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R0057-29470 · q-20261013-advance-20261022-r01-harmonic-oscillator-model-0045
Use the undamped linear oscillator model m*x''+k*x=0, with mass m=5/1 and spring coefficient k=21/1. 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. Require both parameters positive. Give (mass,spring,k/m), the angular-frequency-square candidate rather than its square root.
G001 — Use the undamped linear oscillator model m*x''+k*x=0, with mass m=5/1 and spring coefficient k=21/1. 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. Require both parameters positive. Give (mass,spring,k/m), the angular-frequency-square candidate rather than its square root.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.
Before
source roles = [qf-str22-oscillator-omega-square, oscillator-omega-square]
After
derived = [[mass, spring, omega_square]]
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.
Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001
Before
source roles = [Oscillator mass, [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.
Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S001
Before
source roles = [Oscillator spring, [21, 1]]
After
derived = [21, 21, 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.
Rule: STR22_RAW_rational_PAIR_profile_and_sign_gcd_normalization · Uses: G001, S002
Before
source roles = [5, 21]
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.
Rule: Oscillator_BOTH_strict_model_qualifications_not_repaired · Uses: G001, S003
Before
Oscillator ORIGINAL spring over mass := div(21, 5)
After
Oscillator ORIGINAL spring over mass = 21/5
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [Oscillator ORIGINAL spring over mass, div, [21, 5]]
After
derived = [[1, 1], [21, 5], 21/5]
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.
Rule: STR22_EACH_SOURCE_rational_gcd_cross_cancel_THEN_RAW_gate · Uses: G001, S005
Before
source roles = [5, 21]
After
derived = [21/5]
Conditions: none
Divide by positive m to get x_second+(k/m)x=0.This asks ONLY for omega²=k/m,not a square root,a proven sin/cos solution or stability.
Rule: Oscillator_omega_SQUARED_only_not_sincos_solution · Uses: G001, S006
S007 · 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.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 — Recompute Oscillator ORIGINAL spring over mass 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.(5, 21, 21/5)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 2742e79c8dbcd4c2d7e2b39eacbbab02d9e324c122437ff48bda32cca81970a2candidate record SHA-256: 5be99532a26ba2f733e9a9c6c08ad5c11af034c09d88931ee37c6be3fc49b117semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5Lean: NOT_RUN · human review: NOT_PERFORMED