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R0022-11814 · q-20261013-advance-20261025-r02-convex-function-0039
Use f(t)=a*t+b with coefficients [a,b]=[1, 1] and interval=[-15, 16]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. 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 f(t)=a*t+b with coefficients [a,b]=[1, 1] and interval=[-15, 16]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. 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=[-15, 16]
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 [-15,16];0<=lambda<=1;z=lambda*x+(1-lambda)*y
After
z--15=lambda*(x--15)+(1-lambda)*(y--15)>=0;16-z=lambda*(16-x)+(1-lambda)*(16-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
f(t)=(1)*t+(1);z=lambda*x+(1-lambda)*y
After
f(z)=lambda*((1)*x+(1))+(1-lambda)*((1)*y+(1))
Conditions: none
Distributivity and lambda+(1−lambda)=1 give the affine Jensen identity,hence equality for every endpoint pair and weight on a qualified convex domain.An unqualified domain record does not invalidate the algebraic identity.
Rule: affine_distributive_all_weights_identity · Uses: G001, S002
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, 1], [-15, 16], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
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: d80cd539b52c9d6671e5c43207ecadbb9ad45f5f482b536e5a5ff6b1c626aa84candidate record SHA-256: 0f0af1f9133962bb73ab447f523077efa239f7a308319e69a012069ea0a4cbcfsemantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8facquestion-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01bLean: NOT_RUN · human review: NOT_PERFORMED