# Question 34318: Counterexample to a Universal Proposition

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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.

R0067-34318 · q-20260921-proof-counterexample-universal-0018



## Original question and givens

The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?


- G001 — The domain is explicitly the integers. Consider the universal proposition “Every integer is even.” and the stated witness n=9. Does this witness refute that universal proposition?





## 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-proof](https://ocme.evemisslab.com/en/mko/mko-proof/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-proof-counterexample-universal, [bound, counterexample_role, domain, predicate_id, universal_statement, witness]]

After

derived = [BASE-no-required-operation, integers]

Conditions: none

Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.

Step identifiers and dependenciesRule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001





S002 · impliesBefore

source roles = [even, null, 9]

After

derived = [[9, 2, 1], No]

Conditions: none

FIRST match the stated predicate/bound and INTEGER domain.THEN substitute THIS integer witness and show failure.Use an exact nonzero remainder for even/multiple and retain strict/non-strict inequalities elsewhere.Do not use a metadata flag as an answer oracle.

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





S003 · impliesBefore

source roles = [even, null, 9, No]

After

derived = [exists integer n with not predicate(n), not forall integers predicate(n), Yes]

Conditions: none

ONE actual counterexample IN the domain proves∃n ¬P(n),therefore refuting THIS∀n P(n).It does not prove every n fails,refute other propositions or make finite successful samples a universal proof.

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









## Reverse checks


- S003 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
- S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
- S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.





## Result

Yes

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: 042a070c38c11ce36b1152c6ffff9004025a1612d126acb29f4fe7cbdd1ae01b
- candidate record SHA-256: 0f7d7516a30ddb2a4a85690b616b141a03b9af94ea8558fff0995e6b2f7733db
- semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
- question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260921-proof-counterexample-universal-0018.json)