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R0022-11896 · q-20261013-advance-20261025-r02-convex-function-0071
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-22, 23] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. 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, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-22, 23] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. 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=[-22, 23]
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 [-22,23];0<=lambda<=1;z=lambda*x+(1-lambda)*y
After
z--22=lambda*(x--22)+(1-lambda)*(y--22)>=0;23-z=lambda*(23-x)+(1-lambda)*(23-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, 8, 3];order=[a,b,c]
After
f(t)=(-1)*t^2+(8)*t+(3)
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+(8)*t+(3);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
negative quadratic endpoint difference := sub(-22, 23)
After
negative quadratic endpoint difference = -45
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
negative quadratic endpoint square := mul(-45, -45)
After
negative quadratic endpoint square = 2025
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
negative quadratic weighted square := mul(-1, 2025)
After
negative quadratic weighted square = -2025
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
negative quadratic midpoint Jensen gap := div(-2025, 4)
After
negative quadratic midpoint Jensen gap = -2025/4
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
x=-22;y=23;lambda=1/2;gap=-2025/4
After
negative Jensen gap refutes convexity on this nondegenerate interval
Conditions: none
A nondegenerate interval supplies distinct endpoints.The midpoint weight gives a strictly negative gap:an exact counterexample,not just a coefficient mnemonic.
Rule: negative_leading_coefficient_actual_counterexample · Uses: G001, S009
Before
interval=[-22, 23];nondegenerate=Yes;leading=-1
After
qualified universal quadratic certificate=No
Conditions: none
If a≥0,the nonnegative weight product and square make every Jensen gap nonnegative.This certificate separately requires a nondegenerate interval.No for a singleton or reversed record does not classify a singleton function as nonconvex.
Rule: quadratic_global_sign_and_nondegenerate_guard · Uses: G001, S010
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 negative quadratic midpoint Jensen gap with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute negative quadratic weighted square with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute negative quadratic endpoint square with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute negative quadratic endpoint difference 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, 8, 3], [-22, 23], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)
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: 2e0234432693a47a363717042450827b341a642c27bd76f511e999b73ee843f2candidate record SHA-256: b55ba19cc6a256848d1974fc30fe08e8da6ab573435f01d906832b2b6acff9e9semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED