# Question 27991: Squared Distance with One Regular Linear Constraint Has an Exact Solution [Return to catalogue page 140](https://ocme.evemisslab.com/en/questions/page/140/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261119-r04-lagrange-multiplier-condition-foundations/#q-20261015-pulse-20261119-r04-0066) 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. R0054-27991 · q-20261015-pulse-20261119-r04-0066 ## Original question and givens Minimize squared Euclidean distance to center=[0, 1] subject to normal a=[1, 16], a dot x=18; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope). - G001 — Minimize squared Euclidean distance to center=[0, 1] subject to normal a=[1, 16], a dot x=18; require a nonzero normal. Give (center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope). ## 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-lagrange-multiplier-condition](https://ocme.evemisslab.com/en/mko/mko-lagrange-multiplier-condition/) ## Step-by-step derivation S001 · impliesBefore source roles = [lagrange-single-linear-constraint] After derived = [[center, constraint_gradient, constraint_rhs, normal_displacement_coefficient, solution_point, multiplier, objective_gradient, feasible, stationary, specified_problem_scope]] Conditions: none List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Lagrange single ORIGINAL normal squared norm ORIGINAL product0 := mul(1, 1) After Lagrange single ORIGINAL normal squared norm ORIGINAL product0 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 := sum(0, 1) After Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 = 1 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Lagrange single ORIGINAL normal squared norm ORIGINAL product1 := mul(16, 16) After Lagrange single ORIGINAL normal squared norm ORIGINAL product1 = 256 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 := sum(1, 256) After Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 = 257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Lagrange single ORIGINAL normal dot center ORIGINAL product0 := mul(1, 0) After Lagrange single ORIGINAL normal dot center ORIGINAL product0 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 := sum(0, 0) After Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 = 0 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Lagrange single ORIGINAL normal dot center ORIGINAL product1 := mul(16, 1) After Lagrange single ORIGINAL normal dot center ORIGINAL product1 = 16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 := sum(0, 16) After Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 = 16 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Lagrange single ORIGINAL rhs minus center value := sub(18, 16) After Lagrange single ORIGINAL rhs minus center value = 2 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Lagrange single ORIGINAL normal displacement := div(2, 257) After Lagrange single ORIGINAL normal displacement = 2/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Lagrange single ORIGINAL normal shift0 := mul(2/257, 1) After Lagrange single ORIGINAL normal shift0 = 2/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Lagrange single ORIGINAL solution coordinate0 := sum(0, 2/257) After Lagrange single ORIGINAL solution coordinate0 = 2/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Lagrange single ORIGINAL normal shift1 := mul(2/257, 16) After Lagrange single ORIGINAL normal shift1 = 32/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Lagrange single ORIGINAL solution coordinate1 := sum(1, 32/257) After Lagrange single ORIGINAL solution coordinate1 = 289/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Lagrange single ORIGINAL multiplier2tau := mul(2, 2/257) After Lagrange single ORIGINAL multiplier2tau = 4/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Lagrange single ORIGINAL displacement coordinate0 := sub(2/257, 0) After Lagrange single ORIGINAL displacement coordinate0 = 2/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Lagrange single ORIGINAL objective gradient0 := mul(2, 2/257) After Lagrange single ORIGINAL objective gradient0 = 4/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Lagrange single ORIGINAL displacement coordinate1 := sub(289/257, 1) After Lagrange single ORIGINAL displacement coordinate1 = 32/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Lagrange single ORIGINAL objective gradient1 := mul(2, 32/257) After Lagrange single ORIGINAL objective gradient1 = 64/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Lagrange single ORIGINAL full feasibility ORIGINAL product0 := mul(1, 2/257) After Lagrange single ORIGINAL full feasibility ORIGINAL product0 = 2/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 := sum(0, 2/257) After Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 = 2/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Lagrange single ORIGINAL full feasibility ORIGINAL product1 := mul(16, 289/257) After Lagrange single ORIGINAL full feasibility ORIGINAL product1 = 4624/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 := sum(2/257, 4624/257) After Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 = 18 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Lagrange single ORIGINAL lambda times normal0 := mul(4/257, 1) After Lagrange single ORIGINAL lambda times normal0 = 4/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Lagrange single ORIGINAL lambda times normal1 := mul(4/257, 16) After Lagrange single ORIGINAL lambda times normal1 = 64/257 Conditions: none Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore source roles = [[0, 1], [1, 16], 18, 257, 16, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257]] After derived = [Yes, Yes, ANY feasible y=x+v has a dot v=0;f(y)=f(x)+norm(v)^2] Conditions: none A nonzero normal gives norm²>0 and tau solves a dot(c+tau a)=rhs.The objective is squared distance WITHOUT1/2,so gradient and lambda contain2.For any feasible y=x+v,the cross term2tau(a dot v)=0 and the objective difference is||v||²>=0.This specified problem has a unique minimum,not general sufficiency of multiplier equations. Step identifiers and dependenciesRule: Lagrange_specified_squared_distance_complete_orthogonality_certificate · Uses: G001, S026 ## Reverse checks - S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. - S026 · premise_and_result_audit — Recompute Lagrange single ORIGINAL lambda times normal1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Lagrange single ORIGINAL lambda times normal0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL product1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Lagrange single ORIGINAL full feasibility ORIGINAL product0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Lagrange single ORIGINAL objective gradient1 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Lagrange single ORIGINAL displacement coordinate1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Lagrange single ORIGINAL objective gradient0 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Lagrange single ORIGINAL displacement coordinate0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Lagrange single ORIGINAL multiplier2tau with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Lagrange single ORIGINAL solution coordinate1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal shift1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Lagrange single ORIGINAL solution coordinate0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal shift0 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal displacement with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Lagrange single ORIGINAL rhs minus center value with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL product1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal dot center ORIGINAL product0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL running sum1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL product1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL running sum0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Lagrange single ORIGINAL normal squared norm ORIGINAL product0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct. ## Result ([0, 1], [1, 16], 18, 2/257, [2/257, 289/257], 4/257, [4/257, 64/257], Yes, Yes, squared_distance_with_one_regular_linear_constraint_solved_exactly) 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: 2db6253c11bd7f6187aa46b77c4bbb47dd861af6a6e5f78e38c43149cc2a5fc1 - candidate record SHA-256: 66677326a95c6cbef21b69784eecf0e438b873ef35ca95bb093e3789ed85c4ab - semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54 - question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-pulse-20261119-r04-0066.json)