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R0022-11929 · q-20261013-advance-20261025-r03-convex-optimization-problem-0054
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 6] belongs to the function data. point=0/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. 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]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 6] belongs to the function data. point=0/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. 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=[-5, 6]
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=[3, -9, 0];order=[a,b,c]
After
f(t)=(3)*t^2+(-9)*t+(0)
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
candidate lower slack := sub(0, -5)
After
candidate lower 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, S002
Before
candidate upper slack := sub(6, 0)
After
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, S003
Before
candidate:interval=[-5, 6];point=0;slacks=[5, 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, S004
Before
candidate objective point squared := mul(0, 0)
After
candidate objective point squared = 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, S005
Before
candidate objective quadratic term := mul(3, 0)
After
candidate 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, S006
Before
candidate objective linear term := mul(-9, 0)
After
candidate objective linear 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, S007
Before
candidate objective polynomial value := sum(0, 0, 0)
After
candidate objective polynomial value = 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, S008
Before
point=0;feasible=Yes;polynomial value=0
After
retain value even for infeasible point;evaluation is not optimization
Conditions: none
Retain the polynomial value even when the point is infeasible,as specified.An outside value does not become a feasible solution or optimum.
Rule: outside_domain_value_not_feasibility_or_optimality · Uses: G001, S009
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 — Recompute candidate objective polynomial value with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute candidate objective linear term with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute candidate objective quadratic term with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute candidate objective point squared 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 candidate upper slack with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute 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.([3, -9, 0], [-5, 6], 0, Yes, 0, evaluation_only)
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: bf81427e708819793d73c9358abdf4fedec1f5aea6daacd97182be011aaff19fcandidate record SHA-256: a77c1602591134c57ebfad792589210539e367955584526982cacf17b1f9c22csemantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED