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R0022-11980 · q-20261013-advance-20261025-r03-convex-optimization-problem-0005
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. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. 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. Give (coefficients,interval,global quadratic certificate a≥0?,qualified nonempty ordered feasible interval?,both qualifications?,qualification_only_no_solution_claim). The objective flag uses the named global quadratic criterion; failure does not decide every possible restricted singleton case. 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
f(t)=(0)*t^2+(-8)*t+(1);z=lambda*x+(1-lambda)*y
After
rhs-lhs=(0)*(lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2)
Conditions: none
Expand both Jensen sides.The unit weight sum cancels the linear and constant terms exactly,leaving only the quadratic part.
Rule: jensen_linear_and_constant_terms_cancel · Uses: G001, S002
Before
lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2
After
lambda*(1-lambda)*(x-y)^2;full gap=(0)*lambda*(1-lambda)*(x-y)^2
Conditions: none
Expand the square and collect x²,xy,y².The coefficients are lambda(1−lambda),−2lambda(1−lambda),lambda(1−lambda),yielding the(x−y)^2 factor for all real endpoints.
Rule: jensen_gap_full_symbolic_factorization · Uses: G001, S003
Before
arbitrary x,y in [-6,7];0<=lambda<=1;z=lambda*x+(1-lambda)*y
After
z--6=lambda*(x--6)+(1-lambda)*(y--6)>=0;7-z=lambda*(7-x)+(1-lambda)*(7-y)>=0
Conditions: none
Both slacks are nonnegative sums,hence lower≤z≤upper for every endpoint pair and every permitted lambda.No sampling is used;endpoints and singleton cases are included.
Rule: interval_all_segments_slack_identity · Uses: G001, S004
Before
global quadratic leading coefficient=0;interval record=[-6, 7]
After
global objective certificate=Yes;feasible interval certificate=Yes;qualified=Yes
Conditions: none
The objective flag is the global quadratic a≥0 criterion:for a<0,endpoints0,1 and half weights give a negative gap.The domain flag separately qualifies a nonempty ordered interval.Joint qualification proves no solution,uniqueness,KKT or solver convergence;failure of the global criterion does not classify every restricted singleton problem.
Rule: optimization_two_qualifications_not_solver · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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], Yes, Yes, Yes, qualification_only_no_solution_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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 56d534f093f879bc7c201c61b1fa7ae75efd981a123d211f8c1c3f86d973d3c8candidate record SHA-256: 2a3b055b355eb79e5965a0540ddb45fc9fb5b1f8a8ca0b3aaae01149c32049bdsemantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED