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.
Check THIS bank interface:BASE equations use form/domain;BASE events/complement have no operation;factorials have no domain.ext18 uses local hyphens,other legacy operations underscores.Do not force every answer into a tuple or fraction.
Factorial requested n ORIGINAL factorial factor2 := mul(1, 2)
After
Factorial requested n ORIGINAL factorial factor2 = 2
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Factorial multiplies the COMPLETE2..n factors;at n0/1 the empty product is1.Use arbitrary-precision integers,not Number/scientific notation or another family n64 cap.
Recompute the WHOLE n! and compare the GIVEN decimal STRING first;do not treat an unchecked hint as an answer oracle.Require the exact string,retaining every digit.
Factorial ORIGINAL successor index n plus1 := sum(2, 1)
After
Factorial ORIGINAL successor index n plus1 = 3
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Factorial ORIGINAL successor factor times verified n factorial := mul(3, 2)
After
Factorial ORIGINAL successor factor times verified n factorial = 6
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Factorial missing ORIGINAL verified successor over positive n factorial := div(6, 2)
After
Factorial missing ORIGINAL verified successor over positive n factorial = 3
Conditions: none
Use ORIGINAL operands and declared order for exact integer/rational arithmetic,not floats.Bounds are family-specific:do not revalidate derived values as RAW inputs or add blanket SAFE/n64 caps to BASE families.
Since n!>0 and the GIVEN(n+1)!was also checked by the complete product/recurrence,□n!=(n+1)! uniquely gives□=n+1.Do not trust an unchecked successor hint.
S008 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S007 · premise_and_result_audit — Recompute Factorial missing ORIGINAL verified successor over positive n factorial with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute Factorial ORIGINAL successor factor times verified n factorial with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Factorial ORIGINAL successor index n plus1 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
Result
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.