{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0001","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b4f015560feebaa838d636107c2e4a696007f9b1f6160ba4f0f79c73bc7d585","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173001,"semantic_template":"uniform-sup-certificate|{base_intercept:[1,1],base_slope:[-3,1],error_scale:[1,1],n:1}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=1,E=1>0,n=1≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-3, 1, 1, 1, f_n(x)=-3x+1+1x/2, 1/2, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 1, 1, 1, f_n(x)=-3x+1+1x/2, 1/2, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0002","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76d70cbcf9c4766e8f863736b91f715f278b2e4a41d41869b1daaa7b63f96a99","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173002,"semantic_template":"uniform-sup-certificate|{base_intercept:[2,1],base_slope:[-2,1],error_scale:[2,1],n:2}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=2,E=2>0,n=2≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-2, 2, 2, 2, f_n(x)=-2x+2+2x/3, 2/3, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 2, 2, 2, f_n(x)=-2x+2+2x/3, 2/3, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0003","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47cd6662766bba922a230747d73e5df5e51eaa2da3fb8bf38097a6e2b3728d3c","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173003,"semantic_template":"uniform-sup-certificate|{base_intercept:[3,1],base_slope:[-1,1],error_scale:[3,1],n:3}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=3,E=3>0,n=3≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-1, 3, 3, 3, f_n(x)=-1x+3+3x/4, 3/4, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 3, 3, 3, f_n(x)=-1x+3+3x/4, 3/4, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0004","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"538cf7eab8457ec9a5e3df1539df5b56e454351a1fce492ec256e748f9111d0d","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173004,"semantic_template":"uniform-sup-certificate|{base_intercept:[4,1],base_slope:[0,1],error_scale:[4,1],n:4}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=4,E=4>0,n=4≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(0, 4, 4, 4, f_n(x)=0x+4+4x/5, 4/5, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 4, 4, 4, f_n(x)=0x+4+4x/5, 4/5, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0005","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81585b9ce278b6279c797d3674d1d8618a0d26990d1acbfc4bdbeeb3dc2b8242","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173005,"semantic_template":"uniform-sup-certificate|{base_intercept:[5,1],base_slope:[1,1],error_scale:[5,1],n:5}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=1,b=5,E=5>0,n=5≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(1, 5, 5, 5, f_n(x)=1x+5+5x/6, 5/6, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (1, 5, 5, 5, f_n(x)=1x+5+5x/6, 5/6, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0006","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b12260cc4e4b32b2bbab582e5803ffeefb055851c30871307a10d7e00088ffa","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173006,"semantic_template":"uniform-sup-certificate|{base_intercept:[6,1],base_slope:[2,1],error_scale:[6,1],n:6}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=6,E=6>0,n=6≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(2, 6, 6, 6, f_n(x)=2x+6+6x/7, 6/7, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (2, 6, 6, 6, f_n(x)=2x+6+6x/7, 6/7, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0007","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1f72ff7acff6bf72dd8828a12efc01e529cc415b59a0a91052dce7ad61ec183","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173007,"semantic_template":"uniform-sup-certificate|{base_intercept:[7,1],base_slope:[3,1],error_scale:[7,1],n:7}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=3,b=7,E=7>0,n=7≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(3, 7, 7, 7, f_n(x)=3x+7+7x/8, 7/8, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (3, 7, 7, 7, f_n(x)=3x+7+7x/8, 7/8, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0008","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7093b44e9bce54f643f5d035b2d7ed18b18a16afcbc5a85b4896cda2a331c82f","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173008,"semantic_template":"uniform-sup-certificate|{base_intercept:[8,1],base_slope:[-3,1],error_scale:[8,1],n:8}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=8,E=8>0,n=8≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-3, 8, 8, 8, f_n(x)=-3x+8+8x/9, 8/9, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 8, 8, 8, f_n(x)=-3x+8+8x/9, 8/9, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0009","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df87c98437577514957c9e78019a02ff9921c0d73e46aed82bb817116b95e2db","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173009,"semantic_template":"uniform-sup-certificate|{base_intercept:[9,1],base_slope:[-2,1],error_scale:[9,1],n:9}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=9,E=9>0,n=9≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-2, 9, 9, 9, f_n(x)=-2x+9+9x/10, 9/10, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 9, 9, 9, f_n(x)=-2x+9+9x/10, 9/10, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0010","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c5b88e0b9a28b8c8da0446021a55e2c47ea6d6f4d16c5341897b1fa29f97ff","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173010,"semantic_template":"uniform-sup-certificate|{base_intercept:[10,1],base_slope:[-1,1],error_scale:[10,1],n:10}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=10,E=10>0,n=10≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-1, 10, 10, 10, f_n(x)=-1x+10+10x/11, 10/11, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 10, 10, 10, f_n(x)=-1x+10+10x/11, 10/11, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0011","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9962315d1575490f357b9080f8b5cbed1ab2471dc7ea2bf9e63b0872678c2b9a","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173011,"semantic_template":"uniform-sup-certificate|{base_intercept:[11,1],base_slope:[0,1],error_scale:[11,1],n:11}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=11,E=11>0,n=11≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(0, 11, 11, 11, f_n(x)=0x+11+11x/12, 11/12, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 11, 11, 11, f_n(x)=0x+11+11x/12, 11/12, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0012","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b69fd02a96f92c9efbcec9b8c581f192be0ba106052794745d658afb58aa0d4","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173012,"semantic_template":"uniform-sup-certificate|{base_intercept:[12,1],base_slope:[1,1],error_scale:[12,1],n:12}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=1,b=12,E=12>0,n=12≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(1, 12, 12, 12, f_n(x)=1x+12+12x/13, 12/13, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (1, 12, 12, 12, f_n(x)=1x+12+12x/13, 12/13, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0013","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c12dfbfa17b89bb052539b608e179c5fd859740e865e69be6994138d7eec62bb","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173013,"semantic_template":"uniform-sup-certificate|{base_intercept:[13,1],base_slope:[2,1],error_scale:[13,1],n:13}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=13,E=13>0,n=13≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(2, 13, 13, 13, f_n(x)=2x+13+13x/14, 13/14, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (2, 13, 13, 13, f_n(x)=2x+13+13x/14, 13/14, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0014","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d434ed4454bee86003f3483368b7fbde97bef6fb3f4f7c5783cfca75e07ad96f","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173014,"semantic_template":"uniform-sup-certificate|{base_intercept:[14,1],base_slope:[3,1],error_scale:[14,1],n:14}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=3,b=14,E=14>0,n=14≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(3, 14, 14, 14, f_n(x)=3x+14+14x/15, 14/15, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (3, 14, 14, 14, f_n(x)=3x+14+14x/15, 14/15, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0015","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddb2ff458d3f4c26b1b0fd8ac602c586567522d02e434766ab02dcf927d79f72","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173015,"semantic_template":"uniform-sup-certificate|{base_intercept:[15,1],base_slope:[-3,1],error_scale:[15,1],n:15}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=15,E=15>0,n=15≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-3, 15, 15, 15, f_n(x)=-3x+15+15x/16, 15/16, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 15, 15, 15, f_n(x)=-3x+15+15x/16, 15/16, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0016","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b47e47902cf1b042e3fa803c2c9229de2a9acf8738005dbcafa1daa41463e0ab","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173016,"semantic_template":"uniform-sup-certificate|{base_intercept:[16,1],base_slope:[-2,1],error_scale:[16,1],n:16}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=16,E=16>0,n=16≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-2, 16, 16, 16, f_n(x)=-2x+16+16x/17, 16/17, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 16, 16, 16, f_n(x)=-2x+16+16x/17, 16/17, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0017","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2850026d5a278b5e2a994a5d55579afbc9553fc9d7a21e8c3500398b096348e","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173017,"semantic_template":"uniform-sup-certificate|{base_intercept:[17,1],base_slope:[-1,1],error_scale:[17,1],n:17}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=17,E=17>0,n=17≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-1, 17, 17, 17, f_n(x)=-1x+17+17x/18, 17/18, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 17, 17, 17, f_n(x)=-1x+17+17x/18, 17/18, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0018","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b021840a7d049405b5719e2b40623d78e8734ce3a81c308d5e0edcb81ecb033e","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173018,"semantic_template":"uniform-sup-certificate|{base_intercept:[18,1],base_slope:[0,1],error_scale:[18,1],n:18}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=18,E=18>0,n=18≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(0, 18, 18, 18, f_n(x)=0x+18+18x/19, 18/19, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 18, 18, 18, f_n(x)=0x+18+18x/19, 18/19, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0019","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a0f0fed8eb18d9dae6e9be5e9ad2547bf2f73b7749f47714e723b0ba14b14a7","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173019,"semantic_template":"uniform-sup-certificate|{base_intercept:[19,1],base_slope:[1,1],error_scale:[19,1],n:19}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=1,b=19,E=19>0,n=19≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(1, 19, 19, 19, f_n(x)=1x+19+19x/20, 19/20, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (1, 19, 19, 19, f_n(x)=1x+19+19x/20, 19/20, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0020","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea927c60afffa68090ce163651b3550fae382357a060ea0a9e15a17b3adda8f5","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173020,"semantic_template":"uniform-sup-certificate|{base_intercept:[20,1],base_slope:[2,1],error_scale:[20,1],n:20}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=2,b=20,E=20>0,n=20≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(2, 20, 20, 20, f_n(x)=2x+20+20x/21, 20/21, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (2, 20, 20, 20, f_n(x)=2x+20+20x/21, 20/21, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0021","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"727233ff50f4a033f364ae6f7b286daf6df14e9b226cf6348f22cf33fed38405","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173021,"semantic_template":"uniform-sup-certificate|{base_intercept:[21,1],base_slope:[3,1],error_scale:[21,1],n:21}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=3,b=21,E=21>0,n=21≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(3, 21, 21, 21, f_n(x)=3x+21+21x/22, 21/22, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (3, 21, 21, 21, f_n(x)=3x+21+21x/22, 21/22, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0022","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"956f19bb9f07e72a8921b89bb9764c60076cdfdfc4df50ea98123e5a1ac2ac4f","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173022,"semantic_template":"uniform-sup-certificate|{base_intercept:[22,1],base_slope:[-3,1],error_scale:[22,1],n:22}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-3,b=22,E=22>0,n=22≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-3, 22, 22, 22, f_n(x)=-3x+22+22x/23, 22/23, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-3, 22, 22, 22, f_n(x)=-3x+22+22x/23, 22/23, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0023","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da3daf7ccc02da0a382ba8e5f754f752930c2436746a3dddc0c4b621ccb32837","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173023,"semantic_template":"uniform-sup-certificate|{base_intercept:[23,1],base_slope:[-2,1],error_scale:[23,1],n:23}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-2,b=23,E=23>0,n=23≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-2, 23, 23, 23, f_n(x)=-2x+23+23x/24, 23/24, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0024","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84b82143b74d79af964389e1117f48bdb8cf787ce165c2f643c716dba1b56155","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173024,"semantic_template":"uniform-sup-certificate|{base_intercept:[24,1],base_slope:[-1,1],error_scale:[24,1],n:24}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=-1,b=24,E=24>0,n=24≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(-1, 24, 24, 24, f_n(x)=-1x+24+24x/25, 24/25, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (-1, 24, 24, 24, f_n(x)=-1x+24+24x/25, 24/25, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0025","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ccbbf1f41270003b3565af4f1a595d87761d9475672f6d84084bb5dbd8f462d","family_id":"qf-mod48-uniform-sup-certificate","seed":202611173025,"semantic_template":"uniform-sup-certificate|{base_intercept:[25,1],base_slope:[0,1],error_scale:[25,1],n:25}"},"fields":{"title":"An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum","stem":"On [0,1] let f(x)=m*x+b and f_n(x)=f(x)+E*x/(n+1), with exact m=0,b=25,E=25>0,n=25≥1. Give (base_slope, base_intercept, error_scale, n, function_description, sup_error, attainment_statement, uniform_certificate).","answer":{"kind":"expression","value":"(0, 25, 25, 25, f_n(x)=0x+25+25x/26, 25/26, sup_on_[0,1]_attained_at_x=1, Yes)","explanation":"All functions share the domain and are continuous affine functions. The error is nonnegative and maximized at x=1, giving sup E/(n+1). This is a whole-domain expression certificate, not agreement on a finite grid or the general uniform-limit theorem proof. The exact result, retaining the requested order, is (0, 25, 25, 25, f_n(x)=0x+25+25x/26, 25/26, sup_on_[0,1]_attained_at_x=1, Yes)."},"skills":["An Affine Error on the Whole Unit Interval Has an Exact Attained Supremum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0026","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1cb81117b3898f5100699095a5f9fe2c2456a12fc8a14b257638e8af83771bf","family_id":"qf-mod48-uniform-epsilon-index","seed":202611173026,"semantic_template":"uniform-epsilon-index|{epsilon:[1,7],error_scale:[1,1]}"},"fields":{"title":"One Positive Index Must Give Strict Error Control for Every Point","stem":"Use exact E=1>0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(1, 1/7, 7, 1/8, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(2, 1/4, 8, 2/9, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(3, 1/3, 9, 3/10, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(4, 2/5, 10, 4/11, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(5, 5/11, 11, 5/12, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(6, 1/7, 42, 6/43, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(7, 1/4, 28, 7/29, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(8, 1/3, 24, 8/25, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(9, 2/5, 22, 9/23, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(10, 5/11, 22, 10/23, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(11, 1/7, 77, 11/78, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(12, 1/4, 48, 12/49, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(13, 1/3, 39, 13/40, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(14, 2/5, 35, 7/18, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(15, 5/11, 33, 15/34, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(16, 1/7, 112, 16/113, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(17, 1/4, 68, 17/69, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(18, 1/3, 54, 18/55, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(19, 2/5, 47, 19/48, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(20, 5/11, 44, 4/9, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=1/7>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(21, 1/7, 147, 21/148, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=2/8>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(22, 1/4, 88, 22/89, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=3/9>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(23, 1/3, 69, 23/70, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=4/10>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(24, 2/5, 60, 24/61, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)0 and epsilon=5/11>0 for sup error E/(n+1), n≥1. Give (error_scale, epsilon, N, error_at_N, strict_error_bound, certificate).","answer":{"kind":"expression","value":"(25, 5/11, 55, 25/56, Yes, same_N_controls_every_x_in_[0,1])","explanation":"Choose the first positive integer N with E/(N+1)3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(1, 1, 1/4, 1, 1/3, 5/6, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (1, 1, 1/4, 1, 1/3, 5/6, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0052","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b586a867423bdf239298b23fb9a07d6a3d2f32c932a45af9dc4a27473104d777","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173052,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,5],lipschitz:[2,1],n:2}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=2, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/5≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(2, 2, 1/5, 1, 1/6, 11/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (2, 2, 1/5, 1, 1/6, 11/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0053","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe911f65e49c95559ae43def37c8cb6cad5636dd21d50ac0fc7640e80a09f3b8","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173053,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,6],lipschitz:[3,1],n:3}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=3, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/6≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(3, 3, 1/6, 1, 1/9, 2/3, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (3, 3, 1/6, 1, 1/9, 2/3, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0054","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76c3c2935ce8298691b3f9f02e9a6a6cbcec15eae5b18b63adc99ab3858911cb","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173054,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,7],lipschitz:[4,1],n:4}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=4, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/7≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(4, 4, 1/7, 1, 1/12, 13/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (4, 4, 1/7, 1, 1/12, 13/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0055","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f697bb9d78ce7ca97a8fd322d7688fab3d96a4166269605acdab3f8784cc0352","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173055,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,8],lipschitz:[5,1],n:5}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=5, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/8≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(5, 5, 1/8, 1, 1/15, 7/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (5, 5, 1/8, 1, 1/15, 7/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0056","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2769f6fab7280d0c05f473d8925235dd6eecec1bfd7eb25d625a618868b658ca","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173056,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,9],lipschitz:[1,1],n:6}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=6, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/9≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(6, 1, 1/9, 1, 1/3, 5/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (6, 1, 1/9, 1, 1/3, 5/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0057","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50a0f0e48ef09ec639e2b63d42d0a2a1c7eb8534a521fc1e8fc4f1635b2a21ca","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173057,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,10],lipschitz:[2,1],n:7}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=7, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/10≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(7, 2, 1/10, 1, 1/6, 8/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (7, 2, 1/10, 1, 1/6, 8/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0058","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e551bfb3c414c1ca0e70b859e45c71df25868ca7db56ae8d2f44ea77d863dc47","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173058,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,11],lipschitz:[3,1],n:8}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=8, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/11≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(8, 3, 1/11, 1, 1/9, 17/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (8, 3, 1/11, 1, 1/9, 17/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0059","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaccaee1f4b107f930b848aeeaa0e006ef9e4eb6f0e40a335d0e0d45c7a4d004","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173059,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,12],lipschitz:[4,1],n:9}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=9, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/12≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(9, 4, 1/12, 1, 1/12, 1/2, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (9, 4, 1/12, 1, 1/12, 1/2, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0060","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"952edc8bf48f8a3a8d5dc2d05fe5677b08460398cbf153c1357104e5ad712330","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173060,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,13],lipschitz:[5,1],n:10}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=10, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/13≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(10, 5, 1/13, 1, 1/15, 19/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (10, 5, 1/13, 1, 1/15, 19/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0061","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1d3ff4b00c06b37b05ad3d3170348f38688b94afb57937832b207b328836c8f","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173061,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,14],lipschitz:[1,1],n:11}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=11, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/14≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(11, 1, 1/14, 1, 1/3, 10/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (11, 1, 1/14, 1, 1/3, 10/21, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0062","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f640261fb8b03c7b3888d6db6105ce8dffc9d7f140810fcf5e4e7d05b3f20411","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173062,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,15],lipschitz:[2,1],n:12}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=12, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/15≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(12, 2, 1/15, 1, 1/6, 7/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (12, 2, 1/15, 1, 1/6, 7/15, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0063","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf68f9c1e386212b19f588766de98fcaff72cbafc852c43f817827e796e69bcc","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173063,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,16],lipschitz:[3,1],n:13}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=13, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/16≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(13, 3, 1/16, 1, 1/9, 11/24, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (13, 3, 1/16, 1, 1/9, 11/24, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0064","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d483c6b3df9989d38b6b228d2a6db3318b4dd2263f8ad331d7aba0f68de5c67","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173064,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,17],lipschitz:[4,1],n:14}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=14, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/17≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(14, 4, 1/17, 1, 1/12, 23/51, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (14, 4, 1/17, 1, 1/12, 23/51, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0065","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1a44cb5e025d7e040048bb5be7250d87b50c66b3c440690b404033bb7e67566","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173065,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,18],lipschitz:[5,1],n:15}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=15, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/18≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(15, 5, 1/18, 1, 1/15, 4/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (15, 5, 1/18, 1, 1/15, 4/9, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0066","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"248c3adffb6fa519deba017a606c0437dbb9cb639b49ba5093b4dcbcfb43813a","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173066,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,19],lipschitz:[1,1],n:16}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=16, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/19≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(16, 1, 1/19, 1, 1/3, 25/57, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (16, 1, 1/19, 1, 1/3, 25/57, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0067","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ea9c52e3acdec3da2d8e5aa6da5f1400e77be7ea931167c216bf2538edc48ff","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173067,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,20],lipschitz:[2,1],n:17}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=17, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/20≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(17, 2, 1/20, 1, 1/6, 13/30, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (17, 2, 1/20, 1, 1/6, 13/30, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0068","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aa57a83541364e07f26cc35e2fa952d073ced3e728fb68d746a9159cad42385","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173068,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,21],lipschitz:[3,1],n:18}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=18, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/21≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(18, 3, 1/21, 1, 1/9, 3/7, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (18, 3, 1/21, 1, 1/9, 3/7, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0069","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c161c50109d5bb923e9148b584229f813f437c26d1236dd12040eea5e7399d","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173069,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,22],lipschitz:[4,1],n:19}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=19, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/22≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(19, 4, 1/22, 1, 1/12, 14/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (19, 4, 1/22, 1, 1/12, 14/33, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0070","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8f58434933c9ae1276b486233e5427c385dd2ff0253a356304aa9051333b9b3","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173070,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,23],lipschitz:[5,1],n:20}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=20, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/23≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(20, 5, 1/23, 1, 1/15, 29/69, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (20, 5, 1/23, 1, 1/15, 29/69, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0071","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00d4e2de051408db382fe753c1c2fa404f00b0f10d25d165e56130e93df72aaf","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173071,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,24],lipschitz:[1,1],n:21}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=21, a positive Lipschitz bound L=1, a whole-domain uniform error bound E=1/24≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(21, 1, 1/24, 1, 1/3, 5/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (21, 1, 1/24, 1, 1/3, 5/12, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0072","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"054ed1501aa8c94bfd192b6717d331d859d1e23ccb809a3147d921b1a59a8e1b","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173072,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,25],lipschitz:[2,1],n:22}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=22, a positive Lipschitz bound L=2, a whole-domain uniform error bound E=1/25≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(22, 2, 1/25, 1, 1/6, 31/75, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (22, 2, 1/25, 1, 1/6, 31/75, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0073","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e99310f937699a42c0d30bc10f984870a36470b88cd2c3d3894479df3578f8cc","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173073,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,26],lipschitz:[3,1],n:23}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=23, a positive Lipschitz bound L=3, a whole-domain uniform error bound E=1/26≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(23, 3, 1/26, 1, 1/9, 16/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (23, 3, 1/26, 1, 1/9, 16/39, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0074","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"577fe6a7f05537328f673d058496e3d3559d151d28e1382a379668f000baa9b3","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173074,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,27],lipschitz:[4,1],n:24}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=24, a positive Lipschitz bound L=4, a whole-domain uniform error bound E=1/27≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(24, 4, 1/27, 1, 1/12, 11/27, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (24, 4, 1/27, 1, 1/12, 11/27, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0075","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a5f283e15b23289c65a4910830e8bb3176e2f701e6d2837365839369bbb8b9e","family_id":"qf-mod48-uniform-continuity-transfer","seed":202611173075,"semantic_template":"uniform-continuity-transfer|{epsilon:[1,1],error_bound:[1,28],lipschitz:[5,1],n:25}"},"fields":{"title":"A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer","stem":"Fix N=25, a positive Lipschitz bound L=5, a whole-domain uniform error bound E=1/28≥0 and epsilon=1>3E. Give (N, Lipschitz_constant, uniform_error_bound, epsilon, delta, triangle_bound, strict_triangle_bound, certificate).","answer":{"kind":"expression","value":"(25, 5, 1/28, 1, 1/15, 17/42, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)","explanation":"Use delta=epsilon/(3L) and compare the triangle bound2E+epsilon/3 with epsilon. The given continuity/Lipschitz and uniform-error premises are separate obligations; the index alone is not evidence for them, nor is a common delta across all approximants required. The exact result, retaining the requested order, is (25, 5, 1/28, 1, 1/15, 17/42, Yes, same_N_selects_continuous_f_N_and_epsilon_over_3_transfer)."},"skills":["A Fixed Continuous Approximant Gives the Epsilon-over-Three Transfer"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0076","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34deb9c5516481d613de142610959811ed205cc5e3eaa1b68dc68475196cb090","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173076,"semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:2}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=pointwise_only, positive integer scale=2, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(pointwise_only, 2, f_n(x)=x^(n+2)_on_[0,1], pointwise_limit_discontinuous_at_1, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 2, f_n(x)=x^(n+2)_on_[0,1], pointwise_limit_discontinuous_at_1, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0077","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a7ae81ff581148e577b11d9dbb0398acb3eb3d0bc56a5bfdd82609a990fb9a","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173077,"semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:3}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=discontinuous_terms, positive integer scale=3, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(discontinuous_terms, 3, f_n(x)=1_if_x>=1/4_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 3, f_n(x)=1_if_x>=1/4_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0078","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76615662a7c9a4b14f739f251ab9e8d9731ec9758743b5efa66cd9076d6e41ad","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173078,"semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:4}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=finite_grid, positive integer scale=4, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(finite_grid, 4, grid={k/5:0<=k<=5}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 4, grid={k/5:0<=k<=5}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0079","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a1dbc6cf42b4b152a03f039bed18b43adfa7daf0902a8ebdcabb9fac4cc570a","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173079,"semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:5}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=moving_spike, positive integer scale=5, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(moving_spike, 5, f_n(x)=max(0,1-(n+5)*abs(x-1/(n+5)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 5, f_n(x)=max(0,1-(n+5)*abs(x-1/(n+5)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0080","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7672a709d854e5ecc89a294f47acb32d99cb9511c00ae4e7cecc4302e77d3112","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173080,"semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:6}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=pointwise_only, positive integer scale=6, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(pointwise_only, 6, f_n(x)=x^(n+6)_on_[0,1], pointwise_limit_discontinuous_at_1, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 6, f_n(x)=x^(n+6)_on_[0,1], pointwise_limit_discontinuous_at_1, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0081","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9367fabbb32afd8fb0f9ad8051f9a09c290957cd8f4aced3f8a1e4d6776720b","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173081,"semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:7}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=discontinuous_terms, positive integer scale=7, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(discontinuous_terms, 7, f_n(x)=1_if_x>=1/8_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 7, f_n(x)=1_if_x>=1/8_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0082","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f43abca68201a0a087f6b50983619cc9895dc051f373161bbd5f2aa97e01652","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173082,"semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:8}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=finite_grid, positive integer scale=8, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(finite_grid, 8, grid={k/9:0<=k<=9}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 8, grid={k/9:0<=k<=9}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0083","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84f6bcf110a4bd7a51d0730e351295572c98ea56162f67eb656dc8238dbe3e6c","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173083,"semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:9}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=moving_spike, positive integer scale=9, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(moving_spike, 9, f_n(x)=max(0,1-(n+9)*abs(x-1/(n+9)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 9, f_n(x)=max(0,1-(n+9)*abs(x-1/(n+9)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0084","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddb2e85d27f3c4451b4093a6c0148516c81e3fb5b0d1d028e5cb8d63a48d0551","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173084,"semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:10}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=pointwise_only, positive integer scale=10, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(pointwise_only, 10, f_n(x)=x^(n+10)_on_[0,1], pointwise_limit_discontinuous_at_1, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 10, f_n(x)=x^(n+10)_on_[0,1], pointwise_limit_discontinuous_at_1, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0085","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b84a9c2021c99eb4fc9e5adb6cb59d6a31889c8c6840d116725d28c9fa871288","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173085,"semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:11}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=discontinuous_terms, positive integer scale=11, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(discontinuous_terms, 11, f_n(x)=1_if_x>=1/12_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 11, f_n(x)=1_if_x>=1/12_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0086","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97fb3f659f2e528f658d04c616b1245ead699cb7b7338bc8bb1852640768f8c6","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173086,"semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:12}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=finite_grid, positive integer scale=12, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(finite_grid, 12, grid={k/13:0<=k<=13}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 12, grid={k/13:0<=k<=13}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0087","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92e9e82912adb8769c3fa4093fa4b6a1b59518f9509e017b193ae0f184948d63","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173087,"semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:13}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=moving_spike, positive integer scale=13, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(moving_spike, 13, f_n(x)=max(0,1-(n+13)*abs(x-1/(n+13)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 13, f_n(x)=max(0,1-(n+13)*abs(x-1/(n+13)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0088","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45a99eb91ce5bdb0895e46d5b2f5886213c30f9b6e944cb6212366649e4bcd9d","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173088,"semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:14}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=pointwise_only, positive integer scale=14, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(pointwise_only, 14, f_n(x)=x^(n+14)_on_[0,1], pointwise_limit_discontinuous_at_1, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 14, f_n(x)=x^(n+14)_on_[0,1], pointwise_limit_discontinuous_at_1, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0089","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"808aac0221888cee1c54b79f6d593da85763e963613a8f1aa44d8d5d54c55fc2","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173089,"semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:15}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=discontinuous_terms, positive integer scale=15, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(discontinuous_terms, 15, f_n(x)=1_if_x>=1/16_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 15, f_n(x)=1_if_x>=1/16_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0090","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"885bdcf2490e49ae766c631827569e0ebcc370fc99b00449f5bd4beff9e5026c","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173090,"semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:16}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=finite_grid, positive integer scale=16, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(finite_grid, 16, grid={k/17:0<=k<=17}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 16, grid={k/17:0<=k<=17}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0091","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e51ddb8de4937ee896d522c4384654a312675ab03862629bcb1e2ba4ba4aad90","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173091,"semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:17}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=moving_spike, positive integer scale=17, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(moving_spike, 17, f_n(x)=max(0,1-(n+17)*abs(x-1/(n+17)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 17, f_n(x)=max(0,1-(n+17)*abs(x-1/(n+17)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0092","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eb220a8b3dbfbe6efd11257abfe74479a470de1db07546617187b4cde8963e1","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173092,"semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:18}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=pointwise_only, positive integer scale=18, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(pointwise_only, 18, f_n(x)=x^(n+18)_on_[0,1], pointwise_limit_discontinuous_at_1, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 18, f_n(x)=x^(n+18)_on_[0,1], pointwise_limit_discontinuous_at_1, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0093","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf59462125915d3785b32082a32dabad12168bf5a184b17163d2b32554377938","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173093,"semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:19}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=discontinuous_terms, positive integer scale=19, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 19, f_n(x)=1_if_x>=1/20_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0094","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc508033c689389ee99410b95f5fca0683c818e35dea1ab82a7d4c5f6fcf8d17","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173094,"semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:20}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=finite_grid, positive integer scale=20, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(finite_grid, 20, grid={k/21:0<=k<=21}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 20, grid={k/21:0<=k<=21}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0095","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b28c566a8f12757d13b2bf9849b5e5b5f9ba28da6ba68192d91f9820c0079468","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173095,"semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:21}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=moving_spike, positive integer scale=21, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(moving_spike, 21, f_n(x)=max(0,1-(n+21)*abs(x-1/(n+21)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 21, f_n(x)=max(0,1-(n+21)*abs(x-1/(n+21)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0096","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aa316d62276e6d52760f8b51a4f9beca91bb892c0c9a0c5a80a8668e6b35803","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173096,"semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:22}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=pointwise_only, positive integer scale=22, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(pointwise_only, 22, f_n(x)=x^(n+22)_on_[0,1], pointwise_limit_discontinuous_at_1, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 22, f_n(x)=x^(n+22)_on_[0,1], pointwise_limit_discontinuous_at_1, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0097","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1a1239eb63a707dc80edf9cc398c7cc379b808ba508cadb01c0248357126cce","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173097,"semantic_template":"uniform-limit-boundary|{mode:\"discontinuous_terms\",scale:23}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=discontinuous_terms, positive integer scale=23, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(discontinuous_terms, 23, f_n(x)=1_if_x>=1/24_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (discontinuous_terms, 23, f_n(x)=1_if_x>=1/24_else_0_for_every_n, uniform_but_every_term_and_limit_are_discontinuous, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0098","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"602a8f0df717c655cdd9f6e09e8f365d0681b69442b99d902aa5f65d903c4a0c","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173098,"semantic_template":"uniform-limit-boundary|{mode:\"finite_grid\",scale:24}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=finite_grid, positive integer scale=24, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(finite_grid, 24, grid={k/25:0<=k<=25}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (finite_grid, 24, grid={k/25:0<=k<=25}_with_a_continuous_unit_bump_between_adjacent_grid_points, agreement_on_this_finite_grid_does_not_control_supremum, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0099","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aa070930eb9cc75094a4913a6768b034eb914657380a178126c4dff73db04c2","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173099,"semantic_template":"uniform-limit-boundary|{mode:\"moving_spike\",scale:25}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=moving_spike, positive integer scale=25, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(moving_spike, 25, f_n(x)=max(0,1-(n+25)*abs(x-1/(n+25)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (moving_spike, 25, f_n(x)=max(0,1-(n+25)*abs(x-1/(n+25)))_on_[0,1], continuous_pointwise_to_zero_but_sup_norm_one, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}} {"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-0100","batch_id":"batch-20261015-advance-20261117-r03-uniform-limit-continuity-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"797a5af9c80fb9a2f94114f3c6eaaf52b98ca039113b287acb1e9bd2f2021d19","family_id":"qf-mod48-uniform-limit-boundary","seed":202611173100,"semantic_template":"uniform-limit-boundary|{mode:\"pointwise_only\",scale:26}"},"fields":{"title":"Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms","stem":"Use mode=pointwise_only, positive integer scale=26, and the full sequence/domain description in the result. Give (mode, scale, sequence_description, scope_or_obstruction, theorem_application_claim).","answer":{"kind":"expression","value":"(pointwise_only, 26, f_n(x)=x^(n+26)_on_[0,1], pointwise_limit_discontinuous_at_1, No)","explanation":"x^(n+scale) is pointwise-only with a discontinuous limit at1. A constant discontinuous step sequence is uniform but lacks continuous terms. A continuous bump can miss a fixed grid, and a moving spike can tend pointwise to0 with sup1. The final No withholds applying the theorem, not every separately true convergence fact. The exact result, retaining the requested order, is (pointwise_only, 26, f_n(x)=x^(n+26)_on_[0,1], pointwise_limit_discontinuous_at_1, No)."},"skills":["Pointwise Limits and Finite Grids Cannot Replace Uniform Convergence plus Continuous Terms"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}