# Question 29418: Kinetic and Potential Energy at One State Only [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-0093) 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. R0057-29418 · q-20261013-advance-20261022-r01-harmonic-oscillator-model-0093 ## Original question and givens Use the undamped linear oscillator model m*x''+k*x=0, with mass m=2/1 and spring coefficient k=6/1. 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. position=7/3, velocity=0/2. Give (mass,spring,position,velocity,m*v²/2,k*x²/2,total energy). Signed position and velocity are squared, not clipped. - G001 — Use the undamped linear oscillator model m*x''+k*x=0, with mass m=2/1 and spring coefficient k=6/1. 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. position=7/3, velocity=0/2. Give (mass,spring,position,velocity,m*v²/2,k*x²/2,total energy). Signed position and velocity are squared, not clipped. ## 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-energy-snapshot, oscillator-energy-snapshot] After derived = [[mass, spring, position, velocity, kinetic, potential, total]] 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, [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, S001 S003 · impliesBefore source roles = [Oscillator spring, [6, 1]] After derived = [6, 6, 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 = [2, 6] 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 · impliesBefore source roles = [Oscillator position, [7, 3]] After derived = [7/3, 7, 3] 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, S004 S006 · impliesBefore source roles = [Oscillator velocity, [0, 2]] After derived = [0, 0, 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, S005 S007 · calculationBefore Oscillator ORIGINAL velocity squared := mul(0, 0) After Oscillator ORIGINAL velocity squared = 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, S006 S008 · impliesBefore source roles = [Oscillator ORIGINAL velocity squared, mul, [0, 0]] After derived = [[1, 1], [0, 1], 0] 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, S007 S009 · calculationBefore Oscillator ORIGINAL mass times velocity squared := mul(2, 0) After Oscillator ORIGINAL mass times velocity squared = 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 · impliesBefore source roles = [Oscillator ORIGINAL mass times velocity squared, mul, [2, 0]] After derived = [[1, 1], [0, 1], 0] 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, S009 S011 · calculationBefore Oscillator ORIGINAL kinetic half := div(0, 2) After Oscillator ORIGINAL kinetic half = 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 · impliesBefore source roles = [Oscillator ORIGINAL kinetic half, div, [0, 2]] After derived = [[2, 1], [0, 1], 0] 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, S011 S013 · calculationBefore Oscillator ORIGINAL position squared := mul(7/3, 7/3) After Oscillator ORIGINAL position squared = 49/9 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 · impliesBefore source roles = [Oscillator ORIGINAL position squared, mul, [7/3, 7/3]] After derived = [[1, 1], [49, 9], 49/9] 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, S013 S015 · calculationBefore Oscillator ORIGINAL spring times position squared := mul(6, 49/9) After Oscillator ORIGINAL spring times position squared = 98/3 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, S014 S016 · impliesBefore source roles = [Oscillator ORIGINAL spring times position squared, mul, [6, 49/9]] After derived = [[3, 1], [98, 3], 98/3] 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, S015 S017 · calculationBefore Oscillator ORIGINAL potential half := div(98/3, 2) After Oscillator ORIGINAL potential half = 49/3 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, S016 S018 · impliesBefore source roles = [Oscillator ORIGINAL potential half, div, [98/3, 2]] After derived = [[2, 1], [49, 3], 49/3] 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, S017 S019 · calculationBefore Oscillator ORIGINAL snapshot total energy := sum(0, 49/3) After Oscillator ORIGINAL snapshot total energy = 49/3 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, S018 S020 · impliesBefore source roles = [Oscillator ORIGINAL snapshot total energy, sum, [0, 49/3]] After derived = [[1], [49, 3], 49/3] 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, S019 S021 · impliesBefore source roles = [2, 6, 7/3, 0] After derived = [0, 49/3, 49/3] Conditions: none Compute T=mv²/2,V=kx²/2 and T+V in original operation order.Square signed x,v rather than clipping them.A SINGLE-state snapshot does not prove trajectory conservation,solution existence or stability. Step identifiers and dependenciesRule: Oscillator_ENERGY_at_ONEstate_not_conservation · Uses: G001, S020 ## Reverse checks - 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 — 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. - S019 · premise_and_result_audit — Recompute Oscillator ORIGINAL snapshot total energy with exact rational arithmetic, not rounded decimals. - S018 · 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. - S017 · premise_and_result_audit — Recompute Oscillator ORIGINAL potential half with exact rational arithmetic, not rounded decimals. - S016 · 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. - S015 · premise_and_result_audit — Recompute Oscillator ORIGINAL spring times position squared with exact rational arithmetic, not rounded decimals. - S014 · 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. - S013 · premise_and_result_audit — Recompute Oscillator ORIGINAL position squared with exact rational arithmetic, not rounded decimals. - S012 · 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. - S011 · premise_and_result_audit — Recompute Oscillator ORIGINAL kinetic half with exact rational arithmetic, not rounded decimals. - S010 · 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. - S009 · premise_and_result_audit — Recompute Oscillator ORIGINAL mass times velocity squared 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 velocity squared with exact rational arithmetic, not rounded decimals. - S006 · 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. - S005 · 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. - 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 (2, 6, 7/3, 0, 0, 49/3, 49/3) 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: 6d093a3570d178869931782b86bcf0fa7200c25aad169ed85977ed394e48452e - candidate record SHA-256: 8056245c9be31c4c6132d71810c3f9b74097f6d91174e394f42d1ec5be3c5fcf - 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-0093.json)