fa","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217085,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[14,1],\"rate\":[0,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=0/1, capacity=14/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 0, 14, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 0, 14, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0086","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acb881a05dc81ac163b81149633035456d5c6ff1e04b6732b8d091700c35b5b3","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217086,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[15,1],\"capacity\":[15,1],\"rate\":[1,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=15/1, rate=1/4, capacity=15/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(15, 1/4, 15, 15, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (15, 1/4, 15, 15, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0087","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5ef5bfe7806d72087108fd00dff7cbb446d70c194bf9fad3742177f36ac1ccb","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217087,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[16,1],\"rate\":[1,2]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=1/2, capacity=16/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 1/2, 16, 47/32, No)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 1/2, 16, 47/32, No)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0088","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7da2c8050eba4375d4aca848cba1f1e191e7b41e7c6195f265f1373a629007f4","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217088,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[17,1],\"rate\":[3,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=3/4, capacity=17/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 3/4, 17, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 3/4, 17, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0089","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9e8b369bcc47e951e0269bd331f70750aa67f331d286bc8f877b20f70ea19d1e","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217089,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[18,1],\"capacity\":[18,1],\"rate\":[1,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=18/1, rate=1/1, capacity=18/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(18, 1, 18, 18, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (18, 1, 18, 18, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0090","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f435a76ae13045e02835c8c5da65130c0b55e15d9157b8f412913bb2f25a000","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217090,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[19,1],\"rate\":[0,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=0/1, capacity=19/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 0, 19, 1, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 0, 19, 1, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0091","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f93cccd227910092d1fbc00e38ea7be153bb234105d25dc845ae35945c3a6a23","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217091,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[20,1],\"rate\":[1,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=1/4, capacity=20/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 1/4, 20, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 1/4, 20, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0092","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55ec19d092c9cf1560bea7ffd93c2b74d8ecb22086b100c1cde74d56daab111e","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217092,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[21,1],\"capacity\":[21,1],\"rate\":[1,2]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=21/1, rate=1/2, capacity=21/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(21, 1/2, 21, 21, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (21, 1/2, 21, 21, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0093","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6bd70d2b01354c99fd16f90c692a6cd34182bc2112d78da71525f9a705ec49f","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217093,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[22,1],\"rate\":[3,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=3/4, capacity=22/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 3/4, 22, 151/88, No)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 3/4, 22, 151/88, No)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0094","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12d1663e319916c60bc0af8fe587b84dd8fa625ae91179153c8baadb97ac201d","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217094,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[23,1],\"rate\":[1,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=1/1, capacity=23/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 1, 23, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 1, 23, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0095","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e56c823b7fe539532f50aef131fa263728e3d6191cfa238c0920c296f785302","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217095,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[24,1],\"capacity\":[24,1],\"rate\":[0,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=24/1, rate=0/1, capacity=24/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(24, 0, 24, 24, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (24, 0, 24, 24, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0096","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"232bff5bc62faaa57d80ce9be9978f29fa268d2523ad689e6f2da72709eda3b5","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217096,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[25,1],\"rate\":[1,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=1/4, capacity=25/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 1/4, 25, 31/25, No)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 1/4, 25, 31/25, No)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0097","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25a4b9144e49ba99a9254d977ee38791adbdd3d63ec47ba2810de72cd0b24f63","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217097,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[26,1],\"rate\":[1,2]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=1/2, capacity=26/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 1/2, 26, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 1/2, 26, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0098","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b457700f5e1ad2b88353323e4f64ca34b66481d37074f79f95c9e2090c8e8cd","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217098,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[27,1],\"capacity\":[27,1],\"rate\":[3,4]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=27/1, rate=3/4, capacity=27/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(27, 3/4, 27, 27, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (27, 3/4, 27, 27, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0099","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e062900be4882a46704fb24e871d7c2c796d61120c6fd9ba739b46070288abc2","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217099,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[1,1],\"capacity\":[28,1],\"rate\":[1,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=1/1, rate=1/1, capacity=28/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(1, 1, 28, 55/28, No)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (1, 1, 28, 55/28, No)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261021-r07-discrete-logistic-growth-0100","batch_id":"batch-20261013-advance-20261021-r07-discrete-logistic-growth-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d83736092b65ff93a72c12e8ec459e859effb1f6a6166cbf4ba0d0eb037aab74","family_id":"qf-app21-logistic-equilibrium-audit","seed":202610217100,"semantic_template":"qf-app21-logistic-equilibrium-audit/{\"candidate\":[0,1],\"capacity\":[29,1],\"rate\":[0,1]}"},"fields":{"title":"A Fixed Candidate, Including the Zero-Rate Degeneracy","stem":"Use exactly the fixed discrete increment model next=x+r*x*(1−x/K), with K>0, 0≤r≤1 and 0≤x≤K, at fixed time steps. Reduce exact rational outputs and omit /1 for integers. This is neither another logistic-map convention nor a continuous logistic solution; a finite orbit proves no stability, convergence or empirical population fit. candidate=0/1, rate=0/1, capacity=29/1. Give (candidate,rate,capacity,next,next=candidate?). Both capacity endpoints are fixed; at rate=0 every allowed candidate is fixed.","answer":{"kind":"expression","value":"(0, 0, 29, 0, Yes)","explanation":"Test exact equality after one update. Being fixed is separate from stability, attraction, convergence or empirical fit. This explicit finite/exact slice is neither a general theorem nor a judgment about a person. The exact result, retaining the requested order, is (0, 0, 29, 0, Yes)."},"skills":["A Fixed Candidate, Including the Zero-Rate Degeneracy"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}