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R0022-11918 · q-20261013-advance-20261025-r03-convex-optimization-problem-0043
Declared nonempty interval record=[-19, 20], candidate=0/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. 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 — Declared nonempty interval record=[-19, 20], candidate=0/1. Give (interval,normalized candidate,within both inclusive bounds?,feasibility_only). A reversed record has no qualified feasible point in this protocol. 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=[-19, 20]
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
candidate lower slack := sub(0, -19)
After
candidate lower slack = 19
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, S001
Before
candidate upper slack := sub(20, 0)
After
candidate upper slack = 20
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:interval=[-19, 20];point=0;slacks=[19, 20]
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, S003
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 — Recompute candidate upper slack with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute candidate lower slack with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([-19, 20], 0, Yes, feasibility_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: 0e3f9cbcdc5d4c61d83119500e8e17858ca96f055351ea2486813446f51cbc2fcandidate record SHA-256: 9eeb68427d6f76ba488ed38d21ea1f8a026d6a19aaa31c183a3b1e51ffbbb318semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED