{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0001","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"889db04c1e537847d5846b0462a5ae1e4666fadbde8182b29867265588a9dcf6","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195001,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-4,1],\"intercept\":[1,1],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-3, intercept=1, approximation=-4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-3, 1, -4, 1/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 1, -4, 1/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0002","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"530dfb98877198084cd417e25ef265d9435977b4b4e732bfc0fe8f83c394961f","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195002,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-3,1],\"intercept\":[2,1],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-2, intercept=2, approximation=-3 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-2, 2, -3, 1, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 2, -3, 1, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0003","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7060fa1b21abffeff3f237ccc1f045ffa1aa1c11bafb744afd0dc117edc4ba35","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195003,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-2,1],\"intercept\":[3,1],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-1, intercept=3, approximation=-2 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-1, 3, -2, 3, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 3, -2, 3, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0004","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2cf443605518a3de15be51380c5d419fcbaf4d1b02e780ae9ac34d3941aa782","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195004,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[-1,1],\"intercept\":[4,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=2, intercept=4, approximation=-1 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(2, 4, -1, -2, 2, 1, 2, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 4, -1, -2, 2, 1, 2, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0005","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec472f672ddfbcf40fbbd5df6b008ff78fbce67958e6a523fa1b40d31423aab4","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195005,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[0,1],\"intercept\":[5,1],\"slope\":[1,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=1, intercept=5, approximation=0 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(1, 5, 0, -5, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 5, 0, -5, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0006","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0852d48ca47234b1b3554538c51769e5a9677cc09f079d1920f871d86d24fe96","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195006,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[1,1],\"intercept\":[6,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=2, intercept=6, approximation=1 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(2, 6, 1, -3, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 6, 1, -3, 8, 4, 8, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0007","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c42e040f5467716f41190d35f4203afbea0086516387778c89a5cbe9f9affd8","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195007,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[2,1],\"intercept\":[7,1],\"slope\":[3,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=3, intercept=7, approximation=2 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(3, 7, 2, -7/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 7, 2, -7/3, 13, 13/3, 13, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0008","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af7bba91a5f09c920e366903d8d5e701597dfd2bf60dbfb7449c0f59d90108ec","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195008,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[3,1],\"intercept\":[8,1],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-3, intercept=8, approximation=3 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-3, 8, 3, 8/3, 1, 1/3, 1, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 8, 3, 8/3, 1, 1/3, 1, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0009","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00c1ab9ea8bd7f75c398b974689203308a0f1cdabef471f3ad3855ead1bda567","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195009,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[4,1],\"intercept\":[9,1],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-2, intercept=9, approximation=4 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 9, 4, 9/2, 1, 1/2, 1, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0010","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"233b8985beb01b96faa8de1115ca7e904285fee1cd32c40f5aebb1279bf7e400","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195010,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[5,1],\"intercept\":[10,1],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-1, intercept=10, approximation=5 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-1, 10, 5, 10, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 10, 5, 10, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0011","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01a47ac8982582381920953eede8ecb6243f3c0eb9ce93ad21983c45cbd153e4","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195011,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[6,1],\"intercept\":[11,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=2, intercept=11, approximation=6 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(2, 11, 6, -11/2, 23, 23/2, 23, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 11, 6, -11/2, 23, 23/2, 23, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0012","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf7b315c984bd2a0e52b3fd0e5e852556f48343ca39aa0f27b78bd69e2ec34d3","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195012,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[7,1],\"intercept\":[12,1],\"slope\":[1,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=1, intercept=12, approximation=7 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(1, 12, 7, -12, 19, 19, 19, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 12, 7, -12, 19, 19, 19, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0013","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"334cbc993041406863f5d60f660288e843970237caa1bce97a2a38c825e31f24","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195013,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[8,1],\"intercept\":[13,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=2, intercept=13, approximation=8 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(2, 13, 8, -13/2, 29, 29/2, 29, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 13, 8, -13/2, 29, 29/2, 29, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0014","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3615dafd37161036f34e350a04d671e9be6c3d62b0eecc3c9f02aebce29c4880","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195014,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[9,1],\"intercept\":[14,1],\"slope\":[3,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=3, intercept=14, approximation=9 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(3, 14, 9, -14/3, 41, 41/3, 41, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 14, 9, -14/3, 41, 41/3, 41, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0015","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e5ebf81627cebcd59dae16fed5747dcca3a886fc237e33173077177a8e84f32","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195015,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[10,1],\"intercept\":[15,1],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-3, intercept=15, approximation=10 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-3, 15, 10, 5, 15, 5, 15, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 15, 10, 5, 15, 5, 15, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0016","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa4c8c71569446f471ebe5e08e0f3a93d34a28422b2adeea1420cb8bc8ae6c6e","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195016,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[11,1],\"intercept\":[16,1],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-2, intercept=16, approximation=11 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-2, 16, 11, 8, 6, 3, 6, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 16, 11, 8, 6, 3, 6, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0017","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf1fad10b4447f13785cc629078a0655e0a77da6068c116f70de65196a6b1533","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195017,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[12,1],\"intercept\":[17,1],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-1, intercept=17, approximation=12 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-1, 17, 12, 17, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 17, 12, 17, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0018","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3f6119f97c2661f3926050f180373a8bf755a827c2723bd88d603d99f63eb1f","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195018,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[13,1],\"intercept\":[18,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=2, intercept=18, approximation=13 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(2, 18, 13, -9, 44, 22, 44, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 18, 13, -9, 44, 22, 44, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0019","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8deb9d16173372350a0f40b835efbec268231575cfa1c0c449468f4fc26c3dbd","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195019,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[14,1],\"intercept\":[19,1],\"slope\":[1,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=1, intercept=19, approximation=14 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(1, 19, 14, -19, 33, 33, 33, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (1, 19, 14, -19, 33, 33, 33, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0020","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55857bb621fa8f24973163167af429438d2a3ea8f7698bcc3be3030c752b4cc0","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195020,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[15,1],\"intercept\":[20,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=2, intercept=20, approximation=15 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(2, 20, 15, -10, 50, 25, 50, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 20, 15, -10, 50, 25, 50, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0021","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14e88335dffee363f36e69d1f02484891f10ab0ee9228d1e79cc5e722b00e935","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195021,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[16,1],\"intercept\":[21,1],\"slope\":[3,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=3, intercept=21, approximation=16 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(3, 21, 16, -7, 69, 23, 69, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (3, 21, 16, -7, 69, 23, 69, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0022","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0facaf5cf254515d6f7c4c85b88c2d50d1dae58e83e64b064dd4ad147d2e6d10","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195022,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[17,1],\"intercept\":[22,1],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-3, intercept=22, approximation=17 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-3, 22, 17, 22/3, 29, 29/3, 29, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0023","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39601184efabe9867f4434751cac662e86d53e3f1b2f461aa086bf1a74602ca8","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195023,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[18,1],\"intercept\":[23,1],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-2, intercept=23, approximation=18 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-2, 23, 18, 23/2, 13, 13/2, 13, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-2, 23, 18, 23/2, 13, 13/2, 13, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0024","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15d0f4967788d64b9e17de258fc3302a8dee4681c636c0e6f3b8b9624babc4d2","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195024,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[19,1],\"intercept\":[24,1],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=-1, intercept=24, approximation=19 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(-1, 24, 19, 24, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (-1, 24, 19, 24, 5, 5, 5, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0025","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39c78fdbe39a8133b829a7d5fe64317b7e4aabb91279bfae6f1a92d1a9644427","family_id":"qf-mod50-numerr-affine-residual-forward","seed":202611195025,"semantic_template":"numerr-affine-residual-forward|{\"approximation\":[20,1],\"intercept\":[25,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Residual Equals Absolute Slope Times Forward Error","stem":"Use nonzero slope=2, intercept=25, approximation=20 for f(x)=slope*x+intercept. Give (slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction).","answer":{"kind":"expression","value":"(2, 25, 20, -25/2, 65, 65/2, 65, Yes, residual_and_forward_error_are_distinct_quantities)","explanation":"Solve the exact affine root and calculate |f(approximation)| and |approximation−root| separately. Verify the scale identity without identifying residual with forward error. The affine model supplies the true root here; an arbitrary residual alone would not. The exact result, retaining the requested order, is (2, 25, 20, -25/2, 65, 65/2, 65, Yes, residual_and_forward_error_are_distinct_quantities)."},"skills":["An Affine Residual Equals Absolute Slope Times Forward Error"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0026","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d4cb0b7786604a02f21e98c8a363a3392af6a5208246c70d3bb94cee11c5052","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195026,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[1,3],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-3 and nonnegative absolute input error bound=1/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-3, 1/3, 1, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 1/3, 1, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0027","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f0bc36b610d76abc2a6e09741798d9e9681039d00a15ff34b1287c5e63b9d4","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195027,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[2,3],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-2 and nonnegative absolute input error bound=2/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-2, 2/3, 4/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 2/3, 4/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0028","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca3cd85e0d876821b579005fd5f0a8c5fe8ba5ab3aec98f7b9e10199e03e35ac","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195028,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[1,1],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-1 and nonnegative absolute input error bound=3/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-1, 1, 1, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 1, 1, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0029","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45b0e673b82a6f716882b59043d19fa971d27df83e4a05357d92b09004e3ee7e","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195029,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[4,3],\"slope\":[2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=2 and nonnegative absolute input error bound=4/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(2, 4/3, 8/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 4/3, 8/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0030","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f928b96171d7a7f96887fc54f9cceaff5018332efbfb39c74223c77798d1485","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195030,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[5,3],\"slope\":[1,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=1 and nonnegative absolute input error bound=5/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(1, 5/3, 5/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 5/3, 5/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0031","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9726054d0d846641702708db7dde7e4f240d9351d475b9f9733dec663d40a0ad","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195031,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[2,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=2 and nonnegative absolute input error bound=6/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(2, 2, 4, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 2, 4, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0032","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45a945a2bb0e03a6868baf108d4f26f82713a4cf5701aa013a886bea0473aeee","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195032,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[7,3],\"slope\":[3,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=3 and nonnegative absolute input error bound=7/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(3, 7/3, 7, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 7/3, 7, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0033","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06f5393fd70a0d61bcef8e0e1634c014288cb79d8ea1ff28c1945c4d29249bae","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195033,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[8,3],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-3 and nonnegative absolute input error bound=8/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-3, 8/3, 8, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 8/3, 8, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0034","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8308346169bd856cfbd11c0afed10b5ec95979290b3550fd7311763fa41338f","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195034,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[3,1],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-2 and nonnegative absolute input error bound=9/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-2, 3, 6, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 3, 6, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0035","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6616ab41e350ab06003f8742ff015722e9914cc6fff529082ec561a565c62157","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195035,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[10,3],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-1 and nonnegative absolute input error bound=10/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-1, 10/3, 10/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 10/3, 10/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0036","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5e5432b4c0d0eccf3d17f027d17a6f61fb97deae8aed042293a00f22e644419","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195036,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[11,3],\"slope\":[2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=2 and nonnegative absolute input error bound=11/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(2, 11/3, 22/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 11/3, 22/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0037","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeb373ba3aa03892548f9ebce091b788f2fd68c585ca93bc84bde4d775e5dac6","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195037,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[4,1],\"slope\":[1,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=1 and nonnegative absolute input error bound=12/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(1, 4, 4, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 4, 4, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0038","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9cc0372c44ba64048e2b17fe08da01b32e38de57ab315c38d412333fb1038cc","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195038,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[13,3],\"slope\":[2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=2 and nonnegative absolute input error bound=13/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(2, 13/3, 26/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 13/3, 26/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0039","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8529962e07a2726fd8f7bc69e1c544dca3b078af14f79f64823d161bf59e5f4f","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195039,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[14,3],\"slope\":[3,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=3 and nonnegative absolute input error bound=14/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(3, 14/3, 14, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 14/3, 14, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0040","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dc93cc20f7f11d7b9a41d9da880c21169f5111f19eaf99457946e1ff84266ea","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195040,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[5,1],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-3 and nonnegative absolute input error bound=15/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-3, 5, 15, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 5, 15, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0041","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66452a4af6d7f0309ca086f8f75a5ba429a85112329ca264994d41f157cfaf31","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195041,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[16,3],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-2 and nonnegative absolute input error bound=16/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-2, 16/3, 32/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 16/3, 32/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0042","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0604898e16daaa94730b8145559db827bfe5350fc0faa137c7614a88cec331df","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195042,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[17,3],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-1 and nonnegative absolute input error bound=17/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-1, 17/3, 17/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 17/3, 17/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0043","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72be563d95a18f1d48dc330b6c2461a75fc2ae752c1258b14ef642115ec3fb28","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195043,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[6,1],\"slope\":[2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=2 and nonnegative absolute input error bound=18/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(2, 6, 12, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 6, 12, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0044","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83d6add36feaa8b7e93bcc19bfeca07c71a34938561eb85e74de51939d3d6cdc","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195044,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[19,3],\"slope\":[1,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=1 and nonnegative absolute input error bound=19/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(1, 19/3, 19/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (1, 19/3, 19/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0045","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e761027500deb607e8153f49b8ac68bc077bf372e6f62ea169e61270d4e0d1f","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195045,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[20,3],\"slope\":[2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=2 and nonnegative absolute input error bound=20/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(2, 20/3, 40/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 20/3, 40/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0046","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a12ce2442019f33fe219b7ba17ab5a7f9d1d324ff03ad25f29c6470054c61f3","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195046,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[7,1],\"slope\":[3,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=3 and nonnegative absolute input error bound=21/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(3, 7, 21, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (3, 7, 21, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0047","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c8bededd4688b8f619c06cb18b0eb6b8aab08a37b96d2fd554c90dc10aef015","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195047,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[22,3],\"slope\":[-3,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-3 and nonnegative absolute input error bound=22/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-3, 22/3, 22, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-3, 22/3, 22, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0048","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37843d511c3caf8a77679195c07d626a55aedf4dcfd7184be932e19100d5a89f","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195048,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[23,3],\"slope\":[-2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-2 and nonnegative absolute input error bound=23/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-2, 23/3, 46/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-2, 23/3, 46/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0049","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"596acfbf76930881691ee01c7b786735fd18cdaff261c0e65f20c086227212d1","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195049,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[8,1],\"slope\":[-1,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=-1 and nonnegative absolute input error bound=24/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(-1, 8, 8, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (-1, 8, 8, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0050","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fed3250fede0e2a5f2400930f101c3b8acb7c0b455d2003c1d764bf3d725d1d7","family_id":"qf-mod50-numerr-affine-propagation","seed":202611195050,"semantic_template":"numerr-affine-propagation|{\"input_error_bound\":[25,3],\"slope\":[2,1]}"},"fields":{"title":"An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope","stem":"Use slope=2 and nonnegative absolute input error bound=25/3. Give (slope, input_absolute_error_bound, output_absolute_error_bound, scope).","answer":{"kind":"expression","value":"(2, 25/3, 50/3, affine_output_absolute_error_bound)","explanation":"Multiply the input bound by abs(slope), including a zero slope or zero bound. This exact affine output-error bound does not identify rounding, truncation or measurement sources and does not establish algorithmic convergence. The exact result, retaining the requested order, is (2, 25/3, 50/3, affine_output_absolute_error_bound)."},"skills":["An Affine Map Propagates an Absolute Input Bound by Its Absolute Slope"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0051","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb98a93bd74bc6b344e9e62a17a8658a7b40e2839b045a56a1440bf112b7519","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195051,"semantic_template":"numerr-conditioning-witness|{\"residual\":[1,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=1, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(1, 1, 1, 2, 1/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (1, 1, 1, 2, 1/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0052","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebc7051d44ad4467b126cec5f57c820c6e2c68a331e87d26e90ffe7d59ca5445","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195052,"semantic_template":"numerr-conditioning-witness|{\"residual\":[2,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=2, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(2, 1, 2, 3, 2/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (2, 1, 2, 3, 2/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0053","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"037b734a49efe97ab355ceeb1cb6bf19beb1b63bad251644363d5c3fae424bb4","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195053,"semantic_template":"numerr-conditioning-witness|{\"residual\":[3,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=3, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(3, 1, 3, 4, 3/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (3, 1, 3, 4, 3/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0054","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bca7f7839da9bbb5e46a960db21f98db03ae489d2acaca4f763e5ac052436293","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195054,"semantic_template":"numerr-conditioning-witness|{\"residual\":[4,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=4, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(4, 1, 4, 5, 4/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (4, 1, 4, 5, 4/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0055","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e68d194581614110344292d1d257d2b4cd8f3c41dfd26fd77c13a9ab134cf360","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195055,"semantic_template":"numerr-conditioning-witness|{\"residual\":[5,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=5, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(5, 1, 5, 6, 5/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (5, 1, 5, 6, 5/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0056","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fee3346b33b32a19ab5b6379dd3364de5e83c89f77b14199dec833bfa258bbac","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195056,"semantic_template":"numerr-conditioning-witness|{\"residual\":[6,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=6, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(6, 1, 6, 2, 3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (6, 1, 6, 2, 3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0057","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"306558656283ee868b9c14816503044b4e943628ad8a4f37f31263184d4650a5","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195057,"semantic_template":"numerr-conditioning-witness|{\"residual\":[7,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=7, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(7, 1, 7, 3, 7/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (7, 1, 7, 3, 7/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0058","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c4ff97abb1f4dbee46449d82394713140dbfdc561a2cd5feb65875225e61cef","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195058,"semantic_template":"numerr-conditioning-witness|{\"residual\":[8,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=8, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(8, 1, 8, 4, 2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (8, 1, 8, 4, 2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0059","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d48ed7a2cbba4fed8b17018fd91173f35b180d747d33c5c041d7e2cde2fde96","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195059,"semantic_template":"numerr-conditioning-witness|{\"residual\":[9,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=9, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(9, 1, 9, 5, 9/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (9, 1, 9, 5, 9/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0060","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fd22d32aa9d796a059e87e2c7ed92b2c6ad7e350b30f8f25e3391bf015744f5","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195060,"semantic_template":"numerr-conditioning-witness|{\"residual\":[10,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=10, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(10, 1, 10, 6, 5/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (10, 1, 10, 6, 5/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0061","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8d2aa9650940aea6f5cffaa46ed084e2e1581fe60150b2d4dee94db083ee261","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195061,"semantic_template":"numerr-conditioning-witness|{\"residual\":[11,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=11, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(11, 1, 11, 2, 11/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (11, 1, 11, 2, 11/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0062","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b6614e87af1d84e6a81d38fb9c1d707e9c94afc3a4a7e3f4506ac67ebc3e4fd","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195062,"semantic_template":"numerr-conditioning-witness|{\"residual\":[12,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=12, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(12, 1, 12, 3, 4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (12, 1, 12, 3, 4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0063","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59dc4495837661d78bb459910aa312faa5787c9d67aee1775bb15f6b640709fb","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195063,"semantic_template":"numerr-conditioning-witness|{\"residual\":[13,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=13, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (13, 1, 13, 4, 13/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0064","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cf84991991f928bdf068b662c8b3b89d6a6a7b606450d9357e555e8a43dc649","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195064,"semantic_template":"numerr-conditioning-witness|{\"residual\":[14,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=14, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(14, 1, 14, 5, 14/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (14, 1, 14, 5, 14/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0065","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17197039dacf622a72e51cfe1b9b4d410869b5ef82703ffc1af408be0a4a2769","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195065,"semantic_template":"numerr-conditioning-witness|{\"residual\":[15,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=15, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(15, 1, 15, 6, 5/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (15, 1, 15, 6, 5/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0066","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"581a0204d3ec5303362d89bdbf3c1e08bf455f48b671eeb783d474ad594a8e65","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195066,"semantic_template":"numerr-conditioning-witness|{\"residual\":[16,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=16, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(16, 1, 16, 2, 8, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (16, 1, 16, 2, 8, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0067","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb5e467a6e2cda6e74ad34b84d21ff48b07d936786190a23c5cdbf3786960e4","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195067,"semantic_template":"numerr-conditioning-witness|{\"residual\":[17,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=17, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(17, 1, 17, 3, 17/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (17, 1, 17, 3, 17/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0068","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ccff7fea08756e2bf65a6350f77fda78f655e20892db067bb98a17daa8b1dde","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195068,"semantic_template":"numerr-conditioning-witness|{\"residual\":[18,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=18, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(18, 1, 18, 4, 9/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (18, 1, 18, 4, 9/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0069","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dd2d6d4e314b4e8f7ec9f28c62e9d58b53aa11a440e52662f6aa5aa503e0d85","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195069,"semantic_template":"numerr-conditioning-witness|{\"residual\":[19,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=19, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(19, 1, 19, 5, 19/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (19, 1, 19, 5, 19/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0070","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3741e8c5489d38bf6f4f042be2db2d8b6b3a1d2ec644a38d340a376770c3d24","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195070,"semantic_template":"numerr-conditioning-witness|{\"residual\":[20,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=20, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(20, 1, 20, 6, 10/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (20, 1, 20, 6, 10/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0071","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9245e2cbb580c524fd34b7481107d95cee3bc7d067e9fac0137f36217d5cc87b","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195071,"semantic_template":"numerr-conditioning-witness|{\"residual\":[21,1],\"slope_a\":[1,1],\"slope_b\":[2,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=21, nonzero first slope=1 and nonzero second slope=2, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(21, 1, 21, 2, 21/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (21, 1, 21, 2, 21/2, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0072","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"955e1c0e6091b9f36fb3d12251f3fdbcfbc7c82f6c1aac2af6674d06def73b67","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195072,"semantic_template":"numerr-conditioning-witness|{\"residual\":[22,1],\"slope_a\":[1,1],\"slope_b\":[3,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=22, nonzero first slope=1 and nonzero second slope=3, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(22, 1, 22, 3, 22/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (22, 1, 22, 3, 22/3, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0073","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89a6dd6788f5760740211ba6445dc289e95884217f05353627ab0881e3310f86","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195073,"semantic_template":"numerr-conditioning-witness|{\"residual\":[23,1],\"slope_a\":[1,1],\"slope_b\":[4,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=23, nonzero first slope=1 and nonzero second slope=4, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(23, 1, 23, 4, 23/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (23, 1, 23, 4, 23/4, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0074","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e83a3717bb4499064328b6f47dd46065734efa8320ce1c476c9776e49cd4f168","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195074,"semantic_template":"numerr-conditioning-witness|{\"residual\":[24,1],\"slope_a\":[1,1],\"slope_b\":[5,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=24, nonzero first slope=1 and nonzero second slope=5, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(24, 1, 24, 5, 24/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (24, 1, 24, 5, 24/5, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0075","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"583dcee882f7f63cb9477ae6997011cbcef29f517af5041abae246c8c6d2a3ad","family_id":"qf-mod50-numerr-conditioning-witness","seed":202611195075,"semantic_template":"numerr-conditioning-witness|{\"residual\":[25,1],\"slope_a\":[1,1],\"slope_b\":[6,1]}"},"fields":{"title":"The Same Residual Can Correspond to Different Forward Errors","stem":"Use positive absolute residual=25, nonzero first slope=1 and nonzero second slope=6, with different absolute slopes. Give (absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope).","answer":{"kind":"expression","value":"(25, 1, 25, 6, 25/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)","explanation":"Compute each affine forward error as residual/abs(slope). Their difference witnesses sensitivity to the model scale. It does not turn one residual into a model-independent forward-error guarantee or a general condition-number theory. The exact result, retaining the requested order, is (25, 1, 25, 6, 25/6, Yes, same_residual_can_have_different_forward_error_under_different_conditioning)."},"skills":["The Same Residual Can Correspond to Different Forward Errors"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0076","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9590413d4c8d504e6ed3b2c90afe923fac51fc6692046b05e2d1c432987edb2e","family_id":"qf-mod50-numerr-boundary","seed":202611195076,"semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[1,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=zero_reference, complete model={\"approximation\":1}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).","answer":{"kind":"expression","value":"(zero_reference, 1, reference=0, relative_error=undefined, No)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 1, reference=0, relative_error=undefined, No)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0077","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e977b97b02899367bfce35eec4afddcfaaaabdb48febdc43aa5b7c81fb23764","family_id":"qf-mod50-numerr-boundary","seed":202611195077,"semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[1,1],\"tolerance\":[1,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=inclusive_tolerance, complete model={\"error\":[2,2],\"tolerance\":[2,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).","answer":{"kind":"expression","value":"(inclusive_tolerance, 1, 1, Yes, error<=tolerance_is_true_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 1, 1, Yes, error<=tolerance_is_true_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0078","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49b30b49b8b42ddbfcce998c6dfda39e252bc73fcbb0f5d0fe04a15998b3a0ab","family_id":"qf-mod50-numerr-boundary","seed":202611195078,"semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[3,2],\"tolerance\":[3,2]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=strict_tolerance, complete model={\"error\":[3,2],\"tolerance\":[3,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).","answer":{"kind":"expression","value":"(strict_tolerance, 3/2, 3/2, No, error<tolerance_is_false_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 3/2, 3/2, No, error<tolerance_is_false_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0079","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af84875b8a5bf8b3b5fd46d7487faccf46f1527a689a1ff8b9f08e1a9cd2a432","family_id":"qf-mod50-numerr-boundary","seed":202611195079,"semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[4,1],\"reference\":[-4,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=signed_vs_absolute, complete model={\"approximation\":4,\"reference\":-4}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).","answer":{"kind":"expression","value":"(signed_vs_absolute, 4, -4, 8, 8, No, signed_error_and_absolute_error_preserve_different_information)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 4, -4, 8, 8, No, signed_error_and_absolute_error_preserve_different_information)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0080","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfa9db17505820f5d4fbdfde6b6e0a4495b09149b262ace3b6082d71014ff953","family_id":"qf-mod50-numerr-boundary","seed":202611195080,"semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[5,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=zero_reference, complete model={\"approximation\":5}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).","answer":{"kind":"expression","value":"(zero_reference, 5, reference=0, relative_error=undefined, No)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 5, reference=0, relative_error=undefined, No)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0081","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1699ecdf146eb8ce2bc0453969f00df85d003071d86687cf5d76d44f60aecb71","family_id":"qf-mod50-numerr-boundary","seed":202611195081,"semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[3,1],\"tolerance\":[3,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=inclusive_tolerance, complete model={\"error\":[6,2],\"tolerance\":[6,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).","answer":{"kind":"expression","value":"(inclusive_tolerance, 3, 3, Yes, error<=tolerance_is_true_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 3, 3, Yes, error<=tolerance_is_true_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0082","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ea5e0d543228365ed640eb08a449331f673fff601f54473e3e4c4c8b1d3773","family_id":"qf-mod50-numerr-boundary","seed":202611195082,"semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[7,2],\"tolerance\":[7,2]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=strict_tolerance, complete model={\"error\":[7,2],\"tolerance\":[7,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).","answer":{"kind":"expression","value":"(strict_tolerance, 7/2, 7/2, No, error<tolerance_is_false_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 7/2, 7/2, No, error<tolerance_is_false_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0083","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f74efc8b004106c8ffd3cd221376b2780d75fc5fb5707dc93962f0123f51cd3f","family_id":"qf-mod50-numerr-boundary","seed":202611195083,"semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[8,1],\"reference\":[-8,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=signed_vs_absolute, complete model={\"approximation\":8,\"reference\":-8}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).","answer":{"kind":"expression","value":"(signed_vs_absolute, 8, -8, 16, 16, No, signed_error_and_absolute_error_preserve_different_information)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 8, -8, 16, 16, No, signed_error_and_absolute_error_preserve_different_information)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0084","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"672fb5a5020f4235f9041fe838f4b09abd86e259349f718f1b48c87a5d732d79","family_id":"qf-mod50-numerr-boundary","seed":202611195084,"semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[9,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=zero_reference, complete model={\"approximation\":9}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).","answer":{"kind":"expression","value":"(zero_reference, 9, reference=0, relative_error=undefined, No)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 9, reference=0, relative_error=undefined, No)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0085","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36bf3910cae5dc765ebd6b077c83626d114af0f528d3849eefdef542be7833ea","family_id":"qf-mod50-numerr-boundary","seed":202611195085,"semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[5,1],\"tolerance\":[5,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=inclusive_tolerance, complete model={\"error\":[10,2],\"tolerance\":[10,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).","answer":{"kind":"expression","value":"(inclusive_tolerance, 5, 5, Yes, error<=tolerance_is_true_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 5, 5, Yes, error<=tolerance_is_true_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0086","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11b58dad5971c1e923ec8ff4e9f2362d7a5bc4180d2645b26475a6d71d5842eb","family_id":"qf-mod50-numerr-boundary","seed":202611195086,"semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[11,2],\"tolerance\":[11,2]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=strict_tolerance, complete model={\"error\":[11,2],\"tolerance\":[11,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).","answer":{"kind":"expression","value":"(strict_tolerance, 11/2, 11/2, No, error<tolerance_is_false_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 11/2, 11/2, No, error<tolerance_is_false_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0087","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97f476361452fb042304de20791e5d49698f72fb8d0a904a11c169c232733397","family_id":"qf-mod50-numerr-boundary","seed":202611195087,"semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[12,1],\"reference\":[-12,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=signed_vs_absolute, complete model={\"approximation\":12,\"reference\":-12}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).","answer":{"kind":"expression","value":"(signed_vs_absolute, 12, -12, 24, 24, No, signed_error_and_absolute_error_preserve_different_information)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 12, -12, 24, 24, No, signed_error_and_absolute_error_preserve_different_information)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0088","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6920f3131ad2c8144a9c05f7777f364076c0571ffbe884d528412dd4825c26d5","family_id":"qf-mod50-numerr-boundary","seed":202611195088,"semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[13,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=zero_reference, complete model={\"approximation\":13}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).","answer":{"kind":"expression","value":"(zero_reference, 13, reference=0, relative_error=undefined, No)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 13, reference=0, relative_error=undefined, No)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0089","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06644700e02a5bbf9cc44f49a8be78f97460ebf4ad0e4a4d257326fca4554eea","family_id":"qf-mod50-numerr-boundary","seed":202611195089,"semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[7,1],\"tolerance\":[7,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=inclusive_tolerance, complete model={\"error\":[14,2],\"tolerance\":[14,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).","answer":{"kind":"expression","value":"(inclusive_tolerance, 7, 7, Yes, error<=tolerance_is_true_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 7, 7, Yes, error<=tolerance_is_true_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0090","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"041385819d013eeb22b115f1d3b8f6504a08791fdaf4bbeaaead82dd9fb22324","family_id":"qf-mod50-numerr-boundary","seed":202611195090,"semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[15,2],\"tolerance\":[15,2]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=strict_tolerance, complete model={\"error\":[15,2],\"tolerance\":[15,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).","answer":{"kind":"expression","value":"(strict_tolerance, 15/2, 15/2, No, error<tolerance_is_false_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 15/2, 15/2, No, error<tolerance_is_false_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0091","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ff8e1f768721ff79eab1a393b0fcffa51879f6f2e37d414945d58a142523b58","family_id":"qf-mod50-numerr-boundary","seed":202611195091,"semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[16,1],\"reference\":[-16,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=signed_vs_absolute, complete model={\"approximation\":16,\"reference\":-16}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).","answer":{"kind":"expression","value":"(signed_vs_absolute, 16, -16, 32, 32, No, signed_error_and_absolute_error_preserve_different_information)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 16, -16, 32, 32, No, signed_error_and_absolute_error_preserve_different_information)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0092","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3642d98e75e9d481d282e9ddf361590247d7ee18ab536580a9d3f01593975f1","family_id":"qf-mod50-numerr-boundary","seed":202611195092,"semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[17,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=zero_reference, complete model={\"approximation\":17}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).","answer":{"kind":"expression","value":"(zero_reference, 17, reference=0, relative_error=undefined, No)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 17, reference=0, relative_error=undefined, No)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0093","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"821e7e4c7e6bea5aa6b848ba3e8f91f32aadc43ebfaaf6c84fb5b911a10ea73f","family_id":"qf-mod50-numerr-boundary","seed":202611195093,"semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[9,1],\"tolerance\":[9,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=inclusive_tolerance, complete model={\"error\":[18,2],\"tolerance\":[18,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).","answer":{"kind":"expression","value":"(inclusive_tolerance, 9, 9, Yes, error<=tolerance_is_true_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 9, 9, Yes, error<=tolerance_is_true_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0094","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cbdb5238e9b12a703fd62d4844173c799c6cd9d6d04cb7df781d211c5e643e8","family_id":"qf-mod50-numerr-boundary","seed":202611195094,"semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[19,2],\"tolerance\":[19,2]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=strict_tolerance, complete model={\"error\":[19,2],\"tolerance\":[19,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).","answer":{"kind":"expression","value":"(strict_tolerance, 19/2, 19/2, No, error<tolerance_is_false_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 19/2, 19/2, No, error<tolerance_is_false_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0095","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3e52763cbc5b2345763c501ce8aafa03a4c1c07359802d698d2739afadd18df","family_id":"qf-mod50-numerr-boundary","seed":202611195095,"semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[20,1],\"reference\":[-20,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=signed_vs_absolute, complete model={\"approximation\":20,\"reference\":-20}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).","answer":{"kind":"expression","value":"(signed_vs_absolute, 20, -20, 40, 40, No, signed_error_and_absolute_error_preserve_different_information)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 20, -20, 40, 40, No, signed_error_and_absolute_error_preserve_different_information)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0096","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"691b52d16a061176526adb3a0187a2ecc5ae6abc1d4ac406ca94c10a629f7ea9","family_id":"qf-mod50-numerr-boundary","seed":202611195096,"semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[21,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=zero_reference, complete model={\"approximation\":21}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).","answer":{"kind":"expression","value":"(zero_reference, 21, reference=0, relative_error=undefined, No)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 21, reference=0, relative_error=undefined, No)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0097","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f057896783f8f7d72af492a4d07e4042ea345dca3d8930da0d7262f00d19468f","family_id":"qf-mod50-numerr-boundary","seed":202611195097,"semantic_template":"numerr-boundary|{\"mode\":\"inclusive_tolerance\",\"model\":{\"error\":[11,1],\"tolerance\":[11,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=inclusive_tolerance, complete model={\"error\":[22,2],\"tolerance\":[22,2]}. Give (mode, absolute_error, tolerance, inclusive_test_result, equality_convention).","answer":{"kind":"expression","value":"(inclusive_tolerance, 11, 11, Yes, error<=tolerance_is_true_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (inclusive_tolerance, 11, 11, Yes, error<=tolerance_is_true_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0098","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a262a9cc29acf093d3ee21552d007bcc412470d1f5b4ed8bae13bc45b88b26e3","family_id":"qf-mod50-numerr-boundary","seed":202611195098,"semantic_template":"numerr-boundary|{\"mode\":\"strict_tolerance\",\"model\":{\"error\":[23,2],\"tolerance\":[23,2]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=strict_tolerance, complete model={\"error\":[23,2],\"tolerance\":[23,2]}. Give (mode, absolute_error, tolerance, strict_test_result, equality_convention).","answer":{"kind":"expression","value":"(strict_tolerance, 23/2, 23/2, No, error<tolerance_is_false_at_equality)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (strict_tolerance, 23/2, 23/2, No, error<tolerance_is_false_at_equality)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0099","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81de47e8c248d7a3909be0d59c12886539695fbfcf9487761808d3b4afe84504","family_id":"qf-mod50-numerr-boundary","seed":202611195099,"semantic_template":"numerr-boundary|{\"mode\":\"signed_vs_absolute\",\"model\":{\"approximation\":[24,1],\"reference\":[-24,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=signed_vs_absolute, complete model={\"approximation\":24,\"reference\":-24}. Give (mode, approximation, reference, signed_error, absolute_error, quantity_identification_claim, boundary).","answer":{"kind":"expression","value":"(signed_vs_absolute, 24, -24, 48, 48, No, signed_error_and_absolute_error_preserve_different_information)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (signed_vs_absolute, 24, -24, 48, 48, No, signed_error_and_absolute_error_preserve_different_information)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-pulse-20261119-r05-0100","batch_id":"batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28620a85b252e8c4ccc8ef03dda63b62d1902215cb71ff5feb7edfa7eca0d120","family_id":"qf-mod50-numerr-boundary","seed":202611195100,"semantic_template":"numerr-boundary|{\"mode\":\"zero_reference\",\"model\":{\"approximation\":[25,1]}}"},"fields":{"title":"Zero Reference and Equal Tolerances Require Explicit Error Conventions","stem":"Use mode=zero_reference, complete model={\"approximation\":25}. Give (mode, approximation, reference_description, relative_error_status, relative_error_defined_claim).","answer":{"kind":"expression","value":"(zero_reference, 25, reference=0, relative_error=undefined, No)","explanation":"Relative error is undefined at a zero reference. At error=tolerance the inclusive≤ test is true while the strict< test is false, despite equal tuple arity. Signed error preserves direction and absolute error does not; retain both without silently replacing the convention. The exact result, retaining the requested order, is (zero_reference, 25, reference=0, relative_error=undefined, No)."},"skills":["Zero Reference and Equal Tolerances Require Explicit Error Conventions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}