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R0058-29993 · q-20261007-advance-20261009-r05-antiderivative-0018
On all R, f has ascending coefficients [6, 19, -20, 21], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, f has ascending coefficients [6, 19, -20, 21], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.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
source roles = [qf-density-primitive-coefficient, density_primitive_coefficient]
After
derived = [[specified_zero_constant_primitive_coefficient_fraction_retaining1]]
Conditions: none
Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention.
Rule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001
Before
source roles = [integrand f, [6, 19, -20, 21], No]
After
derived = [[6/1, 19/1, -20/1, 21/1]]
Conditions: none
Raw integer f has length1..4 and rational candidate F length1..5.Every raw number is SAFE;rational denominators are STRICTLY positive,not repaired from negatives.Trim only trailing zeros,keeping a singleton zero polynomial.Derived algebra has no1e6 or raw-degree cap.
Rule: Primitive_RAW_arity_SAFE_inputs_POSITIVE_rational_den_trim_KEEPzero · Uses: G001, S001
Before
Primitive ORIGINAL coefficient degree1 := div(6, 1)
After
Primitive ORIGINAL coefficient degree1 = 6
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Primitive ORIGINAL coefficient degree2 := div(19, 2)
After
Primitive ORIGINAL coefficient degree2 = 19/2
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Primitive ORIGINAL coefficient degree3 := div(-20, 3)
After
Primitive ORIGINAL coefficient degree3 = -20/3
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Primitive ORIGINAL coefficient degree4 := div(21, 4)
After
Primitive ORIGINAL coefficient degree4 = 21/4
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Primitive reverse ORIGINAL derivative coefficient0 := mul(6, 1)
After
Primitive reverse ORIGINAL derivative coefficient0 = 6
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Primitive reverse ORIGINAL derivative coefficient1 := mul(19/2, 2)
After
Primitive reverse ORIGINAL derivative coefficient1 = 19
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Primitive reverse ORIGINAL derivative coefficient2 := mul(-20/3, 3)
After
Primitive reverse ORIGINAL derivative coefficient2 = -20
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Primitive reverse ORIGINAL derivative coefficient3 := mul(21/4, 4)
After
Primitive reverse ORIGINAL derivative coefficient3 = 21
Conditions: none
Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
source roles = [[6/1, 19/1, -20/1, 21/1]]
After
derived = [[0/1, 6/1, 19/2, -20/3, 21/4], [6/1, 19/1, -20/1, 21/1], Yes]
Conditions: none
Since d(x^(j+1)/(j+1))/dx=x^j,integrate EACH monomial into the zero-constant representative F0,and differentiate ALL coefficients back to the original f as an identity.Keep[0/1] for zero;do not rely on sampled agreement.
Rule: Primitive_ZEROconstant_ALL_monomial_coefficients_AND_reverse_identity · Uses: G001, S010
Before
source roles = [[6/1, 19/1, -20/1, 21/1], 3]
After
derived = [-20/3]
Conditions: none
The x^k coefficient is the ORIGINAL x^(k-1) coefficient divided by k;an absent coefficient is0.Do not select the constant or shift the index.Integer fractions retain/1.
Rule: Primitive_specified_power_k_ASCENDING_absent_coefficient_ZERO · Uses: G001, S011
S012 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S011 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S010 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient3 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient2 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Primitive reverse ORIGINAL derivative coefficient0 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree4 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree3 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree2 with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Primitive ORIGINAL coefficient degree1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.-20/3
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: 6f705b1e495dadd776295d7bb12e34c4cd00a74321bf1146ff51de86e442f7c4candidate record SHA-256: dd9e4cf4aa1ec8fa982e20c0c4e9e722d29ad4225290a48bdcb1b86f252bcba8semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05dfquestion-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7Lean: NOT_RUN · human review: NOT_PERFORMED