{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0001","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bd62db71ad40aa07f0f2b7875668e652977c8c5cf87bbdaa53bf500d87d11cb","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114001,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-2,0,2],\"values\":[0,[1,2],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-2, 0, 2], aligned exact values=[0, 1/2, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-2, 0, 2], [0, 1/2, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-2, 0, 2], [0, 1/2, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0002","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c3f8a7bf5c767bda554bd92a0b2010676714df1581286a991a59ea4fc962267","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114002,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-3,0,3],\"values\":[0,[2,3],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-3, 0, 3], aligned exact values=[0, 2/3, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-3, 0, 3], [0, 2/3, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-3, 0, 3], [0, 2/3, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0003","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cf2b7c1af298829cb135e2adc812177c7e950f35f875488a92b01d95a8f782e","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114003,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-4,0,4],\"values\":[0,[3,4],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-4, 0, 4], aligned exact values=[0, 3/4, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-4, 0, 4], [0, 3/4, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-4, 0, 4], [0, 3/4, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0004","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90b4f45af35809793dc97749d182ed7916cf242b23a87cd97a4329ab1f5e144f","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114004,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-5,0,5],\"values\":[0,[4,5],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-5, 0, 5], aligned exact values=[0, 4/5, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-5, 0, 5], [0, 4/5, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-5, 0, 5], [0, 4/5, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0005","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38b573f2f860923a4eb40f592d77cafa18eefdc7d08b262fd9d951beb1405a90","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114005,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-6,0,6],\"values\":[0,[5,6],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-6, 0, 6], aligned exact values=[0, 5/6, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-6, 0, 6], [0, 5/6, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-6, 0, 6], [0, 5/6, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0006","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"097ca2db6598196f007ec682c3a838cdba0ae536faaeb3c7927f9809a0616ca0","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114006,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-7,0,7],\"values\":[0,[6,7],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-7, 0, 7], aligned exact values=[0, 6/7, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-7, 0, 7], [0, 6/7, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-7, 0, 7], [0, 6/7, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0007","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fc5e40eb5cb072956fb7a10a793f28b3c78e1d152165ea2f4d7b4598e30bc37","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114007,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-8,0,8],\"values\":[0,[7,8],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-8, 0, 8], aligned exact values=[0, 7/8, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-8, 0, 8], [0, 7/8, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-8, 0, 8], [0, 7/8, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0008","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df58c0e08530c79741eab7645dd95cd4898d0f333c6a3ee7f63ef35406134c3b","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114008,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-9,0,9],\"values\":[0,[8,9],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-9, 0, 9], [0, 8/9, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-9, 0, 9], [0, 8/9, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0009","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23f1bf08403f4ce39ce4def4eee27a58e0d8ec33861d1a19dd421e98a0b44382","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114009,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-10,0,10],\"values\":[0,[9,10],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-10, 0, 10], aligned exact values=[0, 9/10, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-10, 0, 10], [0, 9/10, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-10, 0, 10], [0, 9/10, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0010","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"310453b8ea985b9c12a51c6ec316bb678a19250f1d06d5387725887a83d2b381","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114010,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-11,0,11],\"values\":[0,[10,11],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-11, 0, 11], aligned exact values=[0, 10/11, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-11, 0, 11], [0, 10/11, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-11, 0, 11], [0, 10/11, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0011","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"404a8ef62c99ca7c6e118a9756ec83a5f8311511d8e12cbc28de636b8dd639d6","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114011,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-12,0,12],\"values\":[0,[11,12],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-12, 0, 12], aligned exact values=[0, 11/12, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-12, 0, 12], [0, 11/12, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-12, 0, 12], [0, 11/12, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0012","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"035cda2b76e1de91713f02a9afeb2dff37c9ae818affb43672ac33d15982cb52","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114012,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-13,0,13],\"values\":[0,[12,13],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-13, 0, 13], aligned exact values=[0, 12/13, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-13, 0, 13], [0, 12/13, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-13, 0, 13], [0, 12/13, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0013","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbda86f0c3276378199128102fe8ce5f9438d934f6d8b6651efbb1c03361f41a","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114013,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-14,0,14],\"values\":[0,[13,14],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-14, 0, 14], aligned exact values=[0, 13/14, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-14, 0, 14], [0, 13/14, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-14, 0, 14], [0, 13/14, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0014","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f0e9e6286d6db83c8f3e96a1067eb0fc267b9f7fd9ff6e74d61b8806d376beb","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114014,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-15,0,15],\"values\":[0,[14,15],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-15, 0, 15], aligned exact values=[0, 14/15, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-15, 0, 15], [0, 14/15, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-15, 0, 15], [0, 14/15, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0015","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f9d5301656e633d29f5ad204ef8913e9adfad48be412ed473c8903fef1c4305","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114015,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-16,0,16],\"values\":[0,[15,16],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-16, 0, 16], aligned exact values=[0, 15/16, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-16, 0, 16], [0, 15/16, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-16, 0, 16], [0, 15/16, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0016","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b22598ebd30677866580df726cc30056a959600ae3d3b850ebd491e50397f8af","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114016,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-17,0,17],\"values\":[0,[16,17],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-17, 0, 17], [0, 16/17, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-17, 0, 17], [0, 16/17, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0017","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"105b109afbc7c43399fcdd6cae929ac7121709d9e9df7aa0370ccb37e650a721","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114017,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-18,0,18],\"values\":[0,[17,18],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-18, 0, 18], aligned exact values=[0, 17/18, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-18, 0, 18], [0, 17/18, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-18, 0, 18], [0, 17/18, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0018","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"649996631c93f6e4206cf8fa8c2b6a1821b62a1e3b94313a27514c38bd880761","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114018,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-19,0,19],\"values\":[0,[18,19],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-19, 0, 19], aligned exact values=[0, 18/19, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-19, 0, 19], [0, 18/19, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-19, 0, 19], [0, 18/19, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0019","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33e57b00a2775317851cd7731bbd9ef92c6690c8cd521f704363494d64d981f1","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114019,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-20,0,20],\"values\":[0,[19,20],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-20, 0, 20], aligned exact values=[0, 19/20, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-20, 0, 20], [0, 19/20, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-20, 0, 20], [0, 19/20, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0020","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02d863d2d97392e0fff55c7e29f307d4fe89138aaf3131ef4cf02134d69f84ac","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114020,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-21,0,21],\"values\":[0,[20,21],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-21, 0, 21], aligned exact values=[0, 20/21, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-21, 0, 21], [0, 20/21, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-21, 0, 21], [0, 20/21, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0021","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c24dc39fe1deac244ea26bca94d6d843465ea255f5acc0c9f05f2b9e9fafc4e","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114021,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-22,0,22],\"values\":[0,[21,22],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-22, 0, 22], aligned exact values=[0, 21/22, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-22, 0, 22], [0, 21/22, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-22, 0, 22], [0, 21/22, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0022","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7572e466ba43321847e211112b208ccd680895f0e2a5215768a266b61ffc2af9","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114022,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-23,0,23],\"values\":[0,[22,23],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-23, 0, 23], aligned exact values=[0, 22/23, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-23, 0, 23], [0, 22/23, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-23, 0, 23], [0, 22/23, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0023","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29c5e946cf26c5ee2d206831abdb832dad013bbd2224884c01a66a06afb8a51f","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114023,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-24,0,24],\"values\":[0,[23,24],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-24, 0, 24], aligned exact values=[0, 23/24, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-24, 0, 24], [0, 23/24, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-24, 0, 24], [0, 23/24, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0024","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13d474885e80020b10b39688e9da2d560d2132d675d8b788564e8ab64b08994d","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114024,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-25,0,25],\"values\":[0,[24,25],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-25, 0, 25], aligned exact values=[0, 24/25, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-25, 0, 25], [0, 24/25, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-25, 0, 25], [0, 24/25, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0025","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b8643d315e5843d6091dd776d3e814b68db6e81c313dd0dbb88424222026ac6","family_id":"qf-mod42-contcdf-piecewise-linear-qualification","seed":202611114025,"semantic_template":"qf-mod42-contcdf-piecewise-linear-qualification/{\"knots\":[-26,0,26],\"values\":[0,[25,26],1]}"},"fields":{"title":"A Continuous CDF May Have Flat Intervals","stem":"Use strictly increasing knots=[-26, 0, 26], aligned exact values=[0, 25/26, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Give (knots, values, continuous, nondecreasing, right_continuous, endpoint_limits_0_1, certificate).","answer":{"kind":"expression","value":"([-26, 0, 26], [0, 25/26, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)","explanation":"Audit continuity, nondecreasingness, right-continuity and endpoint limits0/1 for the whole declared piecewise-linear law. This concerns the CDF, not continuity of a random-variable map on an unspecified sample-space topology. The exact result, retaining the requested order, is ([-26, 0, 26], [0, 25/26, 1], Yes, Yes, Yes, Yes, linear_interpolation_is_continuous_and_flat_intervals_are_allowed)."},"skills":["A Continuous CDF May Have Flat Intervals"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0026","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c3c3a06bbf5487db16574ab4bc2ba6a622b4747991c99fb0c5d477eba592799","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114026,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-2,0,2],\"query_points\":[-3,0,3],\"values\":[0,[1,2],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-2, 0, 2], aligned exact values=[0, 1/2, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-3, 0, 3] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-3, 0], [0, 1/2], [3, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-3, 0], [0, 1/2], [3, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0027","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45666f1acdae5e36d014b7d83f2eb1c0d3720dfd04a0c17593ef2ca959134c92","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114027,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-3,0,3],\"query_points\":[-4,0,4],\"values\":[0,[2,3],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-3, 0, 3], aligned exact values=[0, 2/3, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-4, 0, 4] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-4, 0], [0, 2/3], [4, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-4, 0], [0, 2/3], [4, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0028","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d661774f3eec43aeb8d01160dfd4e7a07788dbf59a1b1d5b0757c238028c0e4b","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114028,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-4,0,4],\"query_points\":[-5,0,5],\"values\":[0,[3,4],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-4, 0, 4], aligned exact values=[0, 3/4, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-5, 0, 5] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-5, 0], [0, 3/4], [5, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-5, 0], [0, 3/4], [5, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0029","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95d26aa0af3b38787da81fad2597caa595bed4c25c060b7eb71b9dcf4044ce8a","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114029,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-5,0,5],\"query_points\":[-6,0,6],\"values\":[0,[4,5],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-5, 0, 5], aligned exact values=[0, 4/5, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-6, 0, 6] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-6, 0], [0, 4/5], [6, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-6, 0], [0, 4/5], [6, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0030","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46cf5981b46ea1caa2f515b2b8ec891aa2a709dd25765a79104af548eae72102","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114030,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-6,0,6],\"query_points\":[-7,0,7],\"values\":[0,[5,6],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-6, 0, 6], aligned exact values=[0, 5/6, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-7, 0, 7] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-7, 0], [0, 5/6], [7, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-7, 0], [0, 5/6], [7, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0031","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a3ca6b95f777172542abbf7e20699d102f72bcf1f8d667eb6c42f2afba6c0dc","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114031,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-7,0,7],\"query_points\":[-8,0,8],\"values\":[0,[6,7],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-7, 0, 7], aligned exact values=[0, 6/7, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-8, 0, 8] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-8, 0], [0, 6/7], [8, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-8, 0], [0, 6/7], [8, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0032","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"908b6b5efc33f5e813d472a63084c2eb82197014ee83bbe5161016dcfafc82c0","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114032,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-8,0,8],\"query_points\":[-9,0,9],\"values\":[0,[7,8],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-8, 0, 8], aligned exact values=[0, 7/8, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-9, 0, 9] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-9, 0], [0, 7/8], [9, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-9, 0], [0, 7/8], [9, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0033","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59d111ebcb767a318fdd3cbc8d7f930dfae610cefb8650c5a7d90a36e737e1d5","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114033,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-9,0,9],\"query_points\":[-10,0,10],\"values\":[0,[8,9],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-10, 0, 10] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-10, 0], [0, 8/9], [10, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-10, 0], [0, 8/9], [10, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0034","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dc20a515575d6023fc9446cd411a0f27609b7950966593f0d9f665b0cc9103eb","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114034,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-10,0,10],\"query_points\":[-11,0,11],\"values\":[0,[9,10],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-10, 0, 10], aligned exact values=[0, 9/10, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-11, 0, 11] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-11, 0], [0, 9/10], [11, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-11, 0], [0, 9/10], [11, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0035","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a4951c32508078b9603eeba1c7ce73e4f8962749d761ea4e84812b3b894244d","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114035,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-11,0,11],\"query_points\":[-12,0,12],\"values\":[0,[10,11],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-11, 0, 11], aligned exact values=[0, 10/11, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-12, 0, 12] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-12, 0], [0, 10/11], [12, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-12, 0], [0, 10/11], [12, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0036","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd93e4a530d84012e11763ef8f4397c8571f19b4acb0d455cde68fccb025ba5f","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114036,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-12,0,12],\"query_points\":[-13,0,13],\"values\":[0,[11,12],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-12, 0, 12], aligned exact values=[0, 11/12, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-13, 0, 13] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-13, 0], [0, 11/12], [13, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-13, 0], [0, 11/12], [13, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0037","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec85d566734bd90602a23f5d7ed205bd5c665891dcd5efdc4417f536b692c7e1","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114037,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-13,0,13],\"query_points\":[-14,0,14],\"values\":[0,[12,13],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-13, 0, 13], aligned exact values=[0, 12/13, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-14, 0, 14] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-14, 0], [0, 12/13], [14, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-14, 0], [0, 12/13], [14, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0038","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"713dbdd5c37022745493b44e2c1943d8c835e00b98417f3ade0224dcc0fb36a6","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114038,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-14,0,14],\"query_points\":[-15,0,15],\"values\":[0,[13,14],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-14, 0, 14], aligned exact values=[0, 13/14, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-15, 0, 15] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-15, 0], [0, 13/14], [15, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-15, 0], [0, 13/14], [15, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0039","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eef88361edcd27a0435a0308e046e5340a71be81102dd885cbf3f314f69a1c95","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114039,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-15,0,15],\"query_points\":[-16,0,16],\"values\":[0,[14,15],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-15, 0, 15], aligned exact values=[0, 14/15, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-16, 0, 16] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-16, 0], [0, 14/15], [16, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-16, 0], [0, 14/15], [16, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0040","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2af3a9a789c878dd96f8a5070f7f785d1c6a8dee874def972c3364746d1f9d23","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114040,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-16,0,16],\"query_points\":[-17,0,17],\"values\":[0,[15,16],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-16, 0, 16], aligned exact values=[0, 15/16, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-17, 0, 17] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-17, 0], [0, 15/16], [17, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-17, 0], [0, 15/16], [17, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0041","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"256b6f3a50477f242574834914396f69ead44ed8f7f6b2f6f71385900e57ad4b","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114041,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-17,0,17],\"query_points\":[-18,0,18],\"values\":[0,[16,17],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-18, 0, 18] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-18, 0], [0, 16/17], [18, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-18, 0], [0, 16/17], [18, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0042","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd1602f23d4d5b5f4e09b07ded7fbed2457880b7d48d897fd6c3d123669908e3","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114042,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-18,0,18],\"query_points\":[-19,0,19],\"values\":[0,[17,18],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-18, 0, 18], aligned exact values=[0, 17/18, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-19, 0, 19] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-19, 0], [0, 17/18], [19, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-19, 0], [0, 17/18], [19, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0043","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b8e0dcdce0be0234709895fb2ea2c877cd433645f8773654337cc4035d2728e","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114043,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-19,0,19],\"query_points\":[-20,0,20],\"values\":[0,[18,19],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-19, 0, 19], aligned exact values=[0, 18/19, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-20, 0, 20] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-20, 0], [0, 18/19], [20, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-20, 0], [0, 18/19], [20, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0044","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dca5448ca7aab3b96f14011a3d7c0db3827cfb925b925c9970c23d7b323dc89a","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114044,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-20,0,20],\"query_points\":[-21,0,21],\"values\":[0,[19,20],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-20, 0, 20], aligned exact values=[0, 19/20, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-21, 0, 21] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-21, 0], [0, 19/20], [21, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-21, 0], [0, 19/20], [21, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0045","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbea6208b7dde6047e6e689e277ddd05392ab73d6c9ebdf9bebb4ddc2968b202","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114045,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-21,0,21],\"query_points\":[-22,0,22],\"values\":[0,[20,21],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-21, 0, 21], aligned exact values=[0, 20/21, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-22, 0, 22] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-22, 0], [0, 20/21], [22, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-22, 0], [0, 20/21], [22, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0046","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8851abff11316dd89be09921661dd6913cb90c94d5e714b9ac1d6639d7de9eae","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114046,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-22,0,22],\"query_points\":[-23,0,23],\"values\":[0,[21,22],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-22, 0, 22], aligned exact values=[0, 21/22, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-23, 0, 23] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-23, 0], [0, 21/22], [23, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-23, 0], [0, 21/22], [23, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0047","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"096c9897024ce21c101d5c042a926655e576a7495cc17d21e499e1f56f425c49","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114047,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-23,0,23],\"query_points\":[-24,0,24],\"values\":[0,[22,23],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-23, 0, 23], aligned exact values=[0, 22/23, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-24, 0, 24] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-24, 0], [0, 22/23], [24, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-24, 0], [0, 22/23], [24, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0048","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c6895a3a1aed0b34e90f05e348dee8c86070d164298586a8541474fec918093","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114048,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-24,0,24],\"query_points\":[-25,0,25],\"values\":[0,[23,24],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-24, 0, 24], aligned exact values=[0, 23/24, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-25, 0, 25] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-25, 0], [0, 23/24], [25, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-25, 0], [0, 23/24], [25, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0049","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb61f1eee27ae118334482e200c1ca8085f2bcf65186cc55175fc06e8130ddc","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114049,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-25,0,25],\"query_points\":[-26,0,26],\"values\":[0,[24,25],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-25, 0, 25], aligned exact values=[0, 24/25, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-26, 0, 26] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-26, 0], [0, 24/25], [26, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-26, 0], [0, 24/25], [26, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0050","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42ea0fc20c649938e39ba7fed65761138c5aaf716f791531f41c78359fa78d62","family_id":"qf-mod42-contcdf-monotone-endpoint-audit","seed":202611114050,"semantic_template":"qf-mod42-contcdf-monotone-endpoint-audit/{\"knots\":[-26,0,26],\"query_points\":[-27,0,27],\"values\":[0,[25,26],1]}"},"fields":{"title":"Strictly Ordered Query Points May Have Equal CDF Values","stem":"Use strictly increasing knots=[-26, 0, 26], aligned exact values=[0, 25/26, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Exact query_points=[-27, 0, 27] must be strictly increasing. Give (query_value_rows, nondecreasing_sample, left_tail_value, right_tail_value, certificate).","answer":{"kind":"expression","value":"([[-27, 0], [0, 25/26], [27, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)","explanation":"Evaluate every query and compare successive values non-strictly. Retain left-tail0 and right-tail1 separately. Global properties come from the declared whole piecewise-linear construction, not the finite query list alone. The exact result, retaining the requested order, is ([[-27, 0], [0, 25/26], [27, 1]], Yes, 0, 1, nondecreasing_with_left_zero_and_right_one_limits)."},"skills":["Strictly Ordered Query Points May Have Equal CDF Values"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0051","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ada087bfd721b979597a74d74c8a594c518fa679e21bfd150fbdcc4e05a59ff5","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114051,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-2,0,2],\"point\":0,\"values\":[0,[1,2],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-2, 0, 2], aligned exact values=[0, 1/2, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 1/2, 1/2, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 1/2, 1/2, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0052","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6ee7c5f30ed6c1f1897f195d6eca837b11155727c1494d6c02ff0206080d90b","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114052,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-3,0,3],\"point\":0,\"values\":[0,[2,3],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-3, 0, 3], aligned exact values=[0, 2/3, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 2/3, 2/3, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 2/3, 2/3, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0053","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb4d6bf0278d98f6103db09e2aaea8775ef8bb340d21d4c40fc86df45a02e322","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114053,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-4,0,4],\"point\":0,\"values\":[0,[3,4],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-4, 0, 4], aligned exact values=[0, 3/4, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 3/4, 3/4, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 3/4, 3/4, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0054","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7b234cc891adcd564f10672fa5f0ee5f48086707d2c4b37578ef562936a0ebc","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114054,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-5,0,5],\"point\":0,\"values\":[0,[4,5],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-5, 0, 5], aligned exact values=[0, 4/5, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 4/5, 4/5, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 4/5, 4/5, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0055","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"695fc7e5c24225dc94245e595177eb2239293749c9160515371950805b740766","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114055,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-6,0,6],\"point\":0,\"values\":[0,[5,6],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-6, 0, 6], aligned exact values=[0, 5/6, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 5/6, 5/6, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 5/6, 5/6, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0056","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b50aae1620cfece06c816b66b8b0d386c2cd7a3f3e034816963b4e4290b06cb","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114056,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-7,0,7],\"point\":0,\"values\":[0,[6,7],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-7, 0, 7], aligned exact values=[0, 6/7, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 6/7, 6/7, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 6/7, 6/7, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0057","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55378988e43bdb0d9e859f7b615fed79fd5cadb085817f4ea0cd92f2329505b9","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114057,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-8,0,8],\"point\":0,\"values\":[0,[7,8],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-8, 0, 8], aligned exact values=[0, 7/8, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 7/8, 7/8, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 7/8, 7/8, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0058","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b66b39119cb59032c9b19c2b1fa3cfad40f17a0addf98da53021ed6ed25181dc","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114058,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-9,0,9],\"point\":0,\"values\":[0,[8,9],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-9, 0, 9], aligned exact values=[0, 8/9, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 8/9, 8/9, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 8/9, 8/9, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0059","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e91a6208063a4912d17e0105c24b746a25cd8b1293f57844fd8ccd5fb6204f4a","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114059,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-10,0,10],\"point\":0,\"values\":[0,[9,10],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-10, 0, 10], aligned exact values=[0, 9/10, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 9/10, 9/10, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 9/10, 9/10, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0060","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49cbb2161742d40bdd83094de292db437367ebceaa46fa93e1e11ac25e70f5af","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114060,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-11,0,11],\"point\":0,\"values\":[0,[10,11],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-11, 0, 11], aligned exact values=[0, 10/11, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 10/11, 10/11, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 10/11, 10/11, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0061","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f607dcbdbfdad0ae0949cb919a4d08cf4b5a1148bab655546e3bb13347fc8cd3","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114061,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-12,0,12],\"point\":0,\"values\":[0,[11,12],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-12, 0, 12], aligned exact values=[0, 11/12, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 11/12, 11/12, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 11/12, 11/12, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0062","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ae93a6735c687d8b0497f7aceb017e6ae88ea38f944a81f99f7454ae84ba9e9","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114062,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-13,0,13],\"point\":0,\"values\":[0,[12,13],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-13, 0, 13], aligned exact values=[0, 12/13, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 12/13, 12/13, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 12/13, 12/13, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0063","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2dec908bfb5b0b250ef7ff06d6882e6edd1249f86de9233926117e1145bc645","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114063,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-14,0,14],\"point\":0,\"values\":[0,[13,14],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-14, 0, 14], aligned exact values=[0, 13/14, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 13/14, 13/14, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 13/14, 13/14, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0064","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a1879a1ddae7c5e2102212d17f19ad759052c8fe1ac28642c9b62005f14511b","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114064,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-15,0,15],\"point\":0,\"values\":[0,[14,15],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-15, 0, 15], aligned exact values=[0, 14/15, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 14/15, 14/15, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 14/15, 14/15, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0065","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"023d6ee48b2e6accff7a4ad1e6a90ae84c61558f133999fe9f39aaec0ea39343","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114065,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-16,0,16],\"point\":0,\"values\":[0,[15,16],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-16, 0, 16], aligned exact values=[0, 15/16, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 15/16, 15/16, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 15/16, 15/16, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0066","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1aea3ae1c17cffde6ab6befb62fdf7cadbb551354adea31b79737669bd445c5c","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114066,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-17,0,17],\"point\":0,\"values\":[0,[16,17],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-17, 0, 17], aligned exact values=[0, 16/17, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 16/17, 16/17, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 16/17, 16/17, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0067","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d683c5092460553a07f23a289aa792fa6f7c361d2158e3a4689c011deec5ddba","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114067,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-18,0,18],\"point\":0,\"values\":[0,[17,18],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-18, 0, 18], aligned exact values=[0, 17/18, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 17/18, 17/18, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 17/18, 17/18, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0068","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d551be937de45ef4f6d557880b3981de7b7ffd4cfbc7a66837def8ef7d93d5d5","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114068,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-19,0,19],\"point\":0,\"values\":[0,[18,19],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-19, 0, 19], aligned exact values=[0, 18/19, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 18/19, 18/19, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 18/19, 18/19, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0069","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7b4dde6031336ae768fc02647a0c76976e500013284eb6bf53eb546a83f02d4","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114069,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-20,0,20],\"point\":0,\"values\":[0,[19,20],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-20, 0, 20], aligned exact values=[0, 19/20, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 19/20, 19/20, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 19/20, 19/20, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0070","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7e45fb975d42d1de29190d07649109475abb6e7fcf5b063f6c343ee387a44cb","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114070,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-21,0,21],\"point\":0,\"values\":[0,[20,21],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-21, 0, 21], aligned exact values=[0, 20/21, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 20/21, 20/21, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 20/21, 20/21, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0071","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d70ac37d42a800adcda42035669ef1bfd48b63cd94439da14d1012abbb863562","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114071,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-22,0,22],\"point\":0,\"values\":[0,[21,22],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-22, 0, 22], aligned exact values=[0, 21/22, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 21/22, 21/22, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 21/22, 21/22, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0072","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"755be873b2da19eebfd01b76a43d7dd55a534157b97b42fc04236259a9a8038d","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114072,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-23,0,23],\"point\":0,\"values\":[0,[22,23],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-23, 0, 23], aligned exact values=[0, 22/23, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 22/23, 22/23, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 22/23, 22/23, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0073","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"252380287497d57d37d72cb43158894e4d46d54ec8d773133a46cc71d448ecef","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114073,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-24,0,24],\"point\":0,\"values\":[0,[23,24],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-24, 0, 24], aligned exact values=[0, 23/24, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 23/24, 23/24, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 23/24, 23/24, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0074","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64a36f625cea188d20c18eeb39fade346c900e90cd548cbca74d74a7f8244ddb","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114074,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-25,0,25],\"point\":0,\"values\":[0,[24,25],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-25, 0, 25], aligned exact values=[0, 24/25, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 24/25, 24/25, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 24/25, 24/25, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0075","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"118a3f9886f38d54b98734d020ed16df85ca01e2528c137c14df2f800496e9d1","family_id":"qf-mod42-contcdf-zero-jump-atomlessness","seed":202611114075,"semantic_template":"qf-mod42-contcdf-zero-jump-atomlessness/{\"knots\":[-26,0,26],\"point\":0,\"values\":[0,[25,26],1]}"},"fields":{"title":"The Singleton Mass Is the Jump, Not the CDF Value","stem":"Use strictly increasing knots=[-26, 0, 26], aligned exact values=[0, 25/26, 1], linear interpolation between knots, constant0 to the left and constant1 to the right. Knot values must be nondecreasing from0 to1. This declares a whole continuous CDF, not just finite samples; flat intervals are allowed. Threshold point=0. Give (point, F_at_point, F_left_limit, singleton_probability, zero_jump, certificate).","answer":{"kind":"expression","value":"(0, 25/26, 25/26, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)","explanation":"Continuity makes F(point) and its left limit equal, so their difference and the singleton probability are0 even if F(point)>0. A zero jump does not by itself imply a density or absolute continuity. The exact result, retaining the requested order, is (0, 25/26, 25/26, 0, Yes, continuous_CDF_has_zero_singleton_mass_but_need_not_have_density)."},"skills":["The Singleton Mass Is the Jump, Not the CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0076","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3064b8c3f8fad47d40404655efc3d830dbfafca28f4aaff9c18bdf2ec5b007c5","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114076,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-1,\"right\":2}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-1,2], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-1, 2, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-1, 2, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0077","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fc7d1a13439bf04fee3ba35bcf78d4cb414dc264526f3c5063b95bd28a7d0f2","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114077,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-2,\"right\":3}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-2,3], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-2, 3, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-2, 3, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0078","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f017155c35edd1c261c18e4db89655b2678507441a39c13b6256992bd872e3e","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114078,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-3,\"right\":4}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-3,4], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-3, 4, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-3, 4, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0079","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa6c85c1e642b9acb8cf8c7baca040fd1de95f525a2baa4d9c22f51baeef0fec","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114079,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-4,\"right\":5}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-4,5], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-4, 5, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-4, 5, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0080","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bd43af9e3d5db3b9cc7d4bffb8cbebcf571bcaf30daade4d2710e3e0962fcb4","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114080,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-5,\"right\":6}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-5,6], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-5, 6, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-5, 6, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0081","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c5a54943cad7d146d47ab0b21668b82c5fd4ed423b3d268cd89139d631a8a53","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114081,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-6,\"right\":7}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-6,7], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-6, 7, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-6, 7, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0082","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca9897dbcba96c4fc4dae93b9563c315172bb3b03a00d9a1500f2c64b845e5a8","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114082,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-7,\"right\":8}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-7,8], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-7, 8, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-7, 8, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0083","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6df759909b3de2d59b44e2ba2fcd56ae57bd4121a5161e0b59b7f5b7c5ea1aa","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114083,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-8,\"right\":9}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-8,9], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-8, 9, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-8, 9, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0084","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa8d4053116214b070d163d26f7602a28b6774096f665f808e1039461aec34ee","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114084,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-9,\"right\":10}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-9,10], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-9, 10, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-9, 10, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0085","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab0f77eaa4ba8ac1ee76a8231eacc8bb6f3a219bf057ba0e570ada55a93cbd4d","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114085,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-10,\"right\":11}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-10,11], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-10, 11, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-10, 11, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0086","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b4dbacb40a53dbcd7c90b4369da825c3c533be4be61891e6927473d13c6c9c2","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114086,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-11,\"right\":12}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-11,12], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-11, 12, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-11, 12, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0087","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd9f138740b4abb18ae01417ee5cdc4eef2fa606d439a4e3e3b563243ee33d79","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114087,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-12,\"right\":13}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-12,13], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-12, 13, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-12, 13, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0088","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"232bf805d8f3e535556b90e04dfa1b8f82327146c3ad343984072db1d8a6b360","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114088,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-13,\"right\":14}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-13,14], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-13, 14, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-13, 14, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0089","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05a548c5e1d7e2bcf9cc142051c222f699aedd78505f47b3f2c08a673906fabd","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114089,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-14,\"right\":15}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-14,15], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-14, 15, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-14, 15, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0090","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c302f9e2aa35b85e3ccb19ea7f1772d9e60cb0ab6c3f172714e8932add9a34d","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114090,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-15,\"right\":16}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-15,16], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-15, 16, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-15, 16, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0091","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb647670b788246240b16470671c357b19f88d52c467c505699f4f0a429d3ab","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114091,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-16,\"right\":17}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-16,17], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-16, 17, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-16, 17, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0092","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b1de208593a201ca5e7e763bd4c808899e5e71868533745d97596ec93ae9974","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114092,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-17,\"right\":18}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-17,18], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-17, 18, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-17, 18, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0093","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7fd68473d50c3224b6c12be5d6c31c59ed674926a485f0ba76c164388889f4e","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114093,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-18,\"right\":19}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-18,19], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-18, 19, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-18, 19, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0094","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"feaf693764db24259398d77105f7e167dcf6e53fac215bfcebca094cb23d30c2","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114094,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-19,\"right\":20}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-19,20], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-19, 20, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-19, 20, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0095","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab2d3fcad9eeb78fab32112412645e8d901803f1986d367300c819631d191cf1","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114095,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-20,\"right\":21}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-20,21], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-20, 21, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-20, 21, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0096","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a39b4a8bec4b03ed9105b19655fd0b98ab9205f6e9a63944caa40e22338b5cd","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114096,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-21,\"right\":22}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-21,22], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-21, 22, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-21, 22, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0097","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5b541ab8a92d2a9692383dada103af4df69f80f8628aebc7ce56e88f38ac995","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114097,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-22,\"right\":23}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-22,23], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-22, 23, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-22, 23, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0098","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73bb3a5b13035fd4b2929e454aa7ad5b29c5c2ad0de6b850ee02de2a1f2ab486","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114098,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-23,\"right\":24}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-23,24], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-23, 24, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-23, 24, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0099","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9787a9675c58aec640cea187db3a9496f184fc0dc9d0146849c746c65e91ccf2","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114099,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-24,\"right\":25}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-24,25], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-24, 25, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-24, 25, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261111-r04-continuous-real-distribution-function-0100","batch_id":"batch-20261014-advance-20261111-r04-continuous-real-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8bf7aff1353777ea317995cba7169599efaccd6451df99ecf4f6e1bbc6f8e63","family_id":"qf-mod42-contcdf-singular-no-density-boundary","seed":202611114100,"semantic_template":"qf-mod42-contcdf-singular-no-density-boundary/{\"left\":-25,\"right\":26}"},"fields":{"title":"Continuous and Atomless Does Not Imply a Density","stem":"Use the standard Cantor distribution affinely carried onto [-25,26], with left<right. Give (left, right, CDF_continuous, atomless, has_density, singular, boundary).","answer":{"kind":"expression","value":"(-25, 26, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)","explanation":"Retain its continuous CDF, atomlessness, absence of a density and singularity as distinct standard-law properties. This named boundary is not derived from finitely sampled CDF values and performs no density calculation. The exact result, retaining the requested order, is (-25, 26, Yes, Yes, No, Yes, affine_Cantor_distribution_is_continuous_atomless_and_singular_without_a_density)."},"skills":["Continuous and Atomless Does Not Imply a Density"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}