OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 29492: A Polynomial Oscillator Residual at One Specified Point

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

Return to catalogue page 148

← Question derivation index · Original question batch

R0057-29492 · q-20261013-advance-20261022-r01-harmonic-oscillator-model-0067

Original question and givens

Use the undamped linear oscillator model m*x''+k*x=0, with mass m=1/1 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. Ascending-power coefficients=[4, -1, 1], point=-1. 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=1/1 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. Ascending-power coefficients=[4, -1, 1], point=-1. 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001

S002 · implies

Before

source roles = [Oscillator mass, [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, 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 = [1, 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 · calculation

Before

Oscillator ORIGINAL first derivative coefficient0 := mul(-1, 1)

After

Oscillator ORIGINAL first derivative coefficient0 = -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, S004

S006 · calculation

Before

Oscillator ORIGINAL first derivative coefficient1 := mul(1, 2)

After

Oscillator ORIGINAL first derivative coefficient1 = 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, S005

S007 · calculation

Before

Oscillator ORIGINAL second derivative coefficient0 := mul(2, 1)

After

Oscillator ORIGINAL second derivative coefficient0 = 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, S006

S008 · implies

Before

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

After

derived = [[-1, 2], [2]]

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 dependencies

Rule: Oscillator_WHOLE_polynomial_differentiate_TWICE_ALL_coefficients · Uses: G001, S007

S009 · calculation

Before

Oscillator ORIGINAL position Horner multiply degree2 := mul(0, -1)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Oscillator ORIGINAL position Horner add degree2 := sum(0, 1)

After

Oscillator ORIGINAL position Horner add degree2 = 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, S009

S011 · calculation

Before

Oscillator ORIGINAL position Horner multiply degree1 := mul(1, -1)

After

Oscillator ORIGINAL position Horner multiply degree1 = -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, S010

S012 · calculation

Before

Oscillator ORIGINAL position Horner add degree1 := sum(-1, -1)

After

Oscillator ORIGINAL position Horner add degree1 = -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, S011

S013 · calculation

Before

Oscillator ORIGINAL position Horner multiply degree0 := mul(-2, -1)

After

Oscillator ORIGINAL position Horner multiply degree0 = 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, S012

S014 · calculation

Before

Oscillator ORIGINAL position Horner add degree0 := sum(2, 4)

After

Oscillator ORIGINAL position 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Oscillator ORIGINAL acceleration Horner multiply degree0 := mul(0, -1)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Oscillator ORIGINAL acceleration Horner add degree0 := sum(0, 2)

After

Oscillator ORIGINAL acceleration Horner add degree0 = 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, S015

S017 · implies

Before

source roles = [[4, -1, 1], [2], -1]

After

derived = [[6, 2]]

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 dependencies

Rule: Oscillator_ORIGINAL_Horner_position_AND_acceleration_at_SAMEpoint · Uses: G001, S016

S018 · calculation

Before

Oscillator ORIGINAL mass times acceleration := mul(1, 2)

After

Oscillator ORIGINAL mass times acceleration = 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, S017

S019 · implies

Before

source roles = [Oscillator ORIGINAL mass times acceleration, mul, [1, 2]]

After

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

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, S018

S020 · calculation

Before

Oscillator ORIGINAL spring times position := mul(5, 6)

After

Oscillator ORIGINAL spring times position = 30

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, S019

S021 · implies

Before

source roles = [Oscillator ORIGINAL spring times position, mul, [5, 6]]

After

derived = [[1, 1], [30, 1], 30]

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, S020

S022 · calculation

Before

Oscillator ORIGINAL signed ODE residual := sum(2, 30)

After

Oscillator ORIGINAL signed ODE residual = 32

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, S021

S023 · implies

Before

source roles = [Oscillator ORIGINAL signed ODE residual, sum, [2, 30]]

After

derived = [[1], [32, 1], 32]

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, S022

S024 · implies

Before

source roles = [[4, -1, 1], -1, 1, 5]

After

derived = [6, 2, 32, 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 dependencies

Rule: 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

([4, -1, 1], -1, 6, 2, 1, 5, 32, 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: 1b5e5430194441d7a65110690639b8b4070545961846c3fca00681640e73012d
  • candidate record SHA-256: 0809a894543c6364c6be0b625623ad67d2ed2657bae644c0c87904d39ae4700d
  • semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
  • question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON