Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-1, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
G001 — Use legal integer fraction representations with nonzero original denominators; negative denominators are allowed. Representation and value remain distinct. Original n=-1, d=-3; ordered scale factors [s,t]=[2,-1]. Give (original value, raw [ns,ds] and [nt,dt], each representation’s eligibility, reduced values or undefined). Preserve the raw products using exact integer arithmetic.
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.
original integer numerator/denominator representations;original denominators nonzero
After
representation and rational value are distinct data
Conditions: none
Negative denominators are valid,but reduced values do not replace original representations.Candidate algorithms about raw numerators/denominators must use those raw data.This finite audit is not a general proof of quotient well-definedness.
A nonzero scale cancels and preserves value;zero scaling produces0/0,which is invalid even with zero numerator and cannot be reduced back to the original value.
A nonzero scale cancels and preserves value;zero scaling produces0/0,which is invalid even with zero numerator and cannot be reduced back to the original value.
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Recompute scaled rational value1 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute scaled denominator1 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute scaled numerator1 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Recompute scaled rational value0 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute scaled denominator0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute scaled numerator0 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute original fraction value with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
Result
(1/3, [[-2, -6], [1, 3]], [Yes, Yes], [1/3, 1/3])
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.