# Question 1364: Coefficient Eligibility of a Candidate Antiderivative [Return to catalogue page 7](https://ocme.evemisslab.com/en/questions/page/7/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r06-antiderivative-foundations/#q-20261007-advance-20261008-r06-antiderivative-0064) 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. R0001-01364 · q-20261007-advance-20261008-r06-antiderivative-0064 ## Original question and givens On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[0, 1], [1, 1], [7, 1], [0, 1], [3, 2]], with positive denominators. Is F′=f as a coefficient identity? - G001 — On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[0, 1], [1, 1], [7, 1], [0, 1], [3, 2]], 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=[0, 1, 7, 0, 3/2] After target f coefficients=[1, 14, 0, 2] Conditions: none A candidate may contain any constant,which disappears under differentiation. Step identifiers and dependenciesRule: candidate_vs_target · Uses: G001, S001 S003 · calculationBefore candidate derivative coefficient0 := mul(1, 1) After candidate derivative coefficient0 = 1 Conditions: none Multiply each coefficient by its degree and lower the exponent by1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore candidate derivative coefficient1 := mul(2, 7) After candidate derivative coefficient1 = 14 Conditions: none Multiply each coefficient by its degree and lower the exponent by1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore candidate derivative coefficient2 := mul(3, 0) After candidate derivative coefficient2 = 0 Conditions: none Multiply each coefficient by its degree and lower the exponent by1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore candidate derivative coefficient3 := mul(4, 3/2) After candidate derivative coefficient3 = 6 Conditions: none Multiply each coefficient by its degree and lower the exponent by1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore differentiate all positive-degree terms After F′ coefficients=[1, 14, 0, 6] Conditions: none For a constant polynomial,normalize the derivative list to one zero coefficient. Step identifiers and dependenciesRule: constant_derivative_zero · Uses: G001, S006 S008 · calculationBefore F′ witness multiply at degree3 := mul(0, 1) After F′ witness multiply at degree3 = 0 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore F′ witness add coefficient3 := sum(0, 6) After F′ witness add coefficient3 = 6 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore F′ witness multiply at degree2 := mul(6, 1) After F′ witness multiply at degree2 = 6 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore F′ witness add coefficient2 := sum(6, 0) After F′ witness add coefficient2 = 6 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore F′ witness multiply at degree1 := mul(6, 1) After F′ witness multiply at degree1 = 6 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore F′ witness add coefficient1 := sum(6, 14) After F′ witness add coefficient1 = 20 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore F′ witness multiply at degree0 := mul(20, 1) After F′ witness multiply at degree0 = 20 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore F′ witness add coefficient0 := sum(20, 1) After F′ witness add coefficient0 = 21 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore f witness multiply at degree3 := mul(0, 1) After f witness multiply at degree3 = 0 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore f witness add coefficient3 := sum(0, 2) After f witness add coefficient3 = 2 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore f witness multiply at degree2 := mul(2, 1) After f witness multiply at degree2 = 2 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore f witness add coefficient2 := sum(2, 0) After f witness add coefficient2 = 2 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore f witness multiply at degree1 := mul(2, 1) After f witness multiply at degree1 = 2 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore f witness add coefficient1 := sum(2, 14) After f witness add coefficient1 = 16 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore f witness multiply at degree0 := mul(16, 1) After f witness multiply at degree0 = 16 Conditions: none In Horner order, multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore f witness add coefficient0 := sum(16, 1) After f witness add coefficient0 = 17 Conditions: none Then add the exact rational coefficient for this degree. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore at x=1, F′(x)=21 and f(x)=17 After one explicit mismatch refutes identity Conditions: none One concrete mismatch refutes a global identity;finite agreements would not prove one. Step identifiers and dependenciesRule: nonidentity_witness · Uses: G001, S023 S025 · impliesBefore F′ coefficients=[1, 14, 0, 6]; f coefficients=[1, 14, 0, 2] After F′=f identically: No 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, S024 ## Reverse checks - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute f witness add coefficient0 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute f witness multiply at degree0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute f witness add coefficient1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute f witness multiply at degree1 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute f witness add coefficient2 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute f witness multiply at degree2 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute f witness add coefficient3 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute f witness multiply at degree3 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute F′ witness add coefficient0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute F′ witness multiply at degree0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute F′ witness add coefficient1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute F′ witness multiply at degree1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute F′ witness add coefficient2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute F′ witness multiply at degree2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute F′ witness add coefficient3 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute F′ witness multiply at degree3 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute candidate derivative coefficient3 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute candidate derivative coefficient2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute candidate derivative coefficient1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute candidate derivative coefficient0 with exact rational arithmetic, not rounded decimals. - 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 No 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: 3593e0774e7faba081f12d9e51a30ba55eaf7cbef0996d37679165ed98178af4 - candidate record SHA-256: 1e31e686c9b20f5a1d2366a6410eb886891b3a90628fdc4e03fd0bd5b05eb7dd - 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-0064.json)