# Question 29483: A Polynomial Oscillator Residual at One Specified Point [Return to catalogue page 148](https://ocme.evemisslab.com/en/questions/page/148/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r01-harmonic-oscillator-model-foundations/#q-20261013-advance-20261022-r01-harmonic-oscillator-model-0058) 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. R0057-29483 · q-20261013-advance-20261022-r01-harmonic-oscillator-model-0058 ## Original question and givens Use the undamped linear oscillator model m*x''+k*x=0, with mass m=4/1 and spring coefficient k=2/1. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Positive m and k qualify this model; this slice establishes no sin/cos closed-form solution or stability theorem. Ascending-power coefficients=[-5, 0, -3], point=0. Give (coefficients,point,x(point),x''(point),mass,spring,m*x''+k*x,zero at this point?). - G001 — Use the undamped linear oscillator model m*x''+k*x=0, with mass m=4/1 and spring coefficient k=2/1. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Positive m and k qualify this model; this slice establishes no sin/cos closed-form solution or stability theorem. Ascending-power coefficients=[-5, 0, -3], point=0. Give (coefficients,point,x(point),x''(point),mass,spring,m*x''+k*x,zero at this point?). ## 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-harmonic-oscillator-model](https://ocme.evemisslab.com/en/mko/mko-harmonic-oscillator-model/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-str22-oscillator-polynomial-residual, oscillator-polynomial-residual] After derived = [[coefficients, x, position, acceleration, mass, spring, residual, zero_at_x]] Conditions: none Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning. Step identifiers and dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001 S002 · impliesBefore source roles = [Oscillator mass, [4, 1]] After derived = [4, 4, 1] Conditions: none A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds. Step identifiers and dependenciesRule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S001 S003 · impliesBefore source roles = [Oscillator spring, [2, 1]] After derived = [2, 2, 1] Conditions: none A raw rational is a PAIR with each component within±1e6 and nonzero denominator.Normalize denominator sign and gcd.Do not accept a scalar or reduce before checking the RAW input bounds. Step identifiers and dependenciesRule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S002 S004 · impliesBefore source roles = [4, 2] After derived = [[Yes, Yes], Yes] Conditions: none For the undamped linear oscillator m*x_second+k*x=0 require BOTH m,k strictly positive.Qualification may legally contain0/negative candidates;do not repair them with abs or replacement. Step identifiers and dependenciesRule: Oscillator_BOTH_strict_model_qualifications_not_repaired · Uses: G001, S003 S005 · calculationBefore Oscillator ORIGINAL first derivative coefficient0 := mul(0, 1) After Oscillator ORIGINAL first derivative coefficient0 = 0 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Oscillator ORIGINAL first derivative coefficient1 := mul(-3, 2) After Oscillator ORIGINAL first derivative coefficient1 = -6 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Oscillator ORIGINAL second derivative coefficient0 := mul(-6, 1) After Oscillator ORIGINAL second derivative coefficient0 = -6 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [[-5, 0, -3]] After derived = [[0, -6], [-6]] Conditions: none Differentiate EVERY ascending-power monomial twice using d(c_i*x^i)=i*c_i*x^(i-1).Keep the original coefficient vector/trailing zeros;an empty derivative means0.Do not require nonconstancy or impose raw1000 bounds on derived coefficients. Step identifiers and dependenciesRule: Oscillator_WHOLE_polynomial_differentiate_TWICE_ALL_coefficients · Uses: G001, S007 S009 · calculationBefore Oscillator ORIGINAL position Horner multiply degree2 := mul(0, 0) After Oscillator ORIGINAL position Horner multiply degree2 = 0 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Oscillator ORIGINAL position Horner add degree2 := sum(0, -3) After Oscillator ORIGINAL position Horner add degree2 = -3 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Oscillator ORIGINAL position Horner multiply degree1 := mul(-3, 0) After Oscillator ORIGINAL position Horner multiply degree1 = 0 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Oscillator ORIGINAL position Horner add degree1 := sum(0, 0) After Oscillator ORIGINAL position Horner add degree1 = 0 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Oscillator ORIGINAL position Horner multiply degree0 := mul(0, 0) After Oscillator ORIGINAL position Horner multiply degree0 = 0 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Oscillator ORIGINAL position Horner add degree0 := sum(0, -5) After Oscillator ORIGINAL position Horner add degree0 = -5 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Oscillator ORIGINAL acceleration Horner multiply degree0 := mul(0, 0) After Oscillator ORIGINAL acceleration Horner multiply degree0 = 0 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Oscillator ORIGINAL acceleration Horner add degree0 := sum(0, -6) After Oscillator ORIGINAL acceleration Horner add degree0 = -6 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [[-5, 0, -3], [-6], 0] After derived = [[-5, -6]] Conditions: none Evaluate the original polynomial AND its second derivative by complete Horner at the SAME given point.Integer evaluation itself has no1e6 cap;subsequent SOURCE rational products/sums retain their own profile. Step identifiers and dependenciesRule: Oscillator_ORIGINAL_Horner_position_AND_acceleration_at_SAMEpoint · Uses: G001, S016 S018 · calculationBefore Oscillator ORIGINAL mass times acceleration := mul(4, -6) After Oscillator ORIGINAL mass times acceleration = -24 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [Oscillator ORIGINAL mass times acceleration, mul, [4, -6]] After derived = [[1, 1], [-24, 1], -24] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S018 S020 · calculationBefore Oscillator ORIGINAL spring times position := mul(2, -5) After Oscillator ORIGINAL spring times position = -10 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore source roles = [Oscillator ORIGINAL spring times position, mul, [2, -5]] After derived = [[1, 1], [-10, 1], -10] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S020 S022 · calculationBefore Oscillator ORIGINAL signed ODE residual := sum(-24, -10) After Oscillator ORIGINAL signed ODE residual = -34 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore source roles = [Oscillator ORIGINAL signed ODE residual, sum, [-24, -10]] After derived = [[1], [-34, 1], -34] Conditions: none SOURCE addition first uses denominator gcd;multiplication cross-cancels.Then BOTH raw result components must be within1e6 BEFORE final reduction.Division also checks the nonzero reciprocal.Do not apply an uncancelled-product profile or reduce the final result before the raw gate. Step identifiers and dependenciesRule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S022 S024 · impliesBefore source roles = [[-5, 0, -3], 0, 4, 2] After derived = [-5, -6, -34, No] Conditions: none Substitute R(x)=m*p_second(x)+k*p(x),retaining the SIGNED residual.R(x)=0 at THIS specified point does not establish an interval-wide ODE solution or stability. Step identifiers and dependenciesRule: Oscillator_SINGLE_point_residual_zero_NOT_interval_solution · Uses: G001, S023 ## Reverse checks - S024 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S023 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S022 · premise_and_result_audit — Recompute Oscillator ORIGINAL signed ODE residual with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S020 · premise_and_result_audit — Recompute Oscillator ORIGINAL spring times position with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S018 · premise_and_result_audit — Recompute Oscillator ORIGINAL mass times acceleration with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S016 · premise_and_result_audit — Recompute Oscillator ORIGINAL acceleration Horner add degree0 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Oscillator ORIGINAL acceleration Horner multiply degree0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Oscillator ORIGINAL position Horner add degree0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Oscillator ORIGINAL position Horner multiply degree0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Oscillator ORIGINAL position Horner add degree1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Oscillator ORIGINAL position Horner multiply degree1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Oscillator ORIGINAL position Horner add degree2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Oscillator ORIGINAL position Horner multiply degree2 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S007 · premise_and_result_audit — Recompute Oscillator ORIGINAL second derivative coefficient0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Oscillator ORIGINAL first derivative coefficient1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Oscillator ORIGINAL first derivative coefficient0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S003 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. ## Result ([-5, 0, -3], 0, -5, -6, 4, 2, -34, No) 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: 03024a20d31647fe331a2bb5b1d7354f353abd50044c5cb01f407a156b3deadc - candidate record SHA-256: a685ce88343043ca9ad3b932996ebefd971b51a943c51118f93925933cd9658e - semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6 - question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r01-harmonic-oscillator-model-0058.json)