# Question 29948: Primitive Constant Selected by an Explicit Anchor [Return to catalogue page 150](https://ocme.evemisslab.com/en/questions/page/150/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r05-antiderivative-foundations/#q-20261007-advance-20261009-r05-antiderivative-0073) 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. R0058-29948 · q-20261007-advance-20261009-r05-antiderivative-0073 ## Original question and givens On all R, f has ascending coefficients [10, 23, 0, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=10/2. Use reduced positive-denominator fractions, retaining /1 for integers. - G001 — On all R, f has ascending coefficients [10, 23, 0, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=10/2. Use reduced positive-denominator fractions, 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-antiderivative](https://ocme.evemisslab.com/en/mko/mko-antiderivative/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density-primitive-anchor-constant, density_primitive_anchor_constant] After derived = [[required_additive_constant_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. Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001 S002 · impliesBefore source roles = [integrand f, [10, 23, 0, 3], No] After derived = [[10/1, 23/1, 0/1, 3/1]] 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. Step identifiers and dependenciesRule: Primitive_RAW_arity_SAFE_inputs_POSITIVE_rational_den_trim_KEEPzero · Uses: G001, S001 S003 · calculationBefore Primitive ORIGINAL coefficient degree1 := div(10, 1) After Primitive ORIGINAL coefficient degree1 = 10 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Primitive ORIGINAL coefficient degree2 := div(23, 2) After Primitive ORIGINAL coefficient degree2 = 23/2 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Primitive ORIGINAL coefficient degree3 := div(0, 3) After Primitive ORIGINAL coefficient degree3 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Primitive ORIGINAL coefficient degree4 := div(3, 4) After Primitive ORIGINAL coefficient degree4 = 3/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Primitive reverse ORIGINAL derivative coefficient0 := mul(10, 1) After Primitive reverse ORIGINAL derivative coefficient0 = 10 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Primitive reverse ORIGINAL derivative coefficient1 := mul(23/2, 2) After Primitive reverse ORIGINAL derivative coefficient1 = 23 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Primitive reverse ORIGINAL derivative coefficient2 := mul(0, 3) After Primitive reverse ORIGINAL derivative coefficient2 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Primitive reverse ORIGINAL derivative coefficient3 := mul(3/4, 4) After Primitive reverse ORIGINAL derivative coefficient3 = 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [[10/1, 23/1, 0/1, 3/1]] After derived = [[0/1, 10/1, 23/2, 0/1, 3/4], [10/1, 23/1, 0/1, 3/1], 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. Step identifiers and dependenciesRule: Primitive_ZEROconstant_ALL_monomial_coefficients_AND_reverse_identity · Uses: G001, S010 S012 · calculationBefore Primitive anchor ORIGINAL Horner product degree4 := mul(0, -3) After Primitive anchor ORIGINAL Horner product degree4 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Primitive anchor ORIGINAL Horner sum degree4 := sum(0, 3/4) After Primitive anchor ORIGINAL Horner sum degree4 = 3/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Primitive anchor ORIGINAL Horner product degree3 := mul(3/4, -3) After Primitive anchor ORIGINAL Horner product degree3 = -9/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Primitive anchor ORIGINAL Horner sum degree3 := sum(-9/4, 0) After Primitive anchor ORIGINAL Horner sum degree3 = -9/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Primitive anchor ORIGINAL Horner product degree2 := mul(-9/4, -3) After Primitive anchor ORIGINAL Horner product degree2 = 27/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Primitive anchor ORIGINAL Horner sum degree2 := sum(27/4, 23/2) After Primitive anchor ORIGINAL Horner sum degree2 = 73/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Primitive anchor ORIGINAL Horner product degree1 := mul(73/4, -3) After Primitive anchor ORIGINAL Horner product degree1 = -219/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Primitive anchor ORIGINAL Horner sum degree1 := sum(-219/4, 10) After Primitive anchor ORIGINAL Horner sum degree1 = -179/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Primitive anchor ORIGINAL Horner product degree0 := mul(-179/4, -3) After Primitive anchor ORIGINAL Horner product degree0 = 537/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Primitive anchor ORIGINAL Horner sum degree0 := sum(537/4, 0) After Primitive anchor ORIGINAL Horner sum degree0 = 537/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Primitive anchor ORIGINAL target minus F0 at anchor := sub(5, 537/4) After Primitive anchor ORIGINAL target minus F0 at anchor = -517/4 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Primitive anchor reverse SAME F0 value plus C := sum(537/4, -517/4) After Primitive anchor reverse SAME F0 value plus C = 5 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore source roles = [[10/1, 23/1, 0/1, 3/1], -3, 5/1, [0/1, 10/1, 23/2, 0/1, 3/4]] After derived = [537/4, -517/4, 5/1, Yes] Conditions: none In the DECLARED family F=F0+C,substitute the ORIGINAL anchor a:F(a)=F0(a)+C=V,so C=V-F0(a).Reverse substitution at the SAME anchor gives V while(F0+C)_prime=f.C changes values although not derivatives;do not drop it or change the anchor.This asks for the constant in this representative family. Step identifiers and dependenciesRule: Primitive_anchor_ORIGINAL_target_C_changes_values_NOT_derivative · Uses: G001, S023 ## Reverse checks - S024 · 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. - S023 · premise_and_result_audit — Recompute Primitive anchor reverse SAME F0 value plus C with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL target minus F0 at anchor with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree2 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree3 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree3 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner sum degree4 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Primitive anchor ORIGINAL Horner product degree4 with exact rational arithmetic, not rounded decimals. - 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. ## Result -517/4 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: 1ccc7d5027f0776c27a33e871e81e9450ae13fad4b6711930ec41b670e6bc041 - candidate record SHA-256: b324bd3f02bd5c64698f1781a16472d08ebd0918021d2248a1ee8b2640e8ec7d - semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df - question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r05-antiderivative-0073.json)