# Closed Truths, Open Bindings, and Candidate Reports

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-20261008-advance-20261012-r03-proposition-foundations · 2026-10-08 · 100 questions

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



## 1. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, False], [0, -2, 2])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [0, -2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0001/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 2. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [-11, -2, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-11, -2, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0002/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 3. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, False], [-10, -3, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-10, -3, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0003/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 4. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [-9, -4, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-9, -4, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0004/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 5. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [-8, -5, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-8, -5, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0005/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 6. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, False], [-7, -6, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-7, -6, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0006/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 7. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [-6, -7, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-6, -7, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0007/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 8. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [-5, -8, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-5, -8, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0008/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 9. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, False], [-4, -9, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-4, -9, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0009/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 10. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [-3, -10, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-3, -10, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0010/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 11. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [-2, -11, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [-2, -11, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0011/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 12. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, False], [-1, -12, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, False], [-1, -12, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0012/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 13. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([True, True, True], [0, -13, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([True, True, True], [0, -13, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0013/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 14. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [1, -14, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [1, -14, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0014/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 15. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, False], [2, -15, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [2, -15, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0015/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 16. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [3, -16, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [3, -16, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0016/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 17. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [4, -17, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [4, -17, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0017/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 18. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, False], [5, -18, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [5, -18, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0018/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 19. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [6, -19, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [6, -19, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0019/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 20. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [7, -20, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [7, -20, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0020/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 21. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, False], [8, -21, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [8, -21, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0021/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 22. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [9, -22, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [9, -22, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0022/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 23. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [10, -23, 0])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [10, -23, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0023/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 24. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, False], [11, -24, 1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, False], [11, -24, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0024/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 25. Closed Comparison Truths and Signed Witnesses

In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. Row i states a_i≤b_i. Give (three truth values, three signed a_i−b_i values).

View answer and explanation([False, True, True], [12, -25, -1])

Compare and subtract each row; negative and zero differences satisfy ≤. A computed truth value is not a registered proof. The exact result, retaining the requested order, is ([False, True, True], [12, -25, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0025/)

Proposition · qf-density12-proposition-comparison-witnesses · candidate





## 26. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-2 with assignment=null (unbound). Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(No, open, not-applicable)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (No, open, not-applicable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0026/)

Proposition · qf-density12-proposition-open-binding · candidate





## 27. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-1 with assignment=-12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -11)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0027/)

Proposition · qf-density12-proposition-open-binding · candidate





## 28. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤0 with assignment=-10. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -10)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0028/)

Proposition · qf-density12-proposition-open-binding · candidate





## 29. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤1 with assignment=-8. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -9)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0029/)

Proposition · qf-density12-proposition-open-binding · candidate





## 30. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤2 with assignment=-6. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -8)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0030/)

Proposition · qf-density12-proposition-open-binding · candidate





## 31. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-2 with assignment=-9. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -7)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0031/)

Proposition · qf-density12-proposition-open-binding · candidate





## 32. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-1 with assignment=-7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -6)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0032/)

Proposition · qf-density12-proposition-open-binding · candidate





## 33. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤0 with assignment=-5. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -5)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0033/)

Proposition · qf-density12-proposition-open-binding · candidate





## 34. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤1 with assignment=-3. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -4)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0034/)

Proposition · qf-density12-proposition-open-binding · candidate





## 35. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤2 with assignment=-1. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -3)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0035/)

Proposition · qf-density12-proposition-open-binding · candidate





## 36. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-2 with assignment=-4. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -2)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0036/)

Proposition · qf-density12-proposition-open-binding · candidate





## 37. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-1 with assignment=-2. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, -1)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0037/)

Proposition · qf-density12-proposition-open-binding · candidate





## 38. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤0 with assignment=0. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, True, 0)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, True, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0038/)

Proposition · qf-density12-proposition-open-binding · candidate





## 39. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤1 with assignment=2. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 1)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0039/)

Proposition · qf-density12-proposition-open-binding · candidate





## 40. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤2 with assignment=4. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 2)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0040/)

Proposition · qf-density12-proposition-open-binding · candidate





## 41. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-2 with assignment=1. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 3)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0041/)

Proposition · qf-density12-proposition-open-binding · candidate





## 42. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-1 with assignment=3. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 4)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0042/)

Proposition · qf-density12-proposition-open-binding · candidate





## 43. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤0 with assignment=5. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 5)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0043/)

Proposition · qf-density12-proposition-open-binding · candidate





## 44. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤1 with assignment=7. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 6)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0044/)

Proposition · qf-density12-proposition-open-binding · candidate





## 45. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤2 with assignment=9. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 7)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0045/)

Proposition · qf-density12-proposition-open-binding · candidate





## 46. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-2 with assignment=6. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 8)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0046/)

Proposition · qf-density12-proposition-open-binding · candidate





## 47. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤-1 with assignment=8. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 9)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0047/)

Proposition · qf-density12-proposition-open-binding · candidate





## 48. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤0 with assignment=10. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 10)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0048/)

Proposition · qf-density12-proposition-open-binding · candidate





## 49. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤1 with assignment=12. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 11)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0049/)

Proposition · qf-density12-proposition-open-binding · candidate





## 50. Open-Sentence Binding and Truth Eligibility

In the classical integer context, consider x≤2 with assignment=14. Give (is the sentence closed?, its bound truth value or open, substitution difference or not-applicable).

View answer and explanation(Yes, False, 12)

Only an integer binding permits substitution and a truth value; an unbound sentence is not False. Truth and proof remain separate. The exact result, retaining the requested order, is (Yes, False, 12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0050/)

Proposition · qf-density12-proposition-open-binding · candidate





## 51. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([True, False, False], [False, True, True], [0, -2, 2])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([True, False, False], [False, True, True], [0, -2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0051/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 52. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [-11, -2, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-11, -2, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0052/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 53. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [-10, -3, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-10, -3, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0053/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 54. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [-9, -4, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-9, -4, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0054/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 55. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [-8, -5, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-8, -5, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0055/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 56. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [-7, -6, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-7, -6, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0056/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 57. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [-6, -7, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-6, -7, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0057/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 58. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [-5, -8, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-5, -8, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0058/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 59. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [-4, -9, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-4, -9, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0059/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 60. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [-3, -10, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-3, -10, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0060/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 61. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [-2, -11, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [-2, -11, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0061/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 62. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [-1, -12, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [-1, -12, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0062/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 63. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([True, False, False], [False, True, True], [0, -13, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([True, False, False], [False, True, True], [0, -13, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0063/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 64. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [1, -14, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [1, -14, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0064/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 65. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [2, -15, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [2, -15, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0065/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 66. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [3, -16, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [3, -16, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0066/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 67. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [4, -17, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [4, -17, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0067/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 68. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [5, -18, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [5, -18, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0068/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 69. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [6, -19, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [6, -19, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0069/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 70. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [7, -20, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [7, -20, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0070/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 71. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [8, -21, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [8, -21, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0071/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 72. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [9, -22, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [9, -22, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0072/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 73. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, True], [True, True, False], [10, -23, 0])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, True], [True, True, False], [10, -23, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0073/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 74. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [11, -24, 1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [11, -24, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0074/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 75. Equality, Negation, and Signed Difference Witnesses

In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. P_i means a_i=b_i. Give (three P truth values, three ¬P truth values, three signed a_i−b_i values).

View answer and explanation([False, False, False], [True, True, True], [12, -25, -1])

A zero difference establishes equality in the instance; then negate each classical truth value. A false proposition is still a proposition, not an unproved open sentence. The exact result, retaining the requested order, is ([False, False, False], [True, True, True], [12, -25, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0075/)

Proposition · qf-density12-proposition-equality-negation-witnesses · candidate





## 76. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[0, 0], [-1, 1], [1, -1]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, False], [Yes, Yes, Yes], [0, -2, 2])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, Yes, Yes], [0, -2, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0076/)

Proposition · qf-density12-proposition-report-defects · candidate





## 77. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-11, 0], [0, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [Yes, No, No], [-11, -2, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, No, No], [-11, -2, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0077/)

Proposition · qf-density12-proposition-report-defects · candidate





## 78. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-10, 0], [-1, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, False], [No, Yes, Yes], [-10, -3, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [No, Yes, Yes], [-10, -3, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0078/)

Proposition · qf-density12-proposition-report-defects · candidate





## 79. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-9, 0], [-2, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [Yes, Yes, No], [-9, -4, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, Yes, No], [-9, -4, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0079/)

Proposition · qf-density12-proposition-report-defects · candidate





## 80. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-8, 0], [-3, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [No, No, Yes], [-8, -5, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, No, Yes], [-8, -5, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0080/)

Proposition · qf-density12-proposition-report-defects · candidate





## 81. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-7, 0], [-4, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, False], [Yes, No, No], [-7, -6, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, No, No], [-7, -6, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0081/)

Proposition · qf-density12-proposition-report-defects · candidate





## 82. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-6, 0], [-5, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [No, Yes, Yes], [-6, -7, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, Yes, Yes], [-6, -7, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0082/)

Proposition · qf-density12-proposition-report-defects · candidate





## 83. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-5, 0], [-6, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [Yes, Yes, Yes], [-5, -8, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, Yes, Yes], [-5, -8, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0083/)

Proposition · qf-density12-proposition-report-defects · candidate





## 84. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-4, 0], [-7, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, False], [No, No, Yes], [-4, -9, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [No, No, Yes], [-4, -9, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0084/)

Proposition · qf-density12-proposition-report-defects · candidate





## 85. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-3, 0], [-8, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [Yes, No, No], [-3, -10, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [Yes, No, No], [-3, -10, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0085/)

Proposition · qf-density12-proposition-report-defects · candidate





## 86. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-2, 0], [-9, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [No, Yes, No], [-2, -11, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, Yes, No], [-2, -11, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0086/)

Proposition · qf-density12-proposition-report-defects · candidate





## 87. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[-1, 0], [-10, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, False], [Yes, Yes, Yes], [-1, -12, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, False], [Yes, Yes, Yes], [-1, -12, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0087/)

Proposition · qf-density12-proposition-report-defects · candidate





## 88. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[0, 0], [-11, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([True, True, True], [No, No, Yes], [0, -13, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([True, True, True], [No, No, Yes], [0, -13, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0088/)

Proposition · qf-density12-proposition-report-defects · candidate





## 89. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[1, 0], [-12, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [No, No, Yes], [1, -14, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, No, Yes], [1, -14, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0089/)

Proposition · qf-density12-proposition-report-defects · candidate





## 90. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[2, 0], [-13, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, False], [Yes, Yes, No], [2, -15, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [Yes, Yes, No], [2, -15, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0090/)

Proposition · qf-density12-proposition-report-defects · candidate





## 91. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[3, 0], [-14, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [No, Yes, Yes], [3, -16, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, Yes, Yes], [3, -16, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0091/)

Proposition · qf-density12-proposition-report-defects · candidate





## 92. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[4, 0], [-15, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [Yes, No, No], [4, -17, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, No, No], [4, -17, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0092/)

Proposition · qf-density12-proposition-report-defects · candidate





## 93. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[5, 0], [-16, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, False], [No, No, Yes], [5, -18, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [No, No, Yes], [5, -18, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0093/)

Proposition · qf-density12-proposition-report-defects · candidate





## 94. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[6, 0], [-17, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [Yes, Yes, No], [6, -19, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, Yes, No], [6, -19, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0094/)

Proposition · qf-density12-proposition-report-defects · candidate





## 95. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[7, 0], [-18, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [No, Yes, No], [7, -20, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, Yes, No], [7, -20, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0095/)

Proposition · qf-density12-proposition-report-defects · candidate





## 96. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[8, 0], [-19, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, False], [Yes, No, No], [8, -21, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [Yes, No, No], [8, -21, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0096/)

Proposition · qf-density12-proposition-report-defects · candidate





## 97. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[9, 0], [-20, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 0, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [No, No, Yes], [9, -22, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [No, No, Yes], [9, -22, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0097/)

Proposition · qf-density12-proposition-report-defects · candidate





## 98. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[10, 0], [-21, 2], [0, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [Yes, Yes, Yes], [10, -23, 0])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, Yes, Yes], [10, -23, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0098/)

Proposition · qf-density12-proposition-report-defects · candidate





## 99. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[11, 0], [-22, 2], [1, 0]]. P_i means a_i≤b_i. Candidate reported=[1, 1, 1], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, False], [No, Yes, No], [11, -24, 1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, False], [No, Yes, No], [11, -24, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0099/)

Proposition · qf-density12-proposition-report-defects · candidate





## 100. Candidate Truth Reports Checked Independently

In a classical integer context, ordered pairs=[[12, 0], [-23, 2], [-1, 0]]. P_i means a_i≤b_i. Candidate reported=[0, 0, 0], with 1 reporting True and 0 False. Give (actual truth values, report-match Yes/No values, signed a_i−b_i values).

View answer and explanation([False, True, True], [Yes, No, No], [12, -25, -1])

Recompute comparisons before checking reports; a candidate self-report is not Evidence, and truth is not a formal proof receipt. The exact result, retaining the requested order, is ([False, True, True], [Yes, No, No], [12, -25, -1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r03-proposition-0100/)

Proposition · qf-density12-proposition-report-defects · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r03-proposition-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261012-r03-proposition-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r03-proposition-foundations/validation-report.json)