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R0011-06164 · q-20261009-advance-20261013-r02-integer-0089
Ordered representations=[1/1, 26/-2, 14/29, -15/-31] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.
G001 — Ordered representations=[1/1, 26/-2, 14/29, -15/-31] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.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
declared exact integer or rational-representation protocol
After
retain signs,ordered operands and original index order
Conditions: none
Check these finite data with exact integer arithmetic,without floating approximations or a claim to prove all integer-ring axioms.
Rule: integer_arithmetic_scope · Uses: G001
Before
position=0;raw fraction=[1, 1]
After
normalized=1/1;integer=Yes
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S001
Before
position=1;raw fraction=[26, -2]
After
normalized=-13/1;integer=Yes
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S002
Before
position=2;raw fraction=[14, 29]
After
normalized=14/29;integer=No
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S003
Before
position=3;raw fraction=[-15, -31]
After
normalized=15/31;integer=No
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([Yes, Yes, No, No], [1, -13], [14/29, 15/31])
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: 89064bc69c922a3a851f457bb59febc67cb6264e389ded9104639b5b0f6caf91candidate record SHA-256: ead20dfe32f2461cc3ba0df39373e9511eaef07bdb4e60deb5c388078f3d7e83semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED