{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0001","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"298ced4be99da4a0fdaf56363f1a8d2dddc19c4a95b8778ddc3fa1dfba859fdc","family_id":"qf-mod48-path-interval-linear","seed":202611176001,"semantic_template":"path-interval-linear|{end:[1,1],left:[-2,1],right:[3,1],start:[0,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-2,3], start=0,end=1, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-2, 3, 0, 1, gamma(t)=0+(1)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-2, 3, 0, 1, gamma(t)=0+(1)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0002","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a8049d9d5c18d1046adf48f9dd0fae8ee507fe17d9709dc842efd9e72f1ef3","family_id":"qf-mod48-path-interval-linear","seed":202611176002,"semantic_template":"path-interval-linear|{end:[2,1],left:[-3,1],right:[4,1],start:[-1,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-3,4], start=-1,end=2, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-3, 4, -1, 2, gamma(t)=-1+(3)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-3, 4, -1, 2, gamma(t)=-1+(3)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0003","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46ad390673f2f6b0919bc3d6638165e1e346b965c3a990592a0c1d734348d260","family_id":"qf-mod48-path-interval-linear","seed":202611176003,"semantic_template":"path-interval-linear|{end:[3,1],left:[-4,1],right:[5,1],start:[-2,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-4,5], start=-2,end=3, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-4, 5, -2, 3, gamma(t)=-2+(5)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-4, 5, -2, 3, gamma(t)=-2+(5)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0004","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48f914605fea42b2ea4d066ec6654c2a2582174187d0f847ac9cc1debdbd3fda","family_id":"qf-mod48-path-interval-linear","seed":202611176004,"semantic_template":"path-interval-linear|{end:[4,1],left:[-5,1],right:[6,1],start:[-3,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-5,6], start=-3,end=4, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-5, 6, -3, 4, gamma(t)=-3+(7)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-5, 6, -3, 4, gamma(t)=-3+(7)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0005","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a10e10bd608bac07cc2a2b7ed3a8746a895a6403b862ecd608f756078e766ac","family_id":"qf-mod48-path-interval-linear","seed":202611176005,"semantic_template":"path-interval-linear|{end:[5,1],left:[-6,1],right:[7,1],start:[-4,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-6,7], start=-4,end=5, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-6, 7, -4, 5, gamma(t)=-4+(9)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0006","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40af3d6398d6c1865292610d47906c7281b2cb4cbb66b7952ccbf2f3629ac96d","family_id":"qf-mod48-path-interval-linear","seed":202611176006,"semantic_template":"path-interval-linear|{end:[6,1],left:[-7,1],right:[8,1],start:[-5,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-7,8], start=-5,end=6, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-7, 8, -5, 6, gamma(t)=-5+(11)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-7, 8, -5, 6, gamma(t)=-5+(11)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0007","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd8b31ae304642423aa261d99c50d3c6189b6b293ba56b3b05bf8fc3af5e18bd","family_id":"qf-mod48-path-interval-linear","seed":202611176007,"semantic_template":"path-interval-linear|{end:[7,1],left:[-8,1],right:[9,1],start:[-6,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-8,9], start=-6,end=7, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-8, 9, -6, 7, gamma(t)=-6+(13)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-8, 9, -6, 7, gamma(t)=-6+(13)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0008","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21a9c668cbcfd76a8d5852f19a0964f00c327ce664ffe7e28aa51ee6e561b515","family_id":"qf-mod48-path-interval-linear","seed":202611176008,"semantic_template":"path-interval-linear|{end:[8,1],left:[-9,1],right:[10,1],start:[-7,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-9,10], start=-7,end=8, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-9, 10, -7, 8, gamma(t)=-7+(15)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-9, 10, -7, 8, gamma(t)=-7+(15)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0009","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b1cb75448dc75de58cd52f044a6585cbcd89dd07430ceb2dc4ee2d4a9c6b303","family_id":"qf-mod48-path-interval-linear","seed":202611176009,"semantic_template":"path-interval-linear|{end:[9,1],left:[-10,1],right:[11,1],start:[-8,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-10,11], start=-8,end=9, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-10, 11, -8, 9, gamma(t)=-8+(17)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-10, 11, -8, 9, gamma(t)=-8+(17)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0010","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cd69f0e17e5fdfbae581e2acc897857b130e750b8f5cc87e4f7866bfd9943a8","family_id":"qf-mod48-path-interval-linear","seed":202611176010,"semantic_template":"path-interval-linear|{end:[10,1],left:[-11,1],right:[12,1],start:[-9,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-11,12], start=-9,end=10, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-11, 12, -9, 10, gamma(t)=-9+(19)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-11, 12, -9, 10, gamma(t)=-9+(19)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0011","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4830da0799bc0cfae41e5a0a4f504dc58f567e91c03517721efb04e63a0e7562","family_id":"qf-mod48-path-interval-linear","seed":202611176011,"semantic_template":"path-interval-linear|{end:[11,1],left:[-12,1],right:[13,1],start:[-10,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-12,13], start=-10,end=11, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-12, 13, -10, 11, gamma(t)=-10+(21)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-12, 13, -10, 11, gamma(t)=-10+(21)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0012","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ab86194866fed14bf83417ed2150944e0c9b1f1d16435f7883b676a05d7618f8","family_id":"qf-mod48-path-interval-linear","seed":202611176012,"semantic_template":"path-interval-linear|{end:[12,1],left:[-13,1],right:[14,1],start:[-11,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-13,14], start=-11,end=12, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-13, 14, -11, 12, gamma(t)=-11+(23)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0013","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"610190fbaf0d3006153353981cd3cd31dd85e64ce71541741aa08fafea2db301","family_id":"qf-mod48-path-interval-linear","seed":202611176013,"semantic_template":"path-interval-linear|{end:[13,1],left:[-14,1],right:[15,1],start:[-12,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-14,15], start=-12,end=13, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-14, 15, -12, 13, gamma(t)=-12+(25)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-14, 15, -12, 13, gamma(t)=-12+(25)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0014","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d9a4daf9d35e7fa612ace8ce1c7290412c421967d1e42f6744558344375faf4","family_id":"qf-mod48-path-interval-linear","seed":202611176014,"semantic_template":"path-interval-linear|{end:[14,1],left:[-15,1],right:[16,1],start:[-13,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-15,16], start=-13,end=14, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-15, 16, -13, 14, gamma(t)=-13+(27)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-15, 16, -13, 14, gamma(t)=-13+(27)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0015","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3fa0cdb1b4612ade92c1f29d5bf20232b16c4c234b64f60c8ab852730264aa4","family_id":"qf-mod48-path-interval-linear","seed":202611176015,"semantic_template":"path-interval-linear|{end:[15,1],left:[-16,1],right:[17,1],start:[-14,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-16,17], start=-14,end=15, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-16, 17, -14, 15, gamma(t)=-14+(29)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-16, 17, -14, 15, gamma(t)=-14+(29)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0016","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2d5da4007827a4c279a0b4b2f907f3d0d70bc68c31f5b3ee024244d277da3c0","family_id":"qf-mod48-path-interval-linear","seed":202611176016,"semantic_template":"path-interval-linear|{end:[16,1],left:[-17,1],right:[18,1],start:[-15,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-17,18], start=-15,end=16, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-17, 18, -15, 16, gamma(t)=-15+(31)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-17, 18, -15, 16, gamma(t)=-15+(31)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0017","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04c9102e122eea63eac5501ad74b0fb6693bed09fc3ffbf1c635bdcb25c48a7e","family_id":"qf-mod48-path-interval-linear","seed":202611176017,"semantic_template":"path-interval-linear|{end:[17,1],left:[-18,1],right:[19,1],start:[-16,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-18,19], start=-16,end=17, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-18, 19, -16, 17, gamma(t)=-16+(33)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-18, 19, -16, 17, gamma(t)=-16+(33)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0018","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6263d357045dc0788f7f801c33377fde1e507436e9f05c7c0b50e9db3c9e46f3","family_id":"qf-mod48-path-interval-linear","seed":202611176018,"semantic_template":"path-interval-linear|{end:[18,1],left:[-19,1],right:[20,1],start:[-17,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-19,20], start=-17,end=18, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-19, 20, -17, 18, gamma(t)=-17+(35)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-19, 20, -17, 18, gamma(t)=-17+(35)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0019","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46edf255828469a2ec29f433c931a2e89c9e694e15c6e25f781a2b6e4f40c591","family_id":"qf-mod48-path-interval-linear","seed":202611176019,"semantic_template":"path-interval-linear|{end:[19,1],left:[-20,1],right:[21,1],start:[-18,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-20,21], start=-18,end=19, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-20, 21, -18, 19, gamma(t)=-18+(37)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-20, 21, -18, 19, gamma(t)=-18+(37)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0020","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"898389736153ed81e31a048d50028e05341d26640bb4708012fc9ba84a0846f5","family_id":"qf-mod48-path-interval-linear","seed":202611176020,"semantic_template":"path-interval-linear|{end:[20,1],left:[-21,1],right:[22,1],start:[-19,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-21,22], start=-19,end=20, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-21, 22, -19, 20, gamma(t)=-19+(39)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-21, 22, -19, 20, gamma(t)=-19+(39)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0021","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b94a8ee02b1bb5ad1a7119a9dd70d9b44ced446cc119c1cf7d135faa27dc36a8","family_id":"qf-mod48-path-interval-linear","seed":202611176021,"semantic_template":"path-interval-linear|{end:[21,1],left:[-22,1],right:[23,1],start:[-20,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-22,23], start=-20,end=21, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-22, 23, -20, 21, gamma(t)=-20+(41)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-22, 23, -20, 21, gamma(t)=-20+(41)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0022","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e65c72babbd2c00aa361b2b4a8da3401e3583087ff4f78c472b1edca9cc6c2f0","family_id":"qf-mod48-path-interval-linear","seed":202611176022,"semantic_template":"path-interval-linear|{end:[22,1],left:[-23,1],right:[24,1],start:[-21,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-23,24], start=-21,end=22, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-23, 24, -21, 22, gamma(t)=-21+(43)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-23, 24, -21, 22, gamma(t)=-21+(43)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0023","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b66ab41a626529a401c185c82961f5685193a0a3df4a5a044f413cc13871e2e7","family_id":"qf-mod48-path-interval-linear","seed":202611176023,"semantic_template":"path-interval-linear|{end:[23,1],left:[-24,1],right:[25,1],start:[-22,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-24,25], start=-22,end=23, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-24, 25, -22, 23, gamma(t)=-22+(45)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-24, 25, -22, 23, gamma(t)=-22+(45)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0024","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b92c28ada50ac2ccc28fb72bbbd7c7d8bde185bb89bff5c42424049525533d95","family_id":"qf-mod48-path-interval-linear","seed":202611176024,"semantic_template":"path-interval-linear|{end:[24,1],left:[-25,1],right:[26,1],start:[-23,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-25,26], start=-23,end=24, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-25, 26, -23, 24, gamma(t)=-23+(47)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-25, 26, -23, 24, gamma(t)=-23+(47)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0025","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"184f314dbe756732705f97da0290b7c18ba72cac7bb66fe1e8e7885f52e17c24","family_id":"qf-mod48-path-interval-linear","seed":202611176025,"semantic_template":"path-interval-linear|{end:[25,1],left:[-26,1],right:[27,1],start:[-24,1]}"},"fields":{"title":"An Affine Path Has Exact Endpoints and Stays in Its Convex Interval","stem":"Use closed real interval [-26,27], start=-24,end=25, with the ordinary unit interval as path domain. Give (left, right, start, end, path_formula, start_statement, end_statement, range_certificate, certificate).","answer":{"kind":"expression","value":"(-26, 27, -24, 25, gamma(t)=-24+(49)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)","explanation":"The continuous affine path (1−t)*start+t*end has the requested endpoints and lies between them for all0≤t≤1. The full convex-weight argument, not finite samples, establishes range containment. Degenerate source intervals are permitted when both endpoints coincide. The exact result, retaining the requested order, is (-26, 27, -24, 25, gamma(t)=-24+(49)t, gamma(0)=start, gamma(1)=end, Yes, convex_combination_stays_in_interval)."},"skills":["An Affine Path Has Exact Endpoints and Stays in Its Convex Interval"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261117-r06-path-connected-space-0026","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"458d8f3dc584461f8d0016a9c0964953ad03bd0282e04d8fc5fe2ba1aea2a35b","family_id":"qf-mod48-path-box-linear","seed":202611176026,"semantic_template":"path-box-linear|{end:[[2,1],[3,1]],lower:[[-2,1],[-3,1]],start:[[-1,1],[-2,1]],upper:[[3,1],[4,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-2, -3],upper=[3, 4], start=[-1, -2],end=[2, 3]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-2, -3], [3, 4], [-1, -2], [2, 3], [-1+(3)t, -2+(5)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-2, -3], [3, 4], [-1, -2], [2, 3], [-1+(3)t, -2+(5)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0027","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27c7cc9433bd3ac0836526fea70e42e003828e616476b420b688a1bdfb9f9c2f","family_id":"qf-mod48-path-box-linear","seed":202611176027,"semantic_template":"path-box-linear|{end:[[3,1],[4,1],[5,1]],lower:[[-3,1],[-4,1],[-5,1]],start:[[-2,1],[-3,1],[-4,1]],upper:[[4,1],[5,1],[6,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-3, -4, -5],upper=[4, 5, 6], start=[-2, -3, -4],end=[3, 4, 5]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-3, -4, -5], [4, 5, 6], [-2, -3, -4], [3, 4, 5], [-2+(5)t, -3+(7)t, -4+(9)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-3, -4, -5], [4, 5, 6], [-2, -3, -4], [3, 4, 5], [-2+(5)t, -3+(7)t, -4+(9)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0028","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81594d5a31fc2800afdb2299186c6f45846748b2ae35fe4f161a558baa210d8f","family_id":"qf-mod48-path-box-linear","seed":202611176028,"semantic_template":"path-box-linear|{end:[[4,1],[5,1],[6,1],[7,1]],lower:[[-4,1],[-5,1],[-6,1],[-7,1]],start:[[-3,1],[-4,1],[-5,1],[-6,1]],upper:[[5,1],[6,1],[7,1],[8,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-4, -5, -6, -7],upper=[5, 6, 7, 8], start=[-3, -4, -5, -6],end=[4, 5, 6, 7]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-4, -5, -6, -7], [5, 6, 7, 8], [-3, -4, -5, -6], [4, 5, 6, 7], [-3+(7)t, -4+(9)t, -5+(11)t, -6+(13)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-4, -5, -6, -7], [5, 6, 7, 8], [-3, -4, -5, -6], [4, 5, 6, 7], [-3+(7)t, -4+(9)t, -5+(11)t, -6+(13)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0029","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9aac562e4ed58707281f9dc7585fc8cf1da0fc0cd0181a951e14537d7274e402","family_id":"qf-mod48-path-box-linear","seed":202611176029,"semantic_template":"path-box-linear|{end:[[5,1],[6,1]],lower:[[-5,1],[-6,1]],start:[[-4,1],[-5,1]],upper:[[6,1],[7,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-5, -6],upper=[6, 7], start=[-4, -5],end=[5, 6]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-5, -6], [6, 7], [-4, -5], [5, 6], [-4+(9)t, -5+(11)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-5, -6], [6, 7], [-4, -5], [5, 6], [-4+(9)t, -5+(11)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0030","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d1df4d5734230c606736a85933c9d6b884a1b515cbd2c6ed1cb22665dbe03c1","family_id":"qf-mod48-path-box-linear","seed":202611176030,"semantic_template":"path-box-linear|{end:[[6,1],[7,1],[8,1]],lower:[[-6,1],[-7,1],[-8,1]],start:[[-5,1],[-6,1],[-7,1]],upper:[[7,1],[8,1],[9,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-6, -7, -8],upper=[7, 8, 9], start=[-5, -6, -7],end=[6, 7, 8]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-6, -7, -8], [7, 8, 9], [-5, -6, -7], [6, 7, 8], [-5+(11)t, -6+(13)t, -7+(15)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-6, -7, -8], [7, 8, 9], [-5, -6, -7], [6, 7, 8], [-5+(11)t, -6+(13)t, -7+(15)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0031","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1276900b4831e3a15eca0eb7bedfe544c38d9e80ac4d1e3793784a6f0cb560f4","family_id":"qf-mod48-path-box-linear","seed":202611176031,"semantic_template":"path-box-linear|{end:[[7,1],[8,1],[9,1],[10,1]],lower:[[-7,1],[-8,1],[-9,1],[-10,1]],start:[[-6,1],[-7,1],[-8,1],[-9,1]],upper:[[8,1],[9,1],[10,1],[11,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-7, -8, -9, -10],upper=[8, 9, 10, 11], start=[-6, -7, -8, -9],end=[7, 8, 9, 10]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-7, -8, -9, -10], [8, 9, 10, 11], [-6, -7, -8, -9], [7, 8, 9, 10], [-6+(13)t, -7+(15)t, -8+(17)t, -9+(19)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-7, -8, -9, -10], [8, 9, 10, 11], [-6, -7, -8, -9], [7, 8, 9, 10], [-6+(13)t, -7+(15)t, -8+(17)t, -9+(19)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0032","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a0e60c4b397059bd7c7a0317ad3c5fd68c23953f60c90bb173483b57d0a85d3","family_id":"qf-mod48-path-box-linear","seed":202611176032,"semantic_template":"path-box-linear|{end:[[8,1],[9,1]],lower:[[-8,1],[-9,1]],start:[[-7,1],[-8,1]],upper:[[9,1],[10,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-8, -9],upper=[9, 10], start=[-7, -8],end=[8, 9]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-8, -9], [9, 10], [-7, -8], [8, 9], [-7+(15)t, -8+(17)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-8, -9], [9, 10], [-7, -8], [8, 9], [-7+(15)t, -8+(17)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0033","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73f0d4670cde4c4bfb36d70232eb79ea183cb6660983ee7f6bd0113846388408","family_id":"qf-mod48-path-box-linear","seed":202611176033,"semantic_template":"path-box-linear|{end:[[9,1],[10,1],[11,1]],lower:[[-9,1],[-10,1],[-11,1]],start:[[-8,1],[-9,1],[-10,1]],upper:[[10,1],[11,1],[12,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-9, -10, -11],upper=[10, 11, 12], start=[-8, -9, -10],end=[9, 10, 11]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-9, -10, -11], [10, 11, 12], [-8, -9, -10], [9, 10, 11], [-8+(17)t, -9+(19)t, -10+(21)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-9, -10, -11], [10, 11, 12], [-8, -9, -10], [9, 10, 11], [-8+(17)t, -9+(19)t, -10+(21)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0034","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0646a0521b0fc0be5a596ba2e13b307541e8288f96c89f058fa3902fcc75d103","family_id":"qf-mod48-path-box-linear","seed":202611176034,"semantic_template":"path-box-linear|{end:[[10,1],[11,1],[12,1],[13,1]],lower:[[-10,1],[-11,1],[-12,1],[-13,1]],start:[[-9,1],[-10,1],[-11,1],[-12,1]],upper:[[11,1],[12,1],[13,1],[14,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-10, -11, -12, -13],upper=[11, 12, 13, 14], start=[-9, -10, -11, -12],end=[10, 11, 12, 13]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-10, -11, -12, -13], [11, 12, 13, 14], [-9, -10, -11, -12], [10, 11, 12, 13], [-9+(19)t, -10+(21)t, -11+(23)t, -12+(25)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-10, -11, -12, -13], [11, 12, 13, 14], [-9, -10, -11, -12], [10, 11, 12, 13], [-9+(19)t, -10+(21)t, -11+(23)t, -12+(25)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0035","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd57364b4d267312022eade28478c3b7198ddb20a1f0e3a30bd3a3094cd9d207","family_id":"qf-mod48-path-box-linear","seed":202611176035,"semantic_template":"path-box-linear|{end:[[11,1],[12,1]],lower:[[-11,1],[-12,1]],start:[[-10,1],[-11,1]],upper:[[12,1],[13,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-11, -12],upper=[12, 13], start=[-10, -11],end=[11, 12]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-11, -12], [12, 13], [-10, -11], [11, 12], [-10+(21)t, -11+(23)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-11, -12], [12, 13], [-10, -11], [11, 12], [-10+(21)t, -11+(23)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0036","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b444c6db000d251b54a9b965233bdb78a317258144167ff5e34056ba4e256b2","family_id":"qf-mod48-path-box-linear","seed":202611176036,"semantic_template":"path-box-linear|{end:[[12,1],[13,1],[14,1]],lower:[[-12,1],[-13,1],[-14,1]],start:[[-11,1],[-12,1],[-13,1]],upper:[[13,1],[14,1],[15,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-12, -13, -14],upper=[13, 14, 15], start=[-11, -12, -13],end=[12, 13, 14]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14], [-11+(23)t, -12+(25)t, -13+(27)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-12, -13, -14], [13, 14, 15], [-11, -12, -13], [12, 13, 14], [-11+(23)t, -12+(25)t, -13+(27)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0037","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cea7065359ef4fedb0f394537c4ede4037dd7a8128baac64bfd11ba363e9c5a0","family_id":"qf-mod48-path-box-linear","seed":202611176037,"semantic_template":"path-box-linear|{end:[[13,1],[14,1],[15,1],[16,1]],lower:[[-13,1],[-14,1],[-15,1],[-16,1]],start:[[-12,1],[-13,1],[-14,1],[-15,1]],upper:[[14,1],[15,1],[16,1],[17,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-13, -14, -15, -16],upper=[14, 15, 16, 17], start=[-12, -13, -14, -15],end=[13, 14, 15, 16]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-13, -14, -15, -16], [14, 15, 16, 17], [-12, -13, -14, -15], [13, 14, 15, 16], [-12+(25)t, -13+(27)t, -14+(29)t, -15+(31)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-13, -14, -15, -16], [14, 15, 16, 17], [-12, -13, -14, -15], [13, 14, 15, 16], [-12+(25)t, -13+(27)t, -14+(29)t, -15+(31)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0038","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"760d7838a5decbb251eb70b21e4b8145ddda5ff9022bb31764142876e63fd023","family_id":"qf-mod48-path-box-linear","seed":202611176038,"semantic_template":"path-box-linear|{end:[[14,1],[15,1]],lower:[[-14,1],[-15,1]],start:[[-13,1],[-14,1]],upper:[[15,1],[16,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-14, -15],upper=[15, 16], start=[-13, -14],end=[14, 15]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-14, -15], [15, 16], [-13, -14], [14, 15], [-13+(27)t, -14+(29)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0039","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d136955fff4a163367808edcce32e4731276b5c92683f17fa808c2990b694597","family_id":"qf-mod48-path-box-linear","seed":202611176039,"semantic_template":"path-box-linear|{end:[[15,1],[16,1],[17,1]],lower:[[-15,1],[-16,1],[-17,1]],start:[[-14,1],[-15,1],[-16,1]],upper:[[16,1],[17,1],[18,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-15, -16, -17],upper=[16, 17, 18], start=[-14, -15, -16],end=[15, 16, 17]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17], [-14+(29)t, -15+(31)t, -16+(33)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-15, -16, -17], [16, 17, 18], [-14, -15, -16], [15, 16, 17], [-14+(29)t, -15+(31)t, -16+(33)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0040","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"08833480e0c6104a3929c1bd126251cbceaeaeea81ac1576479d13d7bc8f8883","family_id":"qf-mod48-path-box-linear","seed":202611176040,"semantic_template":"path-box-linear|{end:[[16,1],[17,1],[18,1],[19,1]],lower:[[-16,1],[-17,1],[-18,1],[-19,1]],start:[[-15,1],[-16,1],[-17,1],[-18,1]],upper:[[17,1],[18,1],[19,1],[20,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-16, -17, -18, -19],upper=[17, 18, 19, 20], start=[-15, -16, -17, -18],end=[16, 17, 18, 19]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-16, -17, -18, -19], [17, 18, 19, 20], [-15, -16, -17, -18], [16, 17, 18, 19], [-15+(31)t, -16+(33)t, -17+(35)t, -18+(37)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-16, -17, -18, -19], [17, 18, 19, 20], [-15, -16, -17, -18], [16, 17, 18, 19], [-15+(31)t, -16+(33)t, -17+(35)t, -18+(37)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0041","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"224a4b0b792b39ac5f986723ddb2067d21f83ebfca0e41503ab1a7bf248178b6","family_id":"qf-mod48-path-box-linear","seed":202611176041,"semantic_template":"path-box-linear|{end:[[17,1],[18,1]],lower:[[-17,1],[-18,1]],start:[[-16,1],[-17,1]],upper:[[18,1],[19,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-17, -18],upper=[18, 19], start=[-16, -17],end=[17, 18]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-17, -18], [18, 19], [-16, -17], [17, 18], [-16+(33)t, -17+(35)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-17, -18], [18, 19], [-16, -17], [17, 18], [-16+(33)t, -17+(35)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0042","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c72b29d09af77512c626d5896684be53a7b42a897c96b489abbabb04e369ba9b","family_id":"qf-mod48-path-box-linear","seed":202611176042,"semantic_template":"path-box-linear|{end:[[18,1],[19,1],[20,1]],lower:[[-18,1],[-19,1],[-20,1]],start:[[-17,1],[-18,1],[-19,1]],upper:[[19,1],[20,1],[21,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-18, -19, -20],upper=[19, 20, 21], start=[-17, -18, -19],end=[18, 19, 20]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-18, -19, -20], [19, 20, 21], [-17, -18, -19], [18, 19, 20], [-17+(35)t, -18+(37)t, -19+(39)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-18, -19, -20], [19, 20, 21], [-17, -18, -19], [18, 19, 20], [-17+(35)t, -18+(37)t, -19+(39)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0043","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f591808c5b9a9e212f686f99e2318d7bc2a2fe25dea2da953a7051d1b22be104","family_id":"qf-mod48-path-box-linear","seed":202611176043,"semantic_template":"path-box-linear|{end:[[19,1],[20,1],[21,1],[22,1]],lower:[[-19,1],[-20,1],[-21,1],[-22,1]],start:[[-18,1],[-19,1],[-20,1],[-21,1]],upper:[[20,1],[21,1],[22,1],[23,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-19, -20, -21, -22],upper=[20, 21, 22, 23], start=[-18, -19, -20, -21],end=[19, 20, 21, 22]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22], [-18+(37)t, -19+(39)t, -20+(41)t, -21+(43)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-19, -20, -21, -22], [20, 21, 22, 23], [-18, -19, -20, -21], [19, 20, 21, 22], [-18+(37)t, -19+(39)t, -20+(41)t, -21+(43)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0044","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02dbad70804e524d05459e8170325329c11af50156d0e61688bc83dba29a44b","family_id":"qf-mod48-path-box-linear","seed":202611176044,"semantic_template":"path-box-linear|{end:[[20,1],[21,1]],lower:[[-20,1],[-21,1]],start:[[-19,1],[-20,1]],upper:[[21,1],[22,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-20, -21],upper=[21, 22], start=[-19, -20],end=[20, 21]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-20, -21], [21, 22], [-19, -20], [20, 21], [-19+(39)t, -20+(41)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0045","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40fbfb454c6f4bcf1cb282d98df51b6716b264ad7826da56f2b29e549c9901cf","family_id":"qf-mod48-path-box-linear","seed":202611176045,"semantic_template":"path-box-linear|{end:[[21,1],[22,1],[23,1]],lower:[[-21,1],[-22,1],[-23,1]],start:[[-20,1],[-21,1],[-22,1]],upper:[[22,1],[23,1],[24,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-21, -22, -23],upper=[22, 23, 24], start=[-20, -21, -22],end=[21, 22, 23]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23], [-20+(41)t, -21+(43)t, -22+(45)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-21, -22, -23], [22, 23, 24], [-20, -21, -22], [21, 22, 23], [-20+(41)t, -21+(43)t, -22+(45)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0046","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f7bf4437d9f8b562d84e4ff3746bc0a2352e6b96fedab72ac619fd8af2abd22","family_id":"qf-mod48-path-box-linear","seed":202611176046,"semantic_template":"path-box-linear|{end:[[22,1],[23,1],[24,1],[25,1]],lower:[[-22,1],[-23,1],[-24,1],[-25,1]],start:[[-21,1],[-22,1],[-23,1],[-24,1]],upper:[[23,1],[24,1],[25,1],[26,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-22, -23, -24, -25],upper=[23, 24, 25, 26], start=[-21, -22, -23, -24],end=[22, 23, 24, 25]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-22, -23, -24, -25], [23, 24, 25, 26], [-21, -22, -23, -24], [22, 23, 24, 25], [-21+(43)t, -22+(45)t, -23+(47)t, -24+(49)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-22, -23, -24, -25], [23, 24, 25, 26], [-21, -22, -23, -24], [22, 23, 24, 25], [-21+(43)t, -22+(45)t, -23+(47)t, -24+(49)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0047","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5950114c10d40dae09294612eb713535bc770b58f89ae3f6a30c4986acce772a","family_id":"qf-mod48-path-box-linear","seed":202611176047,"semantic_template":"path-box-linear|{end:[[23,1],[24,1]],lower:[[-23,1],[-24,1]],start:[[-22,1],[-23,1]],upper:[[24,1],[25,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-23, -24],upper=[24, 25], start=[-22, -23],end=[23, 24]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-23, -24], [24, 25], [-22, -23], [23, 24], [-22+(45)t, -23+(47)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-23, -24], [24, 25], [-22, -23], [23, 24], [-22+(45)t, -23+(47)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0048","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"175d6b22d3b0375b1c1e1c4c669368b59a5e1b143fba7c095b9bc95e35a3b8b4","family_id":"qf-mod48-path-box-linear","seed":202611176048,"semantic_template":"path-box-linear|{end:[[24,1],[25,1],[26,1]],lower:[[-24,1],[-25,1],[-26,1]],start:[[-23,1],[-24,1],[-25,1]],upper:[[25,1],[26,1],[27,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-24, -25, -26],upper=[25, 26, 27], start=[-23, -24, -25],end=[24, 25, 26]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-24, -25, -26], [25, 26, 27], [-23, -24, -25], [24, 25, 26], [-23+(47)t, -24+(49)t, -25+(51)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-24, -25, -26], [25, 26, 27], [-23, -24, -25], [24, 25, 26], [-23+(47)t, -24+(49)t, -25+(51)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0049","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6152f5444a706b6721ac5c99ccf90e2f6a4d922919d5540e1d735e93afdb66e","family_id":"qf-mod48-path-box-linear","seed":202611176049,"semantic_template":"path-box-linear|{end:[[25,1],[26,1],[27,1],[28,1]],lower:[[-25,1],[-26,1],[-27,1],[-28,1]],start:[[-24,1],[-25,1],[-26,1],[-27,1]],upper:[[26,1],[27,1],[28,1],[29,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-25, -26, -27, -28],upper=[26, 27, 28, 29], start=[-24, -25, -26, -27],end=[25, 26, 27, 28]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-25, -26, -27, -28], [26, 27, 28, 29], [-24, -25, -26, -27], [25, 26, 27, 28], [-24+(49)t, -25+(51)t, -26+(53)t, -27+(55)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-25, -26, -27, -28], [26, 27, 28, 29], [-24, -25, -26, -27], [25, 26, 27, 28], [-24+(49)t, -25+(51)t, -26+(53)t, -27+(55)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0050","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afcb5670b4e466ca3b3afc63734a4ef1146ec262fc59ff4494a904d3fc00327c","family_id":"qf-mod48-path-box-linear","seed":202611176050,"semantic_template":"path-box-linear|{end:[[26,1],[27,1]],lower:[[-26,1],[-27,1]],start:[[-25,1],[-26,1]],upper:[[27,1],[28,1]]}"},"fields":{"title":"Coordinatewise Convex Weights Keep the Entire Path in the Box","stem":"Use coordinate bounds lower=[-26, -27],upper=[27, 28], start=[-25, -26],end=[26, 27]. Give (lower, upper, start, end, coordinate_formulas, range_certificate, certificate).","answer":{"kind":"expression","value":"([-26, -27], [27, 28], [-25, -26], [26, 27], [-25+(51)t, -26+(53)t], Yes, coordinatewise_convex_combination_stays_in_box)","explanation":"Qualify matching nonzero dimensions and each endpoint coordinate, then retain every affine coordinate formula. Nonnegative weights summing to1 give whole-range containment and coordinate continuity in the finite product, not a graph walk or just sampled values. The exact result, retaining the requested order, is ([-26, -27], [27, 28], [-25, -26], [26, 27], [-25+(51)t, -26+(53)t], Yes, coordinatewise_convex_combination_stays_in_box)."},"skills":["Coordinatewise Convex Weights Keep the Entire Path in the Box"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0051","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b321ca34be62760a801cd13e16ea62da67e0372afe7c2aa0c4a77083d5535b22","family_id":"qf-mod48-path-concatenation","seed":202611176051,"semantic_template":"path-concatenation|{end:[0,1],left:[-3,1],middle:[2,1],right:[4,1],start:[0,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-3,4] and contained start=0,middle=2,end=0. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(0, 2, 0, 0<=t<=1/2:0+2t(2), 1/2<=t<=1:2+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (0, 2, 0, 0<=t<=1/2:0+2t(2), 1/2<=t<=1:2+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0052","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa98b7ad0af430a2a86ee3759b7ebaf155248328b4621b402834cfa7e7966d27","family_id":"qf-mod48-path-concatenation","seed":202611176052,"semantic_template":"path-concatenation|{end:[1,1],left:[-4,1],middle:[3,1],right:[5,1],start:[-1,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-4,5] and contained start=-1,middle=3,end=1. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-1, 3, 1, 0<=t<=1/2:-1+2t(4), 1/2<=t<=1:3+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-1, 3, 1, 0<=t<=1/2:-1+2t(4), 1/2<=t<=1:3+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0053","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"121835b2fc71d3fcb2f48f376dbc18bee05475a0f5cbe5507ca4ce998680c37a","family_id":"qf-mod48-path-concatenation","seed":202611176053,"semantic_template":"path-concatenation|{end:[2,1],left:[-5,1],middle:[4,1],right:[6,1],start:[-2,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-5,6] and contained start=-2,middle=4,end=2. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-2, 4, 2, 0<=t<=1/2:-2+2t(6), 1/2<=t<=1:4+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-2, 4, 2, 0<=t<=1/2:-2+2t(6), 1/2<=t<=1:4+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0054","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90e73849a8b27864e7ead3fa8af43d8725dc0a16151c57a3769915fa9fdaaa1a","family_id":"qf-mod48-path-concatenation","seed":202611176054,"semantic_template":"path-concatenation|{end:[3,1],left:[-6,1],middle:[5,1],right:[7,1],start:[-3,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-6,7] and contained start=-3,middle=5,end=3. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-3, 5, 3, 0<=t<=1/2:-3+2t(8), 1/2<=t<=1:5+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-3, 5, 3, 0<=t<=1/2:-3+2t(8), 1/2<=t<=1:5+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0055","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa3e35cc4fa5e770f51b377e968b2b417f2e78b50402a06ee482107a81d126ce","family_id":"qf-mod48-path-concatenation","seed":202611176055,"semantic_template":"path-concatenation|{end:[4,1],left:[-7,1],middle:[6,1],right:[8,1],start:[-4,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-7,8] and contained start=-4,middle=6,end=4. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-4, 6, 4, 0<=t<=1/2:-4+2t(10), 1/2<=t<=1:6+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-4, 6, 4, 0<=t<=1/2:-4+2t(10), 1/2<=t<=1:6+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0056","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee2814ab298dd780970106f6b652f48b2fb2a04819d1f1a9868c91e90dd056a3","family_id":"qf-mod48-path-concatenation","seed":202611176056,"semantic_template":"path-concatenation|{end:[5,1],left:[-8,1],middle:[7,1],right:[9,1],start:[-5,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-8,9] and contained start=-5,middle=7,end=5. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-5, 7, 5, 0<=t<=1/2:-5+2t(12), 1/2<=t<=1:7+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-5, 7, 5, 0<=t<=1/2:-5+2t(12), 1/2<=t<=1:7+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0057","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1277fd00f2ecd3bab45d07d471e5d2b261ae15bb92485395676122c96a954dc","family_id":"qf-mod48-path-concatenation","seed":202611176057,"semantic_template":"path-concatenation|{end:[6,1],left:[-9,1],middle:[8,1],right:[10,1],start:[-6,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-9,10] and contained start=-6,middle=8,end=6. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-6, 8, 6, 0<=t<=1/2:-6+2t(14), 1/2<=t<=1:8+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-6, 8, 6, 0<=t<=1/2:-6+2t(14), 1/2<=t<=1:8+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0058","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8099a5678809d8ffb4b7147f696e5ea3bcd401185376179b259aadbc88496cb2","family_id":"qf-mod48-path-concatenation","seed":202611176058,"semantic_template":"path-concatenation|{end:[7,1],left:[-10,1],middle:[9,1],right:[11,1],start:[-7,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-10,11] and contained start=-7,middle=9,end=7. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-7, 9, 7, 0<=t<=1/2:-7+2t(16), 1/2<=t<=1:9+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-7, 9, 7, 0<=t<=1/2:-7+2t(16), 1/2<=t<=1:9+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0059","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3091ab23d7c25db7279a30c7f3d4c833328255668307ff0fe0043a20c7a5a772","family_id":"qf-mod48-path-concatenation","seed":202611176059,"semantic_template":"path-concatenation|{end:[8,1],left:[-11,1],middle:[10,1],right:[12,1],start:[-8,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-11,12] and contained start=-8,middle=10,end=8. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-8, 10, 8, 0<=t<=1/2:-8+2t(18), 1/2<=t<=1:10+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-8, 10, 8, 0<=t<=1/2:-8+2t(18), 1/2<=t<=1:10+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0060","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27273a8aac0cd9597d92c8ab612f5b00f5d871383b878b71213838e155df29c5","family_id":"qf-mod48-path-concatenation","seed":202611176060,"semantic_template":"path-concatenation|{end:[9,1],left:[-12,1],middle:[11,1],right:[13,1],start:[-9,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-12,13] and contained start=-9,middle=11,end=9. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-9, 11, 9, 0<=t<=1/2:-9+2t(20), 1/2<=t<=1:11+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-9, 11, 9, 0<=t<=1/2:-9+2t(20), 1/2<=t<=1:11+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0061","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53e225212a3ffd8be807e8e378505e4ab96c21ed8e85b0c3231fdfb31fd74e08","family_id":"qf-mod48-path-concatenation","seed":202611176061,"semantic_template":"path-concatenation|{end:[10,1],left:[-13,1],middle:[12,1],right:[14,1],start:[-10,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-13,14] and contained start=-10,middle=12,end=10. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-10, 12, 10, 0<=t<=1/2:-10+2t(22), 1/2<=t<=1:12+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-10, 12, 10, 0<=t<=1/2:-10+2t(22), 1/2<=t<=1:12+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0062","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"401fd9843458470d3ae13cda6cc406a838eea66e148ad6f563651381e159aff4","family_id":"qf-mod48-path-concatenation","seed":202611176062,"semantic_template":"path-concatenation|{end:[11,1],left:[-14,1],middle:[13,1],right:[15,1],start:[-11,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-14,15] and contained start=-11,middle=13,end=11. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-11, 13, 11, 0<=t<=1/2:-11+2t(24), 1/2<=t<=1:13+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0063","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb39be78f565d5d5e7741bc562841fda494fb7659f8618dc4987f6e5ea4ec63b","family_id":"qf-mod48-path-concatenation","seed":202611176063,"semantic_template":"path-concatenation|{end:[12,1],left:[-15,1],middle:[14,1],right:[16,1],start:[-12,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-15,16] and contained start=-12,middle=14,end=12. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-12, 14, 12, 0<=t<=1/2:-12+2t(26), 1/2<=t<=1:14+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-12, 14, 12, 0<=t<=1/2:-12+2t(26), 1/2<=t<=1:14+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0064","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f1f3680bd507d23f1511ba75d70e14d8f9493eab3aba215d64c8cdb0598c532f","family_id":"qf-mod48-path-concatenation","seed":202611176064,"semantic_template":"path-concatenation|{end:[13,1],left:[-16,1],middle:[15,1],right:[17,1],start:[-13,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-16,17] and contained start=-13,middle=15,end=13. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-13, 15, 13, 0<=t<=1/2:-13+2t(28), 1/2<=t<=1:15+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-13, 15, 13, 0<=t<=1/2:-13+2t(28), 1/2<=t<=1:15+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0065","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ae5e0772e11471f2b126179b2ae2859d8865ad2edc0db9dbf6bb0f9c2cb6037","family_id":"qf-mod48-path-concatenation","seed":202611176065,"semantic_template":"path-concatenation|{end:[14,1],left:[-17,1],middle:[16,1],right:[18,1],start:[-14,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-17,18] and contained start=-14,middle=16,end=14. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-14, 16, 14, 0<=t<=1/2:-14+2t(30), 1/2<=t<=1:16+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-14, 16, 14, 0<=t<=1/2:-14+2t(30), 1/2<=t<=1:16+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0066","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d75b813a92678b3bc385413f6541e257c7df628b775946cada681a3eed2e925","family_id":"qf-mod48-path-concatenation","seed":202611176066,"semantic_template":"path-concatenation|{end:[15,1],left:[-18,1],middle:[17,1],right:[19,1],start:[-15,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-18,19] and contained start=-15,middle=17,end=15. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-15, 17, 15, 0<=t<=1/2:-15+2t(32), 1/2<=t<=1:17+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-15, 17, 15, 0<=t<=1/2:-15+2t(32), 1/2<=t<=1:17+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0067","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c31754101c06acf9f85d259e69a88c49c4ccf68577695dd10cfc84ab5148fcf9","family_id":"qf-mod48-path-concatenation","seed":202611176067,"semantic_template":"path-concatenation|{end:[16,1],left:[-19,1],middle:[18,1],right:[20,1],start:[-16,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-19,20] and contained start=-16,middle=18,end=16. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-16, 18, 16, 0<=t<=1/2:-16+2t(34), 1/2<=t<=1:18+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-16, 18, 16, 0<=t<=1/2:-16+2t(34), 1/2<=t<=1:18+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0068","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"772c18f7c1e360f7694373528f3e7671f01439dad8c8ce55036cac1b21e4b0d0","family_id":"qf-mod48-path-concatenation","seed":202611176068,"semantic_template":"path-concatenation|{end:[17,1],left:[-20,1],middle:[19,1],right:[21,1],start:[-17,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-20,21] and contained start=-17,middle=19,end=17. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-17, 19, 17, 0<=t<=1/2:-17+2t(36), 1/2<=t<=1:19+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-17, 19, 17, 0<=t<=1/2:-17+2t(36), 1/2<=t<=1:19+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0069","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d792e33691b26e201ad88264c620ca72c8e7a029a8d487f061bd06238bc4ba3","family_id":"qf-mod48-path-concatenation","seed":202611176069,"semantic_template":"path-concatenation|{end:[18,1],left:[-21,1],middle:[20,1],right:[22,1],start:[-18,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-21,22] and contained start=-18,middle=20,end=18. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-18, 20, 18, 0<=t<=1/2:-18+2t(38), 1/2<=t<=1:20+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0070","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d69acf018dc69a85a04ea7b2429db369b63e05ae0fd3466ab39c1c6b8717621","family_id":"qf-mod48-path-concatenation","seed":202611176070,"semantic_template":"path-concatenation|{end:[19,1],left:[-22,1],middle:[21,1],right:[23,1],start:[-19,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-22,23] and contained start=-19,middle=21,end=19. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-19, 21, 19, 0<=t<=1/2:-19+2t(40), 1/2<=t<=1:21+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0071","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46fba61cea8514fa0502f553117cf4db8640646b0e1405f89c3575c153bc3522","family_id":"qf-mod48-path-concatenation","seed":202611176071,"semantic_template":"path-concatenation|{end:[20,1],left:[-23,1],middle:[22,1],right:[24,1],start:[-20,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-23,24] and contained start=-20,middle=22,end=20. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-20, 22, 20, 0<=t<=1/2:-20+2t(42), 1/2<=t<=1:22+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-20, 22, 20, 0<=t<=1/2:-20+2t(42), 1/2<=t<=1:22+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0072","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"403e83fa82e4bbf9b97ada0c9ba0e3734f31cd3de6804e2e4169601b98769a7c","family_id":"qf-mod48-path-concatenation","seed":202611176072,"semantic_template":"path-concatenation|{end:[21,1],left:[-24,1],middle:[23,1],right:[25,1],start:[-21,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-24,25] and contained start=-21,middle=23,end=21. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-21, 23, 21, 0<=t<=1/2:-21+2t(44), 1/2<=t<=1:23+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-21, 23, 21, 0<=t<=1/2:-21+2t(44), 1/2<=t<=1:23+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0073","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9398c3718ba1dd0280f46433f8a68aeae64abdf0644a9fa4ac2fb83bc8e78ae","family_id":"qf-mod48-path-concatenation","seed":202611176073,"semantic_template":"path-concatenation|{end:[22,1],left:[-25,1],middle:[24,1],right:[26,1],start:[-22,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-25,26] and contained start=-22,middle=24,end=22. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-22, 24, 22, 0<=t<=1/2:-22+2t(46), 1/2<=t<=1:24+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-22, 24, 22, 0<=t<=1/2:-22+2t(46), 1/2<=t<=1:24+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0074","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36a616c455709b5a50c5caf0b1d08434f9378be324ee29d5d727476e394e2685","family_id":"qf-mod48-path-concatenation","seed":202611176074,"semantic_template":"path-concatenation|{end:[23,1],left:[-26,1],middle:[25,1],right:[27,1],start:[-23,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-26,27] and contained start=-23,middle=25,end=23. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-23, 25, 23, 0<=t<=1/2:-23+2t(48), 1/2<=t<=1:25+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-23, 25, 23, 0<=t<=1/2:-23+2t(48), 1/2<=t<=1:25+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0075","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"780d2cf8469300a4e877d5a0edc9a2182585d973b6c0557e67625b9cfeb93258","family_id":"qf-mod48-path-concatenation","seed":202611176075,"semantic_template":"path-concatenation|{end:[24,1],left:[-27,1],middle:[26,1],right:[28,1],start:[-24,1]}"},"fields":{"title":"Two Rescaled Affine Branches Meet at the Exact Middle Point","stem":"Use real interval [-27,28] and contained start=-24,middle=26,end=24. Give (start, middle, end, first_branch, second_branch, join_statement, path_certificate).","answer":{"kind":"expression","value":"(-24, 26, 24, 0<=t<=1/2:-24+2t(50), 1/2<=t<=1:26+(2t-1)(-2), both_branches_meet_at_middle, Yes)","explanation":"Rescale the first path to[0,1/2] and the second to[1/2,1]. Check that both branches meet at middle, with the correct outer endpoints and full interval range. Piecewise continuity needs the shared join, not merely two endpoint samples. The exact result, retaining the requested order, is (-24, 26, 24, 0<=t<=1/2:-24+2t(50), 1/2<=t<=1:26+(2t-1)(-2), both_branches_meet_at_middle, Yes)."},"skills":["Two Rescaled Affine Branches Meet at the Exact Middle Point"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0076","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcb43ea1c1277a33508500619c1e2a93727763a36b3a8511b884c8ddece8bf79","family_id":"qf-mod48-path-connected-boundary","seed":202611176076,"semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[1,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=disconnected_gap, positive exact scale=1, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(disconnected_gap, 1, X=(-infinity,0] union [1,infinity), IVT_blocks_a_path_between_components, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 1, X=(-infinity,0] union [1,infinity), IVT_blocks_a_path_between_components, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0077","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"756a34a3c2b5f3655b3fc9d54bac75a16bd0ae742ed2dd0426f0012aad852a3f","family_id":"qf-mod48-path-connected-boundary","seed":202611176077,"semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[2,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=discrete_two_point, positive exact scale=2, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(discrete_two_point, 2, X={0,2}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 2, X={0,2}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0078","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fe22438a3b0c376610dabb3cfa8d99400547d76d8e39343c90f9db7b5ea7d7f","family_id":"qf-mod48-path-connected-boundary","seed":202611176078,"semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[3,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=endpoint_only, positive exact scale=3, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(endpoint_only, 3, candidate_samples_gamma(0)=0_gamma(1/2)=3_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 3, candidate_samples_gamma(0)=0_gamma(1/2)=3_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0079","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac8f6e23597543b9bfddcdda8903a382e57ebe0272631090bb21cb2ded373c4f","family_id":"qf-mod48-path-connected-boundary","seed":202611176079,"semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[4,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=connected_not_path, positive exact scale=4, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(connected_not_path, 4, topologists_sine_curve_scaled, connected_but_not_path_connected, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 4, topologists_sine_curve_scaled, connected_but_not_path_connected, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0080","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6ee496e7e08257d69518111d85edd44ad36b2b5ef50dbe32fa1aaaf5dc4474d","family_id":"qf-mod48-path-connected-boundary","seed":202611176080,"semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[5,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=disconnected_gap, positive exact scale=5, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(disconnected_gap, 5, X=(-infinity,0] union [5,infinity), IVT_blocks_a_path_between_components, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 5, X=(-infinity,0] union [5,infinity), IVT_blocks_a_path_between_components, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0081","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95794a84216be81c31fcc69a151196ae1af872ffe33a835ed1214b69f1fda768","family_id":"qf-mod48-path-connected-boundary","seed":202611176081,"semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[6,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=discrete_two_point, positive exact scale=6, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(discrete_two_point, 6, X={0,6}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 6, X={0,6}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0082","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2751344809c7f7603714a3a98fbcc99c7bfa5472ca980d49f4d6197d159c7033","family_id":"qf-mod48-path-connected-boundary","seed":202611176082,"semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[7,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=endpoint_only, positive exact scale=7, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(endpoint_only, 7, candidate_samples_gamma(0)=0_gamma(1/2)=7_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 7, candidate_samples_gamma(0)=0_gamma(1/2)=7_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0083","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"365bab14fb002be818e2a27cd200552b57a12db208721733bef41d82194333a3","family_id":"qf-mod48-path-connected-boundary","seed":202611176083,"semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[8,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=connected_not_path, positive exact scale=8, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(connected_not_path, 8, topologists_sine_curve_scaled, connected_but_not_path_connected, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 8, topologists_sine_curve_scaled, connected_but_not_path_connected, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0084","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e70412a4f706923cf38328b625c2b85c3045a0556436eaa623316a0164b822b4","family_id":"qf-mod48-path-connected-boundary","seed":202611176084,"semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[9,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=disconnected_gap, positive exact scale=9, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(disconnected_gap, 9, X=(-infinity,0] union [9,infinity), IVT_blocks_a_path_between_components, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 9, X=(-infinity,0] union [9,infinity), IVT_blocks_a_path_between_components, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0085","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40cf08ac3b9681d4379594a2e5a1f6b54d3d6adf461589a56b16943ca9eb5b54","family_id":"qf-mod48-path-connected-boundary","seed":202611176085,"semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[10,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=discrete_two_point, positive exact scale=10, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(discrete_two_point, 10, X={0,10}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 10, X={0,10}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0086","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e0d05ed0e76d958dbe3e592e1dbda2a79279e98a3effab2b005ab52abc1a9b5","family_id":"qf-mod48-path-connected-boundary","seed":202611176086,"semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[11,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=endpoint_only, positive exact scale=11, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(endpoint_only, 11, candidate_samples_gamma(0)=0_gamma(1/2)=11_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 11, candidate_samples_gamma(0)=0_gamma(1/2)=11_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0087","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73d79108610ef6a12674a2475d28134da0c3bc040db4e45351d7430f7ae6a7f4","family_id":"qf-mod48-path-connected-boundary","seed":202611176087,"semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[12,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=connected_not_path, positive exact scale=12, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(connected_not_path, 12, topologists_sine_curve_scaled, connected_but_not_path_connected, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 12, topologists_sine_curve_scaled, connected_but_not_path_connected, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0088","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35fdab98c18b68ddd3fb63c97d6f2184eb2bc1620d470d1ac673a7e07b66fc0e","family_id":"qf-mod48-path-connected-boundary","seed":202611176088,"semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[13,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=disconnected_gap, positive exact scale=13, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(disconnected_gap, 13, X=(-infinity,0] union [13,infinity), IVT_blocks_a_path_between_components, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 13, X=(-infinity,0] union [13,infinity), IVT_blocks_a_path_between_components, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0089","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ace279d8534b97c695cfde65fec13614379d36b966b1cb073bd3a57f9061f012","family_id":"qf-mod48-path-connected-boundary","seed":202611176089,"semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[14,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=discrete_two_point, positive exact scale=14, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(discrete_two_point, 14, X={0,14}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 14, X={0,14}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0090","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aeeb2f9b346973eeb71005f58607d401611df9a2294327f05dde7506e8a9869e","family_id":"qf-mod48-path-connected-boundary","seed":202611176090,"semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[15,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=endpoint_only, positive exact scale=15, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(endpoint_only, 15, candidate_samples_gamma(0)=0_gamma(1/2)=15_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 15, candidate_samples_gamma(0)=0_gamma(1/2)=15_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0091","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a843aece304a7f1667d6cdddd577b2b139806b93f6ebf792508b4ca30187dcf5","family_id":"qf-mod48-path-connected-boundary","seed":202611176091,"semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[16,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=connected_not_path, positive exact scale=16, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(connected_not_path, 16, topologists_sine_curve_scaled, connected_but_not_path_connected, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 16, topologists_sine_curve_scaled, connected_but_not_path_connected, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0092","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13684e86729402e1f91a926301debb703e0d2919d1eadc79d22af92bf66692b3","family_id":"qf-mod48-path-connected-boundary","seed":202611176092,"semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[17,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=disconnected_gap, positive exact scale=17, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(disconnected_gap, 17, X=(-infinity,0] union [17,infinity), IVT_blocks_a_path_between_components, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 17, X=(-infinity,0] union [17,infinity), IVT_blocks_a_path_between_components, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0093","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0468e67d3fcb7c60801de631633820b71d1eb8dcec2670ec9613703e45041dbd","family_id":"qf-mod48-path-connected-boundary","seed":202611176093,"semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[18,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=discrete_two_point, positive exact scale=18, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(discrete_two_point, 18, X={0,18}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 18, X={0,18}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0094","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf6ce0850317c2a8c142b2e7a5e7925e28eaa245e79ac6becf908dae448a8205","family_id":"qf-mod48-path-connected-boundary","seed":202611176094,"semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[19,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=endpoint_only, positive exact scale=19, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(endpoint_only, 19, candidate_samples_gamma(0)=0_gamma(1/2)=19_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 19, candidate_samples_gamma(0)=0_gamma(1/2)=19_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0095","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd21157878b1bed0935b952e690092adc57603b7a7703dc0505c2e008fc45570","family_id":"qf-mod48-path-connected-boundary","seed":202611176095,"semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[20,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=connected_not_path, positive exact scale=20, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(connected_not_path, 20, topologists_sine_curve_scaled, connected_but_not_path_connected, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 20, topologists_sine_curve_scaled, connected_but_not_path_connected, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0096","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26091b94749c0eb55705369f051dfa78f5a4d8230417b67f0985455c557e497e","family_id":"qf-mod48-path-connected-boundary","seed":202611176096,"semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[21,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=disconnected_gap, positive exact scale=21, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(disconnected_gap, 21, X=(-infinity,0] union [21,infinity), IVT_blocks_a_path_between_components, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 21, X=(-infinity,0] union [21,infinity), IVT_blocks_a_path_between_components, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0097","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d6556ce71337b0cc5a67d79bad87304665984f6b8003bb842acc837fabd196c","family_id":"qf-mod48-path-connected-boundary","seed":202611176097,"semantic_template":"path-connected-boundary|{mode:\"discrete_two_point\",scale:[22,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=discrete_two_point, positive exact scale=22, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(discrete_two_point, 22, X={0,22}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (discrete_two_point, 22, X={0,22}_with_discrete_topology, continuous_image_of_connected_[0,1]_cannot_meet_both_points, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0098","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e9e4129f25e8bc155aaef7a8f5f47a306a9fc150eaad05914b57145c4d9f124","family_id":"qf-mod48-path-connected-boundary","seed":202611176098,"semantic_template":"path-connected-boundary|{mode:\"endpoint_only\",scale:[23,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=endpoint_only, positive exact scale=23, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(endpoint_only, 23, candidate_samples_gamma(0)=0_gamma(1/2)=23_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (endpoint_only, 23, candidate_samples_gamma(0)=0_gamma(1/2)=23_gamma(1)=0, three_samples_do_not_define_a_continuous_map_or_control_its_full_range, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0099","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6740288bd10c62bfdcb3fdc0eab0878f5ddb4069e36dfaefb9232460d3a4d658","family_id":"qf-mod48-path-connected-boundary","seed":202611176099,"semantic_template":"path-connected-boundary|{mode:\"connected_not_path\",scale:[24,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=connected_not_path, positive exact scale=24, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(connected_not_path, 24, topologists_sine_curve_scaled, connected_but_not_path_connected, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (connected_not_path, 24, topologists_sine_curve_scaled, connected_but_not_path_connected, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-r06-path-connected-space-0100","batch_id":"batch-20261015-advance-20261117-r06-path-connected-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f14a8062bcdfc8479aadd9d9be6c761312bc70d07a0266796fe5d81e0d93f95","family_id":"qf-mod48-path-connected-boundary","seed":202611176100,"semantic_template":"path-connected-boundary|{mode:\"disconnected_gap\",scale:[25,1]}"},"fields":{"title":"A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions","stem":"Use mode=disconnected_gap, positive exact scale=25, and the declared example description in the result. Give (mode, scale, example_description, scope_or_obstruction, path_connectedness_inference_claim).","answer":{"kind":"expression","value":"(disconnected_gap, 25, X=(-infinity,0] union [25,infinity), IVT_blocks_a_path_between_components, No)","explanation":"A continuous image of the connected unit interval cannot cross the disconnected gap or meet both isolated points. Three endpoint/interior samples do not define a full continuous path. connected_not_path records a named topologist-sine-curve scope, not a supplied full curve construction/proof in this finite family. A single pair or graph walk does not prove the universal path-connectedness claim. The exact result, retaining the requested order, is (disconnected_gap, 25, X=(-infinity,0] union [25,infinity), IVT_blocks_a_path_between_components, No)."},"skills":["A Disconnected Gap, Discrete Pair and Endpoint Samples Have Different Obstructions"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}