The valid fractions are 1/(-7) and 4/(-21), with both denominators nonzero. Do they represent the same rational number?
G001 — The valid fractions are 1/(-7) and 4/(-21), with both denominators nonzero. Do they represent the same rational number?
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.
BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.
Validate EVERY raw SAFE integer component,including the numerator at q0/0/0 eligibility.A validity task may legally return No;actual arithmetic requires ALL used denominators nonzero.
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
Fraction ORIGINAL cross product difference := sub(-21, -28)
After
Fraction ORIGINAL cross product difference = 7
Conditions: none
Use ORIGINAL operands and order for exact integer/rational arithmetic.Raw bounds are bank-specific;derived values do not inherit SAFE or foreign small caps.No floating approximation.
With BOTH denominators nonzero,p/q=r/s IFF ps=rq.Negative denominators are legal.Compare complete exact products,not appearance or decimal approximation.
S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S005 · premise_and_result_audit — Recompute Fraction ORIGINAL cross product difference with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Fraction ORIGINAL r times q with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Fraction ORIGINAL p times s with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
Result
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.