# Question 10176: Candidate Antiderivative Premise for the Endpoint Formula

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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.

R0019-10176 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0076



## Original question and givens

On all R, continuous f has ascending coefficients [0], constant term first. Candidate F has ascending rational coefficients [5/1]. Does F′=f satisfy the antiderivative premise for the endpoint formula?


- G001 — On all R, continuous f has ascending coefficients [0], constant term first. Candidate F has ascending rational coefficients [5/1]. Does F′=f satisfy the antiderivative premise for the endpoint formula?





## 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-fundamental-theorem-calculus](https://ocme.evemisslab.com/en/mko/mko-fundamental-theorem-calculus/)





## Step-by-step derivation

S001 · impliesBefore

raw integrand f=[0];ascending powers on all R

After

normalized coefficients=[0/1];declared continuous real polynomial

Conditions: none

The declared integrand is a continuous real polynomial on all R,and endpoints are oriented.Use FTC only under these supplied sufficient hypotheses;finite checks are not a new general formal FTC proof.

Step identifiers and dependenciesRule: declared_FTC_continuity_hypothesis · Uses: G001





S002 · impliesBefore

raw candidate F=[[5, 1]]

After

normalized candidate coefficients=[5/1]

Conditions: none

Every original denominator must be positive.Normalize to exact rationals while retaining the supplied candidate,not replacing it by a constructed primitive.

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





S003 · impliesBefore

candidate F:constant coefficient=5/1

After

derivative coefficients=[0/1]

Conditions: none

A constant differentiates to the zero polynomial,represented as[0/1],not a fabricated negative power.

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





S004 · impliesBefore

power=0;candidate derivative coefficient=0/1;target f coefficient=0/1

After

coefficient equality=Yes

Conditions: none

Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.

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





S005 · impliesBefore

candidate derivative=[0/1];integrand=[0/1]

After

this candidate satisfies F'=f=Yes

Conditions: none

Failure rejects this F only,not other primitives.The candidate constant is free;coefficient identity is the requested endpoint-formula qualification.

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









## Reverse checks


- 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

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: 0bd483b3f9d2f5837fd37bccac028f86f251834f15db8f8245cc18a289600583
- candidate record SHA-256: c3259daecbc91067b6eff9486c8b841dd9e211e1cf33ad0936816a7f0bf15a0e
- semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
- question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0076.json)