# Question 1351: Coefficient Eligibility of a Candidate Antiderivative

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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.

R0001-01351 · q-20261007-advance-20261008-r06-antiderivative-0051



## Original question and givens

On all R, f has ascending coefficients [0], starting with its constant term. The candidate F has ascending rational coefficient pairs [[7, 1]], with positive denominators. Is F′=f as a coefficient identity?


- G001 — On all R, f has ascending coefficients [0], starting with its constant term. The candidate F has ascending rational coefficient pairs [[7, 1]], with positive denominators. Is F′=f as a coefficient identity?





## 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-antiderivative](https://ocme.evemisslab.com/en/mko/mko-antiderivative/)





## Step-by-step derivation

S001 · impliesBefore

domain=R; coefficient lists are in ascending powers

After

constant term first; rational pairs=[numerator,denominator]

Conditions: none

Fix coefficient order;do not read ascending coefficients as descending or swap numerator and denominator.

Step identifiers and dependenciesRule: polynomial_representation · Uses: G001





S002 · impliesBefore

candidate F coefficients=[7]

After

target f coefficients=[0]

Conditions: none

A candidate may contain any constant,which disappears under differentiation.

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





S003 · impliesBefore

differentiate all positive-degree terms

After

F′ coefficients=[0]

Conditions: none

For a constant polynomial,normalize the derivative list to one zero coefficient.

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





S004 · impliesBefore

F′ coefficients=[0]; f coefficients=[0]

After

F′=f identically: Yes

Conditions: none

Positive cases use full coefficient identity;negative cases also have an explicit counterexample point.

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









## Reverse checks


- 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: e86559741f76e43bde2db6b00344014ee50fd1a01190cb442fcf6ed3f47fb960
- candidate record SHA-256: db731b16f279788991952387f772258c6b653ff62f64d297958519e95475dafb
- semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
- question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r06-antiderivative-0051.json)