Question 29500: A Polynomial Oscillator Residual at One Specified Point

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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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R0057-29500 · q-20261013-advance-20261022-r01-harmonic-oscillator-model-0075

Original question and givens

Use the undamped linear oscillator model m*x''+k*x=0, with mass m=1/1 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. Ascending-power coefficients=[12, 2, 3], point=2. 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=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. Ascending-power coefficients=[12, 2, 3], point=2. 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

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, [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 = [1, 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 · calculation

Before

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

After

Oscillator ORIGINAL first 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, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [[12, 2, 3]]

After

derived = [[2, 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 dependencies

Rule: Oscillator_WHOLE_polynomial_differentiate_TWICE_ALL_coefficients · Uses: G001, S007

S009 · calculation

Before

Oscillator ORIGINAL position Horner multiply degree2 := mul(0, 2)

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, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Oscillator ORIGINAL position Horner multiply degree1 := mul(3, 2)

After

Oscillator ORIGINAL position Horner multiply degree1 = 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, S010

S012 · calculation

Before

Oscillator ORIGINAL position Horner add degree1 := sum(6, 2)

After

Oscillator ORIGINAL position Horner add degree1 = 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, S011

S013 · calculation

Before

Oscillator ORIGINAL position Horner multiply degree0 := mul(8, 2)

After

Oscillator ORIGINAL position Horner multiply degree0 = 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 · calculation

Before

Oscillator ORIGINAL position Horner add degree0 := sum(16, 12)

After

Oscillator ORIGINAL position Horner add degree0 = 28

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, 2)

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, 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [[12, 2, 3], [6], 2]

After

derived = [[28, 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 dependencies

Rule: Oscillator_ORIGINAL_Horner_position_AND_acceleration_at_SAMEpoint · Uses: G001, S016

S018 · calculation

Before

Oscillator ORIGINAL mass times acceleration := mul(1, 6)

After

Oscillator ORIGINAL mass times acceleration = 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, S017

S019 · implies

Before

source roles = [Oscillator ORIGINAL mass times acceleration, mul, [1, 6]]

After

derived = [[1, 1], [6, 1], 6]

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(1, 28)

After

Oscillator ORIGINAL spring times position = 28

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, [1, 28]]

After

derived = [[1, 1], [28, 1], 28]

Conditions: none

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(6, 28)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

source roles = [Oscillator ORIGINAL signed ODE residual, sum, [6, 28]]

After

derived = [[1], [34, 1], 34]

Conditions: none

Step identifiers and dependencies

Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S022

S024 · implies

Before

source roles = [[12, 2, 3], 2, 1, 1]

After

derived = [28, 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 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

([12, 2, 3], 2, 28, 6, 1, 1, 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: 6153b522d6b96e74cbdceeb118c8a938912d4d127c4692d5b8b43af697017129
candidate record SHA-256: 19cbedc21997641985e6e544377dabd971fb7b7855c9a88e20a2527e5e184ded
semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
Lean: NOT_RUN · human review: NOT_PERFORMED

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