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Question 29423: Kinetic and Potential Energy at One State Only

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0057-29423 · q-20261013-advance-20261022-r01-harmonic-oscillator-model-0098

Original question and givens

Use the undamped linear oscillator model m*x''+k*x=0, with mass m=3/1 and spring coefficient k=5/1. 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=12/2, velocity=-2/3. 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=3/1 and spring coefficient k=5/1. 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=12/2, velocity=-2/3. 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001

S002 · implies

Before

source roles = [Oscillator mass, [3, 1]]

After

derived = [3, 3, 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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S001

S003 · implies

Before

source roles = [Oscillator spring, [5, 1]]

After

derived = [5, 5, 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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S002

S004 · implies

Before

source roles = [3, 5]

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 dependencies

Rule: Oscillator_BOTH_strict_model_qualifications_not_repaired · Uses: G001, S003

S005 · implies

Before

source roles = [Oscillator position, [12, 2]]

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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S004

S006 · implies

Before

source roles = [Oscillator velocity, [-2, 3]]

After

derived = [-2/3, -2, 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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S005

S007 · calculation

Before

Oscillator ORIGINAL velocity squared := mul(-2/3, -2/3)

After

Oscillator ORIGINAL velocity squared = 4/9

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [Oscillator ORIGINAL velocity squared, mul, [-2/3, -2/3]]

After

derived = [[1, 1], [4, 9], 4/9]

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S007

S009 · calculation

Before

Oscillator ORIGINAL mass times velocity squared := mul(3, 4/9)

After

Oscillator ORIGINAL mass times velocity squared = 4/3

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [Oscillator ORIGINAL mass times velocity squared, mul, [3, 4/9]]

After

derived = [[3, 1], [4, 3], 4/3]

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S009

S011 · calculation

Before

Oscillator ORIGINAL kinetic half := div(4/3, 2)

After

Oscillator ORIGINAL kinetic half = 2/3

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [Oscillator ORIGINAL kinetic half, div, [4/3, 2]]

After

derived = [[2, 1], [2, 3], 2/3]

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S011

S013 · calculation

Before

Oscillator ORIGINAL position squared := mul(6, 6)

After

Oscillator ORIGINAL position squared = 36

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [Oscillator ORIGINAL position squared, mul, [6, 6]]

After

derived = [[1, 1], [36, 1], 36]

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S013

S015 · calculation

Before

Oscillator ORIGINAL spring times position squared := mul(5, 36)

After

Oscillator ORIGINAL spring times position squared = 180

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [Oscillator ORIGINAL spring times position squared, mul, [5, 36]]

After

derived = [[1, 1], [180, 1], 180]

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S015

S017 · calculation

Before

Oscillator ORIGINAL potential half := div(180, 2)

After

Oscillator ORIGINAL potential half = 90

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [Oscillator ORIGINAL potential half, div, [180, 2]]

After

derived = [[2, 1], [90, 1], 90]

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S017

S019 · calculation

Before

Oscillator ORIGINAL snapshot total energy := sum(2/3, 90)

After

Oscillator ORIGINAL snapshot total energy = 272/3

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [Oscillator ORIGINAL snapshot total energy, sum, [2/3, 90]]

After

derived = [[1], [272, 3], 272/3]

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 dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S019

S021 · implies

Before

source roles = [3, 5, 6, -2/3]

After

derived = [2/3, 90, 272/3]

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 dependencies

Rule: 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

(3, 5, 6, -2/3, 2/3, 90, 272/3)

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: bb350cca2de9690ee79a6670ffca9aa057e8e91f991a1e8a4a46ed59abc3e4c5
  • candidate record SHA-256: 9a73ae27507c74e71b18453a50c0f247fe81c25ee6a695f89db183b492783780
  • semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
  • question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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