Explore related knowledge and questions →
← Question derivation index · Original question batch
R0022-11856 · q-20261013-advance-20261025-r02-convex-function-0081
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=2/1, lambda=6/8 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
G001 — Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=2/1, lambda=6/8 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.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, 5]
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
arbitrary x,y in [-5,5];0<=lambda<=1;z=lambda*x+(1-lambda)*y
After
z--5=lambda*(x--5)+(1-lambda)*(y--5)>=0;5-z=lambda*(5-x)+(1-lambda)*(5-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, S001
Before
univariate quadratic coefficients=[1, -7, 2];order=[a,b,c]
After
f(t)=(1)*t^2+(-7)*t+(2)
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, S002
Before
f(t)=(1)*t^2+(-7)*t+(2);z=lambda*x+(1-lambda)*y
After
rhs-lhs=(1)*(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, S003
Before
lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2
After
lambda*(1-lambda)*(x-y)^2;full gap=(1)*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, S004
Before
Jensen complementary weight := sub(1, 3/4)
After
Jensen complementary weight = 1/4
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, S005
Before
Jensen weighted x := mul(3/4, -2)
After
Jensen weighted x = -3/2
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
Jensen weighted y := mul(1/4, 2)
After
Jensen weighted y = 1/2
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
Jensen weighted input := sum(-3/2, 1/2)
After
Jensen weighted input = -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, S008
Before
Jensen lhs point squared := mul(-1, -1)
After
Jensen lhs 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, S009
Before
Jensen lhs quadratic term := mul(1, 1)
After
Jensen lhs quadratic term = 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, S010
Before
Jensen lhs linear term := mul(-7, -1)
After
Jensen lhs linear term = 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, S011
Before
Jensen lhs polynomial value := sum(1, 7, 2)
After
Jensen lhs polynomial value = 10
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
Jensen left endpoint point squared := mul(-2, -2)
After
Jensen left endpoint 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, S013
Before
Jensen left endpoint quadratic term := mul(1, 4)
After
Jensen left endpoint quadratic term = 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, S014
Before
Jensen left endpoint linear term := mul(-7, -2)
After
Jensen left endpoint linear term = 14
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
Jensen left endpoint polynomial value := sum(4, 14, 2)
After
Jensen left endpoint polynomial value = 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, S016
Before
Jensen right endpoint point squared := mul(2, 2)
After
Jensen right endpoint 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, S017
Before
Jensen right endpoint quadratic term := mul(1, 4)
After
Jensen right endpoint quadratic term = 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, S018
Before
Jensen right endpoint linear term := mul(-7, 2)
After
Jensen right endpoint linear term = -14
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, S019
Before
Jensen right endpoint polynomial value := sum(4, -14, 2)
After
Jensen right endpoint polynomial value = -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, S020
Before
Jensen weighted left value := mul(3/4, 20)
After
Jensen weighted left 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, S021
Before
Jensen weighted right value := mul(1/4, -8)
After
Jensen weighted right value = -2
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, S022
Before
Jensen rhs := sum(15, -2)
After
Jensen rhs = 13
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, S023
Before
Jensen rhs minus lhs := sub(13, 10)
After
Jensen rhs minus lhs = 3
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, S024
Before
qualified x=-2;y=2;lambda=3/4;lhs=10;rhs=13;gap=3
After
Jensen instance passes=Yes;scope=single_instance_only
Conditions: none
A negative gap at qualified endpoints and weight is a counterexample.Nonnegative proves only this instance;lambda0 or x=y can pass even for a nonconvex quadratic.
Rule: one_Jensen_instance_not_universal_positive_proof · Uses: G001, S025
S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S025 · premise_and_result_audit — Recompute Jensen rhs minus lhs with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute Jensen rhs with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute Jensen weighted right value with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute Jensen weighted left value with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Recompute Jensen right endpoint polynomial value with exact rational arithmetic, not rounded decimals.S020 · premise_and_result_audit — Recompute Jensen right endpoint linear term with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute Jensen right endpoint quadratic term with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute Jensen right endpoint point squared with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute Jensen left endpoint polynomial value with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute Jensen left endpoint linear term with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute Jensen left endpoint quadratic term with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute Jensen left endpoint point squared with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute Jensen lhs polynomial value with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute Jensen lhs linear term with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute Jensen lhs quadratic term with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Jensen lhs point squared with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Jensen weighted input with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Jensen weighted y with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Jensen weighted x with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Jensen complementary weight 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 — 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.([1, -7, 2], [-5, 5], -2, 2, 3/4, -1, 10, 13, 3, Yes, single_instance_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: 3a6b2ceaffeffb8f577a66d1b9e19680ba419bbe990e5fb671f332353b33ae70candidate record SHA-256: 8318389da72876838af954a1012826471aa5d11ee8f5b3d45b56d0e0d14a7b33semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED