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R0022-11955 · q-20261013-advance-20261025-r03-convex-optimization-problem-0080
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-6, 7] belongs to the function data. left=2/1, right=1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.
G001 — Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-6, 7] belongs to the function data. left=2/1, right=1/1. Give (coefficients,interval,left,left feasible?,left value,right,right feasible?,right value,relation,pairwise_only_no_global_optimum_claim). The relation is not_comparable if either point is infeasible; otherwise left_lower, right_lower or equal. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.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.
Before
declared interval record=[-6, 7]
After
ordered_nonempty=Yes;nondegenerate=Yes
Conditions: none
This nonempty-interval protocol uses lower≤upper;the nondegenerate certificate additionally needs lower<upper.A reversed record is unqualified,not a declaration that the abstract empty set is nonconvex.A singleton interval is convex.
Rule: interval_certificate_qualification_not_empty_set_nonconvexity · Uses: G001
Before
univariate quadratic coefficients=[0, -8, 1];order=[a,b,c]
After
f(t)=(0)*t^2+(-8)*t+(1)
Conditions: none
These are quadratic,linear,constant coefficients in descending order,not the constant-first boundary-polynomial order.Retain zero leading coefficients.
Rule: quadratic_descending_coefficient_order · Uses: G001, S001
Before
left candidate lower slack := sub(2, -6)
After
left candidate lower slack = 8
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
left candidate upper slack := sub(7, 2)
After
left candidate upper slack = 5
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
left candidate:interval=[-6, 7];point=2;slacks=[8, 5]
After
feasible=Yes
Conditions: none
Both inclusive bounds must hold.A reversed record supplies no qualified feasible point.The polynomial may have a value outside the domain,but a value does not establish eligibility.
Rule: inclusive_point_feasibility_separate_evaluation · Uses: G001, S004
Before
right candidate lower slack := sub(1, -6)
After
right candidate lower slack = 7
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
right candidate upper slack := sub(7, 1)
After
right candidate upper slack = 6
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
right candidate:interval=[-6, 7];point=1;slacks=[7, 6]
After
feasible=Yes
Conditions: none
Both inclusive bounds must hold.A reversed record supplies no qualified feasible point.The polynomial may have a value outside the domain,but a value does not establish eligibility.
Rule: inclusive_point_feasibility_separate_evaluation · Uses: G001, S007
Before
left objective point squared := mul(2, 2)
After
left objective point squared = 4
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
left objective quadratic term := mul(0, 4)
After
left objective quadratic term = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
left objective linear term := mul(-8, 2)
After
left objective linear term = -16
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left objective polynomial value := sum(0, -16, 1)
After
left objective polynomial value = -15
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
right objective point squared := mul(1, 1)
After
right objective point squared = 1
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
right objective quadratic term := mul(0, 1)
After
right objective quadratic term = 0
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
right objective linear term := mul(-8, 1)
After
right objective linear term = -8
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
right objective polynomial value := sum(0, -8, 1)
After
right objective polynomial value = -7
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
two candidate value difference := sub(-15, -7)
After
two candidate value difference = -8
Conditions: none
Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left feasible=Yes;right feasible=Yes;left value=-15;right value=-7;difference=-8
After
eligible pair relation=left_lower;scope=pairwise_only_no_global_optimum_claim
Conditions: none
Compare values only if both points are feasible;otherwise they are not comparable.Even a lower point only beats this other candidate.No quantification over all feasible points has established a global optimum.
Rule: two_eligible_points_not_global_optimum · Uses: G001, S017
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Recompute two candidate value difference with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute right objective polynomial value with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute right objective linear term with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute right objective quadratic term with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute right objective point squared with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute left objective polynomial value with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute left objective linear term with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute left objective quadratic term with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute left objective point squared with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute right candidate upper slack with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute right candidate lower slack with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Recompute left candidate upper slack with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute left candidate lower slack with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([0, -8, 1], [-6, 7], 2, Yes, -15, 1, Yes, -7, left_lower, pairwise_only_no_global_optimum_claim)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
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