# Question 11899: A Nondegenerate-Interval Quadratic Jensen-Gap Certificate

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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.

R0022-11899 · q-20261013-advance-20261025-r02-convex-function-0074



## Original question and givens

Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-25, 26] 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]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-25, 26] 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.





## 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=[-25, 26]

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 [-25,26];0<=lambda<=1;z=lambda*x+(1-lambda)*y

After

z--25=lambda*(x--25)+(1-lambda)*(y--25)>=0;26-z=lambda*(26-x)+(1-lambda)*(26-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

univariate quadratic coefficients=[3, 11, -1];order=[a,b,c]

After

f(t)=(3)*t^2+(11)*t+(-1)

Conditions: none

These are quadratic,linear,constant coefficients in descending order,not the constant-first boundary-polynomial order.Retain zero leading coefficients.

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





S004 · impliesBefore

f(t)=(3)*t^2+(11)*t+(-1);z=lambda*x+(1-lambda)*y

After

rhs-lhs=(3)*(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.

Step identifiers and dependenciesRule: jensen_linear_and_constant_terms_cancel · Uses: G001, S003





S005 · impliesBefore

lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2

After

lambda*(1-lambda)*(x-y)^2;full gap=(3)*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.

Step identifiers and dependenciesRule: jensen_gap_full_symbolic_factorization · Uses: G001, S004





S006 · impliesBefore

interval=[-25, 26];nondegenerate=Yes;leading=3

After

qualified universal quadratic certificate=Yes

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.

Step identifiers and dependenciesRule: quadratic_global_sign_and_nondegenerate_guard · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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.





## Result

([3, 11, -1], [-25, 26], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, 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: ca95eee2d7a8a3ef55e8d5149a836cedbf1c3b5334153c9f5f5c5bf4f08bb6f7
- candidate record SHA-256: 85b94e507bebdf847e0c9068a2b1564cbcc501b3c630bc8a11caed1c31f21d32
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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