# Finite-Domain Counterexamples and Explicit Predicate Extensions

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261012-advance-20261017-r07-counterexample-foundations · 2026-10-12 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([], [], Yes, does-not-exist)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes, does-not-exist).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0001/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 2. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[true, true, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, Yes], [2, Yes]], [], Yes, does-not-exist)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes]], [], Yes, does-not-exist).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0002/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 3. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No]], [0, 1, 2], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0003/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 4. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, No], [2, No], [3, No], [4, No]], [0, 1, 2, 3, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, No], [4, No]], [0, 1, 2, 3, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0004/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 5. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, No], [2, No], [3, No], [4, No]], [1, 2, 3, 4], No, 1)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, No], [4, No]], [1, 2, 3, 4], No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0005/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 6. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, Yes], [2, No], [3, No], [4, No]], [0, 2, 3, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, No], [4, No]], [0, 2, 3, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0006/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 7. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, Yes], [2, No], [3, No], [4, No]], [2, 3, 4], No, 2)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, No], [4, No]], [2, 3, 4], No, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0007/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 8. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, No], [2, Yes], [3, No], [4, No]], [0, 1, 3, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, No], [4, No]], [0, 1, 3, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0008/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 9. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, No], [2, Yes], [3, No], [4, No]], [1, 3, 4], No, 1)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, No], [4, No]], [1, 3, 4], No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0009/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 10. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, Yes], [2, Yes], [3, No], [4, No]], [0, 3, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, Yes], [3, No], [4, No]], [0, 3, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0010/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 11. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No]], [3, 4], No, 3)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No]], [3, 4], No, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0011/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 12. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, No], [2, No], [3, Yes], [4, No]], [0, 1, 2, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, Yes], [4, No]], [0, 1, 2, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0012/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 13. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, No], [2, No], [3, Yes], [4, No]], [1, 2, 4], No, 1)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, Yes], [4, No]], [1, 2, 4], No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0013/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 14. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, Yes], [2, No], [3, Yes], [4, No]], [0, 2, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, Yes], [4, No]], [0, 2, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0014/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 15. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No]], [2, 4], No, 2)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No]], [2, 4], No, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0015/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 16. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, No], [2, Yes], [3, Yes], [4, No]], [0, 1, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, Yes], [4, No]], [0, 1, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0016/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 17. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No]], [1, 4], No, 1)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No]], [1, 4], No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0017/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 18. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No]], [0, 4], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No]], [0, 4], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0018/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 19. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No]], [4], No, 4)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No]], [4], No, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0019/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 20. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, No], [2, No], [3, No], [4, Yes]], [0, 1, 2, 3], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, No], [3, No], [4, Yes]], [0, 1, 2, 3], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0020/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 21. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, No], [2, No], [3, No], [4, Yes]], [1, 2, 3], No, 1)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, No], [3, No], [4, Yes]], [1, 2, 3], No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0021/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 22. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, Yes], [2, No], [3, No], [4, Yes]], [0, 2, 3], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, Yes], [2, No], [3, No], [4, Yes]], [0, 2, 3], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0022/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 23. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes]], [2, 3], No, 2)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes]], [2, 3], No, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0023/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 24. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, No], [1, No], [2, Yes], [3, No], [4, Yes]], [0, 1, 3], No, 0)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, No], [1, No], [2, Yes], [3, No], [4, Yes]], [0, 1, 3], No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0024/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 25. Complete Failure Set and Least Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Give (indexed truth table, all failing source labels, universal truth?, least failing label or does-not-exist).

View answer and explanation([[0, Yes], [1, No], [2, Yes], [3, No], [4, Yes]], [1, 3], No, 1)

Inspect every value in the stated finite table and preserve the distinction between a counterexample and its absence. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No], [2, Yes], [3, No], [4, Yes]], [1, 3], No, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0025/)

Counterexamples and domain eligibility · qf-density17-counter-full-failure-min · candidate





## 26. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0026/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 27. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, -1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, No], [outside-domain, outside-domain, outside-domain], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0027/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 28. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, -1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, No, No], [Yes, outside-domain, outside-domain], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, No], [Yes, outside-domain, outside-domain], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0028/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 29. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, -1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, No, No], [No, outside-domain, outside-domain], [Yes, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, No], [No, outside-domain, outside-domain], [Yes, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0029/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 30. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, Yes, outside-domain], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, Yes, outside-domain], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0030/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 31. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 0, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, Yes, outside-domain], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, Yes, outside-domain], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0031/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 32. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 0, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, No], [No, Yes, outside-domain], [Yes, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [No, Yes, outside-domain], [Yes, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0032/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 33. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, No, outside-domain], [No, Yes, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, No], [outside-domain, No, outside-domain], [No, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0033/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 34. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, No, outside-domain], [No, Yes, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [Yes, No, outside-domain], [No, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0034/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 35. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 1, -1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, No], [No, No, outside-domain], [Yes, Yes, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, No], [No, No, outside-domain], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0035/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 36. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, -1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, Yes], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, Yes], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0036/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 37. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, -1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, Yes], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, Yes], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0037/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 38. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, -1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, Yes], [Yes, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, Yes], [Yes, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0038/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 39. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, Yes], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, Yes], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0039/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 40. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 0, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, Yes], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0040/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 41. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 0, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, Yes], [Yes, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0041/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 42. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, No, Yes], [No, Yes, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, No, Yes], [No, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0042/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 43. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0043/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 44. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 1, 0]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, No])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, No, Yes], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0044/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 45. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, -1, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, No], [No, No, Yes])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, No, Yes], [outside-domain, outside-domain, No], [No, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0045/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 46. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, -1, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, No], [No, No, Yes])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [Yes, outside-domain, No], [No, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0046/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 47. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, -1, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, No], [Yes, No, Yes])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, No, Yes], [No, outside-domain, No], [Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0047/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 48. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[-1, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, No], [No, No, Yes])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [No, Yes, Yes], [outside-domain, Yes, No], [No, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0048/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 49. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[0, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, No], [No, No, Yes])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [Yes, Yes, No], [No, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0049/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 50. Three Domain Guards before Counterexample Evaluation

The explicit finite domain is Fin(2)={0,…,1} and the complete Boolean predicate table is truth=[true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Ordered candidates=[1, 0, 1]. Give (indexed truth table, candidate domain masks, predicate values or outside-domain, counterexample flags).

View answer and explanation([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, No], [Yes, No, Yes])

Only a typed candidate with false predicate value is a counterexample; outside candidates never become witnesses. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, Yes], [1, No]], [Yes, Yes, Yes], [No, Yes, No], [Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0050/)

Counterexamples and domain eligibility · qf-density17-counter-three-candidate-guards · candidate





## 51. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([], [], [], No, No, Yes)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0051/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 52. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 2], [], [], Yes, No, Yes)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], [], Yes, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0052/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 53. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 2], [0], [0], Yes, Yes, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [0], [0], Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0053/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 54. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[1], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 2], [1], [], Yes, No, Yes)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [1], [], Yes, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0054/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 55. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(3)={0,…,2} and the complete Boolean predicate table is truth=[true, true, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([], [0, 2], [], No, No, Yes)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [0, 2], [], No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0055/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 56. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0056/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 57. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([1, 2, 3, 4], [0, 2, 4], [2, 4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 4], [0, 2, 4], [2, 4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0057/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 58. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 4], [0, 2, 4], [0, 2, 4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0058/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 59. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([2, 3, 4], [0, 2, 4], [2, 4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 4], [0, 2, 4], [2, 4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0059/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 60. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 1, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 4], [0, 2, 4], [0, 4], Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0060/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 61. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([1, 3, 4], [0, 2, 4], [4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 4], [0, 2, 4], [4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0061/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 62. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=3. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 3, 4], [0, 2, 4], [0, 4], Yes, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 4], [0, 2, 4], [0, 4], Yes, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0062/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 63. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([3, 4], [0, 2, 4], [4], Yes, Yes, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [0, 2, 4], [4], Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0063/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 64. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [0, 2, 4], [0, 2, 4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0064/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 65. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([1, 2, 4], [0, 2, 4], [2, 4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 4], [0, 2, 4], [2, 4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0065/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 66. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 2, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 4], [0, 2, 4], [0, 2, 4], Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0066/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 67. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([2, 4], [0, 2, 4], [2, 4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [0, 2, 4], [2, 4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0067/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 68. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 1, 4], [0, 2, 4], [0, 4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 4], [0, 2, 4], [0, 4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0068/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 69. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=3. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([1, 4], [0, 2, 4], [4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [0, 2, 4], [4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0069/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 70. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=4. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 4], [0, 2, 4], [0, 4], Yes, Yes, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 4], [0, 2, 4], [0, 4], Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0070/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 71. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=5. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([4], [0, 2, 4], [4], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4], [0, 2, 4], [4], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0071/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 72. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=-1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 1, 2, 3], [0, 2, 4], [0, 2], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 2, 4], [0, 2], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0072/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 73. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=0. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([1, 2, 3], [0, 2, 4], [2], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3], [0, 2, 4], [2], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0073/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 74. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=1. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([0, 2, 3], [0, 2, 4], [0, 2], No, No, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3], [0, 2, 4], [0, 2], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0074/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 75. Restriction Can Remove an Original Counterexample

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Restrict to subset=[4, 2, 0], normalized as a set; proposed witness=2. Give (original failures, retained labels, restricted failures, witness fails in original domain?, witness remains in restricted failures?, restricted universal truth?).

View answer and explanation([2, 3], [0, 2, 4], [2], Yes, Yes, No)

Check both the original domain and restricted membership; removing a failing input can eliminate that witness. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [0, 2, 4], [2], Yes, Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0075/)

Counterexamples and domain eligibility · qf-density17-counter-restriction-witness · candidate





## 76. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([], [[0, Yes]], [], Yes, Yes, 0)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, Yes]], [], Yes, Yes, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0076/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 77. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(0)=∅ and the complete Boolean predicate table is truth=[] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=0. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([], [[0, No]], [0], Yes, No, 0)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, No]], [0], Yes, No, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0077/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 78. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(8)={0,…,7} and the complete Boolean predicate table is truth=[true, true, true, true, true, true, true, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=8. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, Yes], [5, Yes], [6, Yes], [7, Yes], [8, No]], [8], Yes, No, 8)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, Yes], [5, Yes], [6, Yes], [7, Yes], [8, No]], [8], Yes, No, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0078/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 79. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(8)={0,…,7} and the complete Boolean predicate table is truth=[false, false, false, false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=8. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, No], [6, No], [7, No], [8, Yes]], [0, 1, 2, 3, 4, 5, 6, 7], No, No, 8)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, No], [6, No], [7, No], [8, Yes]], [0, 1, 2, 3, 4, 5, 6, 7], No, No, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0079/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 80. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 1, 2, 3, 4], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, Yes]], [0, 1, 2, 3, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [[0, No], [1, No], [2, No], [3, No], [4, No], [5, Yes]], [0, 1, 2, 3, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0080/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 81. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([1, 2, 3, 4], [[0, Yes], [1, No], [2, No], [3, No], [4, No], [5, No]], [1, 2, 3, 4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3, 4], [[0, Yes], [1, No], [2, No], [3, No], [4, No], [5, No]], [1, 2, 3, 4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0081/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 82. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 2, 3, 4], [[0, No], [1, Yes], [2, No], [3, No], [4, No], [5, Yes]], [0, 2, 3, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3, 4], [[0, No], [1, Yes], [2, No], [3, No], [4, No], [5, Yes]], [0, 2, 3, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0082/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 83. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([2, 3, 4], [[0, Yes], [1, Yes], [2, No], [3, No], [4, No], [5, No]], [2, 3, 4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3, 4], [[0, Yes], [1, Yes], [2, No], [3, No], [4, No], [5, No]], [2, 3, 4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0083/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 84. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 1, 3, 4], [[0, No], [1, No], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 1, 3, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3, 4], [[0, No], [1, No], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 1, 3, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0084/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 85. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([1, 3, 4], [[0, Yes], [1, No], [2, Yes], [3, No], [4, No], [5, No]], [1, 3, 4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 3, 4], [[0, Yes], [1, No], [2, Yes], [3, No], [4, No], [5, No]], [1, 3, 4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0085/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 86. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 3, 4], [[0, No], [1, Yes], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 3, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 3, 4], [[0, No], [1, Yes], [2, Yes], [3, No], [4, No], [5, Yes]], [0, 3, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0086/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 87. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, false, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([3, 4], [[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No], [5, No]], [3, 4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 4], [[0, Yes], [1, Yes], [2, Yes], [3, No], [4, No], [5, No]], [3, 4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0087/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 88. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 1, 2, 4], [[0, No], [1, No], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 1, 2, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [[0, No], [1, No], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 1, 2, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0088/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 89. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([1, 2, 4], [[0, Yes], [1, No], [2, No], [3, Yes], [4, No], [5, No]], [1, 2, 4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 4], [[0, Yes], [1, No], [2, No], [3, Yes], [4, No], [5, No]], [1, 2, 4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0089/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 90. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 2, 4], [[0, No], [1, Yes], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 2, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 4], [[0, No], [1, Yes], [2, No], [3, Yes], [4, No], [5, Yes]], [0, 2, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0090/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 91. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([2, 4], [[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No], [5, No]], [2, 4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 4], [[0, Yes], [1, Yes], [2, No], [3, Yes], [4, No], [5, No]], [2, 4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0091/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 92. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 1, 4], [[0, No], [1, No], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 1, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 4], [[0, No], [1, No], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 1, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0092/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 93. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([1, 4], [[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No], [5, No]], [1, 4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 4], [[0, Yes], [1, No], [2, Yes], [3, Yes], [4, No], [5, No]], [1, 4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0093/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 94. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 4], [[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 4], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 4], [[0, No], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, Yes]], [0, 4], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0094/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 95. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, true, true, false] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4], [[0, Yes], [1, Yes], [2, Yes], [3, Yes], [4, No], [5, No]], [4, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0095/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 96. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 1, 2, 3], [[0, No], [1, No], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 1, 2, 3], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [[0, No], [1, No], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 1, 2, 3], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0096/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 97. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, false, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([1, 2, 3], [[0, Yes], [1, No], [2, No], [3, No], [4, Yes], [5, No]], [1, 2, 3, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 3], [[0, Yes], [1, No], [2, No], [3, No], [4, Yes], [5, No]], [1, 2, 3, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0097/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 98. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 2, 3], [[0, No], [1, Yes], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 2, 3], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 3], [[0, No], [1, Yes], [2, No], [3, No], [4, Yes], [5, Yes]], [0, 2, 3], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0098/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 99. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[true, true, false, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value false at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([2, 3], [[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes], [5, No]], [2, 3, 5], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 3], [[0, Yes], [1, Yes], [2, No], [3, No], [4, Yes], [5, No]], [2, 3, 5], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0099/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate





## 100. One Declared New Predicate Value and Universal Truth

The explicit finite domain is Fin(5)={0,…,4} and the complete Boolean predicate table is truth=[false, false, true, false, true] in index order. An outside-domain candidate is not evaluated and is not a counterexample. Empty-domain universal quantification is true; finite enumeration is not an infinite-domain proof. Append the explicitly given Boolean value true at new label n=5. Give (original failures, extended indexed truth table, later failures, original universal truth?, later universal truth?, new label).

View answer and explanation([0, 1, 3], [[0, No], [1, No], [2, Yes], [3, No], [4, Yes], [5, Yes]], [0, 1, 3], No, No, 5)

Extend only with the declared value; a previously empty domain or true universal statement does not predict the new value. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 3], [[0, No], [1, No], [2, Yes], [3, No], [4, Yes], [5, Yes]], [0, 1, 3], No, No, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261017-r07-counterexample-0100/)

Counterexamples and domain eligibility · qf-density17-counter-domain-extension · candidate



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