# Question 11813: An Affine Jensen Identity on a Qualified Interval

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0022-11813 · q-20261013-advance-20261025-r02-convex-function-0038



## Original question and givens

Use f(t)=a*t+b with coefficients [a,b]=[0, 0] and interval=[-14, 15]. 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]=[0, 0] and interval=[-14, 15]. 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.





## Definition and scope

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.


- [mko-convex-function](https://ocme.evemisslab.com/en/mko/mko-convex-function/)





## Step-by-step derivation

S001 · impliesBefore

declared interval record=[-14, 15]

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.

Step identifiers and dependenciesRule: interval_certificate_qualification_not_empty_set_nonconvexity · Uses: G001





S002 · impliesBefore

arbitrary x,y in [-14,15];0<=lambda<=1;z=lambda*x+(1-lambda)*y

After

z--14=lambda*(x--14)+(1-lambda)*(y--14)>=0;15-z=lambda*(15-x)+(1-lambda)*(15-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.

Step identifiers and dependenciesRule: interval_all_segments_slack_identity · Uses: G001, S001





S003 · impliesBefore

f(t)=(0)*t+(0);z=lambda*x+(1-lambda)*y

After

f(z)=lambda*((0)*x+(0))+(1-lambda)*((0)*y+(0))

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.

Step identifiers and dependenciesRule: affine_distributive_all_weights_identity · Uses: G001, S002









## Reverse checks


- 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.





## Result

([0, 0], [-14, 15], 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.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: a0c56206ff320f21c733e568c50ae4f2eec8db338eade9b0922e48df240fb95f
- candidate record SHA-256: 1d075e8ef29f26f460521a246d68833729bc3525bcc59f4c42f1a837e5aa99c1
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r02-convex-function-0038.json)