# Question 11175: Linear Halfspace Closure at Qualified Endpoints [Return to catalogue page 56](https://ocme.evemisslab.com/en/questions/page/56/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r07-convex-set-foundations/#q-20261013-advance-20261024-r07-convex-set-0075) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0021-11175 · q-20261013-advance-20261024-r07-convex-set-0075 ## Original question and givens Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=26, left=[0, 0], right=[25, 0], lambda=25/27∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. - G001 — Convexity means every pair of set members and every lambda in [0,1] has lambda*left+(1−lambda)*right in the set. Positive cases use complete algebraic certificates, not sampled lambda values. One exact counterexample suffices to refute convexity; a failed counterexample does not prove convexity. normal=[1, 1], bound=26, left=[0, 0], right=[25, 0], lambda=25/27∈[0,1]. Both endpoints must satisfy normal·point≤bound. Give (normal,bound,left,right,left dot product,right dot product,lambda,weighted point,its dot product,inside?). Use exact rational arithmetic, reduce fractions to positive denominators and omit /1 for integer outputs. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-convex-set](https://ocme.evemisslab.com/en/mko/mko-convex-set/) ## Step-by-step derivation S001 · calculationBefore left endpoint dot term0 := mul(1, 0) After left endpoint dot term0 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore left endpoint dot term1 := mul(1, 0) After left endpoint dot term1 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore left endpoint dot sum := sum(0, 0) After left endpoint dot sum = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore right endpoint dot term0 := mul(1, 25) After right endpoint dot term0 = 25 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore right endpoint dot term1 := mul(1, 0) After right endpoint dot term1 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore right endpoint dot sum := sum(25, 0) After right endpoint dot sum = 25 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore normal=[1, 1];bound=26;left dot=0;right dot=25 After both endpoints satisfy the closed halfspace inequality Conditions: none First qualify both endpoint memberships.Do not infer them from the desired convexity;the same inequality also handles a zero normal. Step identifiers and dependenciesRule: halfspace_endpoint_membership · Uses: G001, S006 S008 · calculationBefore halfspace point complementary weight := sub(1, 25/27) After halfspace point complementary weight = 2/27 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore halfspace point:raw lambda=[25, 27];lambda=25/27;one-minus-lambda=2/27 After in_closed_unit_interval=Yes;lambda weights left Conditions: none Lambda weights the left endpoint and1−lambda the right.Lambda0 selects right,1 selects left.Outside values still define the affine expression,but not a convex combination;do not clip. Step identifiers and dependenciesRule: two_point_direction_and_qualification · Uses: G001, S008 S010 · impliesBefore halfspace point:ordered vectors=[[0, 0], [25, 0]];aligned weights=[25/27, 2/27] After dimension=2;preserve duplicates and order Conditions: none Preserve vector order,duplicates and coefficient pairing.Sum every vector's contribution within the same coordinate. Step identifiers and dependenciesRule: ordered_vector_weight_pairing · Uses: G001, S009 S011 · calculationBefore halfspace point coordinate0 term0 := mul(25/27, 0) After halfspace point coordinate0 term0 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore halfspace point coordinate0 term1 := mul(2/27, 25) After halfspace point coordinate0 term1 = 50/27 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore halfspace point coordinate0 sum := sum(0, 50/27) After halfspace point coordinate0 sum = 50/27 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore halfspace point coordinate1 term0 := mul(25/27, 0) After halfspace point coordinate1 term0 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore halfspace point coordinate1 term1 := mul(2/27, 0) After halfspace point coordinate1 term1 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore halfspace point coordinate1 sum := sum(0, 0) After halfspace point coordinate1 sum = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore combined point dot term0 := mul(1, 50/27) After combined point dot term0 = 50/27 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore combined point dot term1 := mul(1, 0) After combined point dot term1 = 0 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore combined point dot sum := sum(50/27, 0) After combined point dot sum = 50/27 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore halfspace bound slack := sub(26, 50/27) After halfspace bound slack = 652/27 Conditions: none Compute exactly with this step's specified roles,retaining signs,denominators and original pairings. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore normal dot z=50/27;endpoint dots=(0,25);lambda=25/27;bound=26 After normal dot z=lambda*0+(1-lambda)*25<=26;inside=Yes Conditions: none Linearity makes the new dot product a nonnegative unit-weighted sum of endpoint values.Each is≤bound,so the result is≤bound.This algebraic reason is universal;the numbers instantiate it. Step identifiers and dependenciesRule: halfspace_linear_universal_reason · Uses: G001, S020 ## Reverse checks - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute halfspace bound slack with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute combined point dot sum with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute combined point dot term1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute combined point dot term0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute halfspace point coordinate1 sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute halfspace point coordinate1 term1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute halfspace point coordinate1 term0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute halfspace point coordinate0 sum with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute halfspace point coordinate0 term1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute halfspace point coordinate0 term0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute halfspace point complementary weight with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute right endpoint dot sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute right endpoint dot term1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute right endpoint dot term0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute left endpoint dot sum with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute left endpoint dot term1 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute left endpoint dot term0 with exact rational arithmetic, not rounded decimals. ## Result ([1, 1], 26, [0, 0], [25, 0], 0, 25, 25/27, [50/27, 0], 50/27, Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: af916b7bed29aa6d22427d56df99e83097725e6e0d03f520170d7effad05782e - candidate record SHA-256: 775319206209da585444b10d26e9b968bdd480dca30d33530ae4e40b9031489e - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r07-convex-set-0075.json)