# Question 3891: Oddness Eligibility and Symmetric Integral Pair [Return to catalogue page 20](https://ocme.evemisslab.com/en/questions/page/20/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261007-advance-20261009-r06-definite-integral-foundations/#q-20261007-advance-20261009-r06-definite-integral-0041) 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. R0006-03891 · q-20261007-advance-20261009-r06-definite-integral-0041 ## Original question and givens On all R, f has ascending coefficients [3, 4, 16, -1], constant term first. For r=1≥0, give (is f odd?,I(f,−r,r)) as Yes/No followed by its exact signed polynomial moment. Use reduced positive-denominator fractions, retaining /1 for integers. - G001 — On all R, f has ascending coefficients [3, 4, 16, -1], constant term first. For r=1≥0, give (is f odd?,I(f,−r,r)) as Yes/No followed by its exact signed polynomial moment. 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-definite-integral](https://ocme.evemisslab.com/en/mko/mko-definite-integral/) ## Step-by-step derivation S001 · impliesBefore explicit real polynomial moment or constant integrand model After oriented I(a,b)=H(b)−H(a);do not sort endpoints Conditions: none Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum. Step identifiers and dependenciesRule: polynomial_moment_scope · Uses: G001 S002 · impliesBefore even-power coefficient degree0=3 After required zero=No Conditions: none f(−x)+f(x) keeps twice the even-power terms;the polynomial is identically0 exactly when every even coefficient,including the constant,is0. Step identifiers and dependenciesRule: odd_polynomial_even_coefficient · Uses: G001, S001 S003 · impliesBefore even-power coefficient degree2=16 After required zero=No Conditions: none f(−x)+f(x) keeps twice the even-power terms;the polynomial is identically0 exactly when every even coefficient,including the constant,is0. Step identifiers and dependenciesRule: odd_polynomial_even_coefficient · Uses: G001, S002 S004 · calculationBefore symmetric interval primitive coefficient1 := div(3, 1) After symmetric interval primitive coefficient1 = 3 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore symmetric interval primitive coefficient2 := div(4, 2) After symmetric interval primitive coefficient2 = 2 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore symmetric interval primitive coefficient3 := div(16, 3) After symmetric interval primitive coefficient3 = 16/3 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore symmetric interval primitive coefficient4 := div(-1, 4) After symmetric interval primitive coefficient4 = -1/4 Conditions: none The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore [0, 3, 2, 16/3, -1/4] After H′=the supplied polynomial;integrate from -1 to 1 Conditions: none Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference. Step identifiers and dependenciesRule: polynomial_primitive_identity · Uses: G001, S007 S009 · impliesBefore symmetric interval H(b) ascending coefficients=[0, 3, 2, 16/3, -1/4] After evaluate at x=1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S008 S010 · calculationBefore symmetric interval H(b) Horner multiply before degree3 := mul(-1/4, 1) After symmetric interval H(b) Horner multiply before degree3 = -1/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore symmetric interval H(b) Horner add degree3 := sum(-1/4, 16/3) After symmetric interval H(b) Horner add degree3 = 61/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore symmetric interval H(b) Horner multiply before degree2 := mul(61/12, 1) After symmetric interval H(b) Horner multiply before degree2 = 61/12 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore symmetric interval H(b) Horner add degree2 := sum(61/12, 2) After symmetric interval H(b) Horner add degree2 = 85/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore symmetric interval H(b) Horner multiply before degree1 := mul(85/12, 1) After symmetric interval H(b) Horner multiply before degree1 = 85/12 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore symmetric interval H(b) Horner add degree1 := sum(85/12, 3) After symmetric interval H(b) Horner add degree1 = 121/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore symmetric interval H(b) Horner multiply before degree0 := mul(121/12, 1) After symmetric interval H(b) Horner multiply before degree0 = 121/12 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore symmetric interval H(b) Horner add degree0 := sum(121/12, 0) After symmetric interval H(b) Horner add degree0 = 121/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore symmetric interval H(a) ascending coefficients=[0, 3, 2, 16/3, -1/4] After evaluate at x=-1 Conditions: none Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list. Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S017 S019 · calculationBefore symmetric interval H(a) Horner multiply before degree3 := mul(-1/4, -1) After symmetric interval H(a) Horner multiply before degree3 = 1/4 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore symmetric interval H(a) Horner add degree3 := sum(1/4, 16/3) After symmetric interval H(a) Horner add degree3 = 67/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore symmetric interval H(a) Horner multiply before degree2 := mul(67/12, -1) After symmetric interval H(a) Horner multiply before degree2 = -67/12 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore symmetric interval H(a) Horner add degree2 := sum(-67/12, 2) After symmetric interval H(a) Horner add degree2 = -43/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore symmetric interval H(a) Horner multiply before degree1 := mul(-43/12, -1) After symmetric interval H(a) Horner multiply before degree1 = 43/12 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore symmetric interval H(a) Horner add degree1 := sum(43/12, 3) After symmetric interval H(a) Horner add degree1 = 79/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore symmetric interval H(a) Horner multiply before degree0 := mul(79/12, -1) After symmetric interval H(a) Horner multiply before degree0 = -79/12 Conditions: none Multiply by the same evaluation point first. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore symmetric interval H(a) Horner add degree0 := sum(-79/12, 0) After symmetric interval H(a) Horner add degree0 = -79/12 Conditions: none Then add the full coefficient at this position. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore symmetric interval oriented moment := sub(121/12, -79/12) After symmetric interval oriented moment = 50/3 Conditions: none Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · impliesBefore odd polynomial=No;r=1;integral=50/3 After zero symmetric integral alone does not imply oddness Conditions: none An odd function has symmetric integral0,but not conversely:3x²−1 integrates to0 on[−1,1] and is not odd.For r=0 every polynomial has integral0. Step identifiers and dependenciesRule: odd_integral_converse_guard · Uses: G001, S027 ## Reverse checks - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Recompute symmetric interval oriented moment with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute symmetric interval H(a) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute symmetric interval H(a) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree2 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute symmetric interval H(b) Horner add degree3 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute symmetric interval H(b) Horner multiply before degree3 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute symmetric interval primitive coefficient4 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute symmetric interval primitive coefficient3 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute symmetric interval primitive coefficient2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute symmetric interval primitive coefficient1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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. ## Result (No, 50/3) 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: d62a18f6dd2eef0c952070b46e0064d02812710e15b79153d2d4e5c2a658d5fb - candidate record SHA-256: 7bc74676d1bf7f6cc92f78ab347a242cc300319120b2c312c82c7940f33d62a4 - semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d - question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r06-definite-integral-0041.json)