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R0001-01326 · q-20261007-advance-20261008-r06-antiderivative-0026
On all R, f has ascending coefficients [0], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.
G001 — On all R, f has ascending coefficients [0], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.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
f coefficients ascending=[0]
After
seek F₀ with constant coefficient0 and F₀′=f
Conditions: none
Antiderivatives are not generally unique; this task additionally fixes the constant coefficient at0.
Rule: canonical_primitive_domain · Uses: G001, S001
Before
primitive coefficient at degree1 := div(0, 1)
After
primitive coefficient at degree1 = 0
Conditions: none
Since (x^(k+1))′=(k+1)x^k,divide the coefficient by the positive nonzero integer k+1.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
differentiate back coefficient0 := mul(0, 1)
After
differentiate back coefficient0 = 0
Conditions: none
Differentiation multiplies back by k+1 and must recover the original coefficient.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
[0]
After
all derivative coefficients agree with f
Conditions: none
Coefficient agreement proves the polynomial identity on all reals,not merely equality at a few points.
Rule: global_polynomial_identity · Uses: G001, S004
Before
[0]
After
reduced coefficients;integer coefficients retain /1
Conditions: none
Return ascending coefficients,trim redundant leading-in-degree zeros,and retain[0/1] for the zero polynomial.
Rule: coefficient_output_protocol · Uses: G001, S005
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 — Recompute differentiate back coefficient0 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute primitive coefficient at degree1 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.[0/1]
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: 09a0ae2256336c59ea22e0051a06203ae8c53a1cc74392d1a01db50fa54c7fcdcandidate record SHA-256: 22d2524b4ed6ae18f7cbbc9c867623768ae7c131d68e5b4bf7f8963ee463857bsemantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496ccquestion-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7adLean: NOT_RUN · human review: NOT_PERFORMED