Question 29407: Kinetic and Potential Energy at One State Only

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0057-29407 · q-20261013-advance-20261022-r01-harmonic-oscillator-model-0082

Original question and givens

Use the undamped linear oscillator model m*x''+k*x=0, with mass m=3/1 and spring coefficient k=1/1. 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=-4/1, velocity=3/3. 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=1/1. 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=-4/1, velocity=3/3. 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.

mko-harmonic-oscillator-model

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, [1, 1]]

After

derived = [1, 1, 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, 1]

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, [-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 dependencies

Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S004

S006 · implies

Before

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

After

derived = [1, 1, 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, S005

S007 · calculation

Before

Oscillator ORIGINAL velocity squared := mul(1, 1)

After

Oscillator ORIGINAL velocity squared = 1

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, [1, 1]]

After

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

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, 1)

After

Oscillator ORIGINAL mass times velocity squared = 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, 1]]

After

derived = [[1, 1], [3, 1], 3]

Conditions: none

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(3, 2)

After

Oscillator ORIGINAL kinetic half = 3/2

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, [3, 2]]

After

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

Conditions: none

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(-4, -4)

After

Oscillator ORIGINAL position squared = 16

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, [-4, -4]]

After

derived = [[1, 1], [16, 1], 16]

Conditions: none

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(1, 16)

After

Oscillator ORIGINAL spring times position squared = 16

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, [1, 16]]

After

derived = [[1, 1], [16, 1], 16]

Conditions: none

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(16, 2)

After

Oscillator ORIGINAL potential half = 8

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, [16, 2]]

After

derived = [[2, 1], [8, 1], 8]

Conditions: none

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(3/2, 8)

After

Oscillator ORIGINAL snapshot total energy = 19/2

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, [3/2, 8]]

After

derived = [[1], [19, 2], 19/2]

Conditions: none

Step identifiers and dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S019

S021 · implies

Before

source roles = [3, 1, -4, 1]

After

derived = [3/2, 8, 19/2]

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, 1, -4, 1, 3/2, 8, 19/2)

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: 82c16c2c7a9b6626c8f5896b0ac8cc280b052b762af8b00485e9b529bf4e27d6
candidate record SHA-256: 63d14d61953b6d48fe0f1a4a0a57e0344f678ff08355927d3d2093b3ae504b8b
semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
Lean: NOT_RUN · human review: NOT_PERFORMED

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