# Question 32688: Signed Moment Integral of a Polynomial [Return to catalogue page 164](https://ocme.evemisslab.com/en/questions/page/164/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261008-r07-definite-integral-foundations/#q-20261007-advance-20261008-r07-definite-integral-0063) 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. R0064-32688 · q-20261007-advance-20261008-r07-definite-integral-0063 ## Original question and givens On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=-1, b=1. Give a reduced fraction with positive denominator, retaining /1 for integers. - G001 — On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Use coefficientwise exact signed moments on oriented endpoints a=-1, b=1. Give a reduced fraction with positive denominator, retaining /1 for integers. ## 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-definite-integral](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-integral-polynomial, polynomial_integral, reals, []] After derived = [[signed_moment_fraction_RETAIN1]] Conditions: none Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status. Step identifiers and dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001 S002 · impliesBefore source roles = [[0, 13, -1, 1]] After derived = [[0, 13, -1, 1]] Conditions: none RAW ascending coefficient arity1..4 and ALL SAFE values are checked BEFORE trimming trailing zeros,retaining[0].Equal endpoints/zero polynomials do not waive RAW validation. Step identifiers and dependenciesRule: Moment_RAW_ascending_1to4_coefficients_BEFORE_trim_ALL_validated · Uses: G001, S001 S003 · calculationBefore Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 := mul(1, -1) After Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 = -1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Polynomial ORIGINAL coefficient0 lower endpoint, -1, 1] After derived = [-1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S003 S005 · calculationBefore Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 := mul(1, 1) After Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [Polynomial ORIGINAL coefficient0 upper endpoint, 1, 1] After derived = [1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S005 S007 · calculationBefore Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power := sub(1, -1) After Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power = 2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference := mul(0, 2) After Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent := div(0, 1) After Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [Polynomial ORIGINAL coefficient0, 0, 0, -1, 1] After derived = [-1, 1, 2, 0/1] Conditions: none Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence. Step identifiers and dependenciesRule: Moment_GIVEN_signed_polynomial_coefficient_moment_orientation · Uses: G001, S009 S011 · calculationBefore Polynomial moment ORIGINAL sum through coefficient0 := sum(0, 0) After Polynomial moment ORIGINAL sum through coefficient0 = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 := mul(1, -1) After Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 = -1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 := mul(-1, -1) After Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [Polynomial ORIGINAL coefficient1 lower endpoint, -1, 2] After derived = [1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S013 S015 · calculationBefore Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 := mul(1, 1) After Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 := mul(1, 1) After Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [Polynomial ORIGINAL coefficient1 upper endpoint, 1, 2] After derived = [1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S016 S018 · calculationBefore Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power := sub(1, 1) After Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference := mul(13, 0) After Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent := div(0, 2) After Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore source roles = [Polynomial ORIGINAL coefficient1, 13, 1, -1, 1] After derived = [1, 1, 0, 0/1] Conditions: none Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence. Step identifiers and dependenciesRule: Moment_GIVEN_signed_polynomial_coefficient_moment_orientation · Uses: G001, S020 S022 · calculationBefore Polynomial moment ORIGINAL sum through coefficient1 := sum(0, 0) After Polynomial moment ORIGINAL sum through coefficient1 = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 := mul(1, -1) After Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 = -1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 := mul(-1, -1) After Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 := mul(1, -1) After Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 = -1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore source roles = [Polynomial ORIGINAL coefficient2 lower endpoint, -1, 3] After derived = [-1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S025 S027 · calculationBefore Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 := mul(1, 1) After Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 := mul(1, 1) After Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 := mul(1, 1) After Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · impliesBefore source roles = [Polynomial ORIGINAL coefficient2 upper endpoint, 1, 3] After derived = [1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S029 S031 · calculationBefore Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power := sub(1, -1) After Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power = 2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference := mul(-1, 2) After Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference = -2 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent := div(-2, 3) After Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent = -2/3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore source roles = [Polynomial ORIGINAL coefficient2, -1, 2, -1, 1] After derived = [-1, 1, 2, -2/3] Conditions: none Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence. Step identifiers and dependenciesRule: Moment_GIVEN_signed_polynomial_coefficient_moment_orientation · Uses: G001, S033 S035 · calculationBefore Polynomial moment ORIGINAL sum through coefficient2 := sum(0, -2/3) After Polynomial moment ORIGINAL sum through coefficient2 = -2/3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 := mul(1, -1) After Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 = -1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 := mul(-1, -1) After Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 := mul(1, -1) After Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 = -1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · calculationBefore Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 := mul(-1, -1) After Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · impliesBefore source roles = [Polynomial ORIGINAL coefficient3 lower endpoint, -1, 4] After derived = [1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S039 S041 · calculationBefore Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 := mul(1, 1) After Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 := mul(1, 1) After Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 := mul(1, 1) After Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 := mul(1, 1) After Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 = 1 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · impliesBefore source roles = [Polynomial ORIGINAL coefficient3 upper endpoint, 1, 4] After derived = [1] Conditions: none Show EVERY integer-power multiplication,retaining parity signs for negative bases.Derived values are not RAW-SAFE recapped or computed by floating pow. Step identifiers and dependenciesRule: Moment_INTEGER_power_complete_factors_NO_raw_recap · Uses: G001, S044 S046 · calculationBefore Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power := sub(1, 1) After Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference := mul(1, 0) After Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent := div(0, 4) After Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent = 0 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · impliesBefore source roles = [Polynomial ORIGINAL coefficient3, 1, 3, -1, 1] After derived = [1, 1, 0, 0/1] Conditions: none Use the DECLARED signed polynomial moment c_k(b^(k+1)−a^(k+1))/(k+1)exactly.Reversed endpoints are legal;do not take absolute values.This restricted moment model is not general FTC/integrability Evidence. Step identifiers and dependenciesRule: Moment_GIVEN_signed_polynomial_coefficient_moment_orientation · Uses: G001, S048 S050 · calculationBefore Polynomial moment ORIGINAL sum through coefficient3 := sum(-2/3, 0) After Polynomial moment ORIGINAL sum through coefficient3 = -2/3 Conditions: none Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · impliesBefore source roles = [[0, 13, -1, 1], -1, 1] After derived = [-2/3] Conditions: none Sum the COMPLETE coefficientwise moments,not guesses from sampled points.Retain signs and oriented endpoints;reduce to a positive denominator and retain/1. Step identifiers and dependenciesRule: Moment_POLYNOMIAL_complete_coefficientwise_signed_sum · Uses: G001, S050 ## Reverse checks - S051 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S050 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient3 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S048 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S044 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S039 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication4 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient3 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient2 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S033 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S029 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S025 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication3 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient2 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S020 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S016 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S013 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication2 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient1 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Polynomial moment ORIGINAL sum through coefficient0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 ORIGINAL signed moment over positive exponent with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 ORIGINAL coefficient times power difference with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 ORIGINAL upper power minus lower power with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 upper endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Recompute Polynomial ORIGINAL coefficient0 lower endpoint ORIGINAL power multiplication1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result -2/3 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: 6cf64cadfd90d7591e82ea125c9cd7c00bc72358b8d0616419963a74c417f1bf - candidate record SHA-256: 3e0725feea72bbc65bdd3e8a86f96473978f866d227c10b47c9b31d2bb51dc77 - semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b - question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261008-r07-definite-integral-0063.json)