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R0001-01374 · q-20261007-advance-20261008-r06-antiderivative-0074
On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[10, 1], [11, 1], [12, 1], [1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?
G001 — On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[10, 1], [11, 1], [12, 1], [1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?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.
Before
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.
Rule: polynomial_representation · Uses: G001
Before
candidate F coefficients=[10, 11, 12, 1/3, 2]
After
target f coefficients=[11, 24, 1, 4]
Conditions: none
A candidate may contain any constant,which disappears under differentiation.
Rule: candidate_vs_target · Uses: G001, S001
Before
candidate derivative coefficient0 := mul(1, 11)
After
candidate derivative coefficient0 = 11
Conditions: none
Multiply each coefficient by its degree and lower the exponent by1.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
candidate derivative coefficient1 := mul(2, 12)
After
candidate derivative coefficient1 = 24
Conditions: none
Multiply each coefficient by its degree and lower the exponent by1.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
candidate derivative coefficient2 := mul(3, 1/3)
After
candidate derivative coefficient2 = 1
Conditions: none
Multiply each coefficient by its degree and lower the exponent by1.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
candidate derivative coefficient3 := mul(4, 2)
After
candidate derivative coefficient3 = 8
Conditions: none
Multiply each coefficient by its degree and lower the exponent by1.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
differentiate all positive-degree terms
After
F′ coefficients=[11, 24, 1, 8]
Conditions: none
For a constant polynomial,normalize the derivative list to one zero coefficient.
Rule: constant_derivative_zero · Uses: G001, S006
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
F′ witness add coefficient3 := sum(0, 8)
After
F′ witness add coefficient3 = 8
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
F′ witness multiply at degree2 := mul(8, 1)
After
F′ witness multiply at degree2 = 8
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
F′ witness add coefficient2 := sum(8, 1)
After
F′ witness add coefficient2 = 9
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
F′ witness multiply at degree1 := mul(9, 1)
After
F′ witness multiply at degree1 = 9
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
F′ witness add coefficient1 := sum(9, 24)
After
F′ witness add coefficient1 = 33
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
F′ witness multiply at degree0 := mul(33, 1)
After
F′ witness multiply at degree0 = 33
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
F′ witness add coefficient0 := sum(33, 11)
After
F′ witness add coefficient0 = 44
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
f witness add coefficient3 := sum(0, 4)
After
f witness add coefficient3 = 4
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
f witness multiply at degree2 := mul(4, 1)
After
f witness multiply at degree2 = 4
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
f witness add coefficient2 := sum(4, 1)
After
f witness add coefficient2 = 5
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
f witness multiply at degree1 := mul(5, 1)
After
f witness multiply at degree1 = 5
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
f witness add coefficient1 := sum(5, 24)
After
f witness add coefficient1 = 29
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S020
Before
f witness multiply at degree0 := mul(29, 1)
After
f witness multiply at degree0 = 29
Conditions: none
In Horner order, multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
f witness add coefficient0 := sum(29, 11)
After
f witness add coefficient0 = 40
Conditions: none
Then add the exact rational coefficient for this degree.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
at x=1, F′(x)=44 and f(x)=40
After
one explicit mismatch refutes identity
Conditions: none
One concrete mismatch refutes a global identity;finite agreements would not prove one.
Rule: nonidentity_witness · Uses: G001, S023
Before
F′ coefficients=[11, 24, 1, 8]; f coefficients=[11, 24, 1, 4]
After
F′=f identically: No
Conditions: none
Positive cases use full coefficient identity;negative cases also have an explicit counterexample point.
Rule: primitive_candidate_verdict · Uses: G001, S024
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.No
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: a6b0f96367ceb8a386cff904dce89fbd6e8bc3a2397f38bfbc44f40be8a3a76bcandidate record SHA-256: b71b5e221f336918b9106e267614d8cd2d0222e089000c5b1b8ee89df826ccd2semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496ccquestion-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7adLean: NOT_RUN · human review: NOT_PERFORMED