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.
Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.
FIRST verify the raw string and Boolean flag against the declared ASCII syntax,THEN classify the exact value.The frozen No cases are genuine half-integers.A parser rejecting string1.0 does NOT prove real1.0 is outside Z.Ambiguous out-of-slice cases are not false mathematical No answers.
Integer nonmember ORIGINAL floor times reduced denominator := mul(15, 2)
After
Integer nonmember ORIGINAL floor times reduced denominator = 30
Conditions: none
Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.
Integer nonmember ORIGINAL numerator minus floor product := sub(31, 30)
After
Integer nonmember ORIGINAL numerator minus floor product = 1
Conditions: none
Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.
Integer nonmember ORIGINAL positive proper remainder fraction := div(1, 2)
After
Integer nonmember ORIGINAL positive proper remainder fraction = 1/2
Conditions: none
Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.
Integer nonmember ORIGINAL value minus lower integer := sub(31/2, 15)
After
Integer nonmember ORIGINAL value minus lower integer = 1/2
Conditions: none
Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.
Integer nonmember ORIGINAL upper integer minus value := sub(16, 31/2)
After
Integer nonmember ORIGINAL upper integer minus value = 1/2
Conditions: none
Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.
Reduce to n/d and write n=kd+r with0<r<d.The value lies STRICTLY between consecutive integers k,k+1,so cannot be an integer.These five have denominator2 and odd numerators;this does NOT classify every decimal representation as noninteger.
S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
S007 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL upper integer minus value with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL value minus lower integer with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL positive proper remainder fraction with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL numerator minus floor product with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL floor times reduced denominator with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,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 data,field roles,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.