# Question 6161: Rational Candidates Split by Integer Eligibility

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0011-06161 · q-20261009-advance-20261013-r02-integer-0086



## Original question and givens

Ordered representations=[-2/1, 20/-2, 11/23, -12/-25] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.


- G001 — Ordered representations=[-2/1, 20/-2, 11/23, -12/-25] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.





## 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.


- [mko-integer](https://ocme.evemisslab.com/en/mko/mko-integer/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: integer_arithmetic_scope · Uses: G001





S002 · impliesBefore

position=0;raw fraction=[-2, 1]

After

normalized=-2/1;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.

Step identifiers and dependenciesRule: normalized_rational_integer_qualification · Uses: G001, S001





S003 · impliesBefore

position=1;raw fraction=[20, -2]

After

normalized=-10/1;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.

Step identifiers and dependenciesRule: normalized_rational_integer_qualification · Uses: G001, S002





S004 · impliesBefore

position=2;raw fraction=[11, 23]

After

normalized=11/23;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.

Step identifiers and dependenciesRule: normalized_rational_integer_qualification · Uses: G001, S003





S005 · impliesBefore

position=3;raw fraction=[-12, -25]

After

normalized=12/25;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.

Step identifiers and dependenciesRule: normalized_rational_integer_qualification · Uses: G001, S004









## Reverse checks


- 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.





## Result

([Yes, Yes, No, No], [-2, -10], [11/23, 12/25])

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 197a54fd90a86ee5dad0ef83da7b4cec9dcf3063ce8bb66b24f5821dc4706ea1
- candidate record SHA-256: 42ee3ce9d59136ec084e260bf22c64abd06d9fc4a15e7082a9f24d9406cc6f64
- semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
- question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r02-integer-0086.json)