# Complete Finite Certificates, Universal Integer Encodings and Failed-Witness Boundaries

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-20261014-advance-20261031-r01-at-most-countable-set-foundations · 2026-10-14 · 100 questions

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



## 1. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[1, -2, 40] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([1, -2, 40], [[1, 0], [-2, 1], [40, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([1, -2, 40], [[1, 0], [-2, 1], [40, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0001/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 2. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[2, -3, 42] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([2, -3, 42], [[2, 0], [-3, 1], [42, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([2, -3, 42], [[2, 0], [-3, 1], [42, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0002/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 3. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[3, -4, 44] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([3, -4, 44], [[3, 0], [-4, 1], [44, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([3, -4, 44], [[3, 0], [-4, 1], [44, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0003/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 4. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[4, -5, 46] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([4, -5, 46], [[4, 0], [-5, 1], [46, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([4, -5, 46], [[4, 0], [-5, 1], [46, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0004/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 5. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[5, -6, 48] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([5, -6, 48], [[5, 0], [-6, 1], [48, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([5, -6, 48], [[5, 0], [-6, 1], [48, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0005/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 6. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[6, -7, 50] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([6, -7, 50], [[6, 0], [-7, 1], [50, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([6, -7, 50], [[6, 0], [-7, 1], [50, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0006/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 7. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[7, -8, 52] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([7, -8, 52], [[7, 0], [-8, 1], [52, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0007/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 8. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[8, -9, 54] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([8, -9, 54], [[8, 0], [-9, 1], [54, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([8, -9, 54], [[8, 0], [-9, 1], [54, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0008/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 9. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[9, -10, 56] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([9, -10, 56], [[9, 0], [-10, 1], [56, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([9, -10, 56], [[9, 0], [-10, 1], [56, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0009/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 10. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[10, -11, 58] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([10, -11, 58], [[10, 0], [-11, 1], [58, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([10, -11, 58], [[10, 0], [-11, 1], [58, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0010/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 11. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[11, -12, 60] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([11, -12, 60], [[11, 0], [-12, 1], [60, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([11, -12, 60], [[11, 0], [-12, 1], [60, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0011/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 12. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[12, -13, 62] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([12, -13, 62], [[12, 0], [-13, 1], [62, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([12, -13, 62], [[12, 0], [-13, 1], [62, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0012/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 13. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[13, -14, 64] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([13, -14, 64], [[13, 0], [-14, 1], [64, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([13, -14, 64], [[13, 0], [-14, 1], [64, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0013/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 14. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[14, -15, 66] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([14, -15, 66], [[14, 0], [-15, 1], [66, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([14, -15, 66], [[14, 0], [-15, 1], [66, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0014/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 15. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[15, -16, 68] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([15, -16, 68], [[15, 0], [-16, 1], [68, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([15, -16, 68], [[15, 0], [-16, 1], [68, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0015/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 16. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[16, -17, 70] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([16, -17, 70], [[16, 0], [-17, 1], [70, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([16, -17, 70], [[16, 0], [-17, 1], [70, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0016/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 17. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[17, -18, 72] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([17, -18, 72], [[17, 0], [-18, 1], [72, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([17, -18, 72], [[17, 0], [-18, 1], [72, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0017/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 18. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[18, -19, 74] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([18, -19, 74], [[18, 0], [-19, 1], [74, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([18, -19, 74], [[18, 0], [-19, 1], [74, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0018/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 19. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[19, -20, 76] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([19, -20, 76], [[19, 0], [-20, 1], [76, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([19, -20, 76], [[19, 0], [-20, 1], [76, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0019/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 20. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[20, -21, 78] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([20, -21, 78], [[20, 0], [-21, 1], [78, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([20, -21, 78], [[20, 0], [-21, 1], [78, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0020/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 21. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[21, -22, 80] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([21, -22, 80], [[21, 0], [-22, 1], [80, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([21, -22, 80], [[21, 0], [-22, 1], [80, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0021/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 22. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[22, -23, 82] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([22, -23, 82], [[22, 0], [-23, 1], [82, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([22, -23, 82], [[22, 0], [-23, 1], [82, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0022/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 23. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[23, -24, 84] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([23, -24, 84], [[23, 0], [-24, 1], [84, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([23, -24, 84], [[23, 0], [-24, 1], [84, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0023/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 24. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[24, -25, 86] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([24, -25, 86], [[24, 0], [-25, 1], [86, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([24, -25, 86], [[24, 0], [-25, 1], [86, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0024/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 25. A Complete Finite List Gives an At-Most-Countable Certificate

Complete nonempty duplicate-free finite integer list elements=[25, -26, 88] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

View answer and explanation([25, -26, 88], [[25, 0], [-26, 1], [88, 2]], Yes, finite_implies_at_most_countable)

Map each listed element to its unique position in the natural numbers. Completeness of this finite list is the declared set premise; the result is at most countable, not necessarily countably infinite. This family rejects an empty list rather than denying that the empty set is at most countable. The exact result, retaining the requested order, is ([25, -26, 88], [[25, 0], [-26, 1], [88, 2]], Yes, finite_implies_at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0025/)

At-most-countable set · qf-mod31-countable-finite-certificate · candidate





## 26. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=-12, selected left=-1, right=2. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(1, -12, -1, 2, -13, -10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, -12, -1, 2, -13, -10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0026/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 27. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=-11, selected left=-2, right=3. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(2, -11, -2, 3, -15, -5, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, -11, -2, 3, -15, -5, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0027/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 28. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=-10, selected left=-3, right=4. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(3, -10, -3, 4, -19, 2, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, -10, -3, 4, -19, 2, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0028/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 29. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=-9, selected left=-4, right=5. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(4, -9, -4, 5, -25, 11, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, -9, -4, 5, -25, 11, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0029/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 30. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=-8, selected left=-5, right=6. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(5, -8, -5, 6, -33, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, -8, -5, 6, -33, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0030/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 31. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=-7, selected left=-6, right=7. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(1, -7, -6, 7, -13, 0, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, -7, -6, 7, -13, 0, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0031/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 32. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=-6, selected left=-7, right=8. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(2, -6, -7, 8, -20, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, -6, -7, 8, -20, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0032/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 33. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=-5, selected left=-8, right=9. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(3, -5, -8, 9, -29, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, -5, -8, 9, -29, 22, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0033/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 34. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=-4, selected left=-9, right=10. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(4, -4, -9, 10, -40, 36, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, -4, -9, 10, -40, 36, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0034/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 35. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=-3, selected left=-10, right=11. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(5, -3, -10, 11, -53, 52, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, -3, -10, 11, -53, 52, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0035/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 36. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=-2, selected left=-11, right=12. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(1, -2, -11, 12, -13, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, -2, -11, 12, -13, 10, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0036/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 37. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=-1, selected left=-12, right=13. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(2, -1, -12, 13, -25, 25, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, -1, -12, 13, -25, 25, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0037/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 38. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=0, selected left=-13, right=14. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(3, 0, -13, 14, -39, 42, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, 0, -13, 14, -39, 42, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0038/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 39. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=1, selected left=-14, right=15. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(4, 1, -14, 15, -55, 61, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, 1, -14, 15, -55, 61, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0039/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 40. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=2, selected left=-15, right=16. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(5, 2, -15, 16, -73, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, 2, -15, 16, -73, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0040/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 41. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=3, selected left=-16, right=17. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(1, 3, -16, 17, -13, 20, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, 3, -16, 17, -13, 20, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0041/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 42. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=4, selected left=-17, right=18. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(2, 4, -17, 18, -30, 40, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, 4, -17, 18, -30, 40, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0042/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 43. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=5, selected left=-18, right=19. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(3, 5, -18, 19, -49, 62, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, 5, -18, 19, -49, 62, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0043/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 44. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=6, selected left=-19, right=20. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(4, 6, -19, 20, -70, 86, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, 6, -19, 20, -70, 86, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0044/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 45. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=7, selected left=-20, right=21. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(5, 7, -20, 21, -93, 112, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, 7, -20, 21, -93, 112, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0045/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 46. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=1, intercept b=8, selected left=-21, right=22. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(1, 8, -21, 22, -13, 30, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (1, 8, -21, 22, -13, 30, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0046/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 47. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=2, intercept b=9, selected left=-22, right=23. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(2, 9, -22, 23, -35, 55, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (2, 9, -22, 23, -35, 55, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0047/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 48. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=3, intercept b=10, selected left=-23, right=24. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(3, 10, -23, 24, -59, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (3, 10, -23, 24, -59, 82, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0048/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 49. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=4, intercept b=11, selected left=-24, right=25. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(4, 11, -24, 25, -85, 111, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (4, 11, -24, 25, -85, 111, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0049/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 50. A Nonzero Integer Slope Supplies a Universal Injection, Not a Sample Guess

On Z use f(x)=a*x+b, with nonzero slope a=5, intercept b=12, selected left=-25, right=26. Give (slope,intercept,left,right,image_left,image_right,injective_check?,universal_nonzero_slope_injection_Z_to_Z_then_encode_N).

View answer and explanation(5, 12, -25, 26, -113, 142, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N)

A nonzero slope makes a*x+b=a*y+b imply x=y for all integers. The displayed flag checks the chosen pair consistently, including equal inputs, but is not the source of universal injectivity. Compose with the standard integer-to-natural encoding to obtain an at-most-countable certificate. The exact result, retaining the requested order, is (5, 12, -25, 26, -113, 142, Yes, universal_nonzero_slope_injection_Z_to_Z_then_encode_N).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0050/)

At-most-countable set · qf-mod31-countable-affine-injection · candidate





## 51. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[0, 50, -1], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 0, [[0, 0], [1, 50], [2, -1]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 0, [[0, 0], [1, 50], [2, -1]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0051/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 52. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[1, 51, -2], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 51, [[0, 1], [1, 51], [2, -2]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 51, [[0, 1], [1, 51], [2, -2]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0052/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 53. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[2, 52, -3], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -3, [[0, 2], [1, 52], [2, -3]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -3, [[0, 2], [1, 52], [2, -3]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0053/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 54. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[3, 53, -4], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 3, [[0, 3], [1, 53], [2, -4]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 3, [[0, 3], [1, 53], [2, -4]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0054/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 55. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[4, 54, -5], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 54, [[0, 4], [1, 54], [2, -5]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 54, [[0, 4], [1, 54], [2, -5]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0055/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 56. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[5, 55, -6], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -6, [[0, 5], [1, 55], [2, -6]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -6, [[0, 5], [1, 55], [2, -6]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0056/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 57. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[6, 56, -7], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 6, [[0, 6], [1, 56], [2, -7]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 6, [[0, 6], [1, 56], [2, -7]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0057/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 58. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[7, 57, -8], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 57, [[0, 7], [1, 57], [2, -8]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 57, [[0, 7], [1, 57], [2, -8]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0058/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 59. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[8, 58, -9], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -9, [[0, 8], [1, 58], [2, -9]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -9, [[0, 8], [1, 58], [2, -9]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0059/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 60. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[9, 59, -10], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 9, [[0, 9], [1, 59], [2, -10]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 9, [[0, 9], [1, 59], [2, -10]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0060/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 61. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[10, 60, -11], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 60, [[0, 10], [1, 60], [2, -11]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 60, [[0, 10], [1, 60], [2, -11]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0061/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 62. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[11, 61, -12], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -12, [[0, 11], [1, 61], [2, -12]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -12, [[0, 11], [1, 61], [2, -12]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0062/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 63. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[12, 62, -13], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 12, [[0, 12], [1, 62], [2, -13]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 12, [[0, 12], [1, 62], [2, -13]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0063/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 64. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[13, 63, -14], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 63, [[0, 13], [1, 63], [2, -14]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 63, [[0, 13], [1, 63], [2, -14]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0064/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 65. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[14, 64, -15], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -15, [[0, 14], [1, 64], [2, -15]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -15, [[0, 14], [1, 64], [2, -15]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0065/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 66. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[15, 65, -16], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 15, [[0, 15], [1, 65], [2, -16]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 15, [[0, 15], [1, 65], [2, -16]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0066/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 67. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[16, 66, -17], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 66, [[0, 16], [1, 66], [2, -17]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 66, [[0, 16], [1, 66], [2, -17]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0067/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 68. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[17, 67, -18], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -18, [[0, 17], [1, 67], [2, -18]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -18, [[0, 17], [1, 67], [2, -18]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0068/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 69. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[18, 68, -19], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 18, [[0, 18], [1, 68], [2, -19]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0069/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 70. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[19, 69, -20], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 69, [[0, 19], [1, 69], [2, -20]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 69, [[0, 19], [1, 69], [2, -20]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0070/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 71. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[20, 70, -21], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -21, [[0, 20], [1, 70], [2, -21]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -21, [[0, 20], [1, 70], [2, -21]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0071/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 72. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[21, 71, -22], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 21, [[0, 21], [1, 71], [2, -22]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 21, [[0, 21], [1, 71], [2, -22]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0072/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 73. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[22, 72, -23], zero-based index=1. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(1, 72, [[0, 22], [1, 72], [2, -23]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (1, 72, [[0, 22], [1, 72], [2, -23]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0073/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 74. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[23, 73, -24], zero-based index=2. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(2, -24, [[0, 23], [1, 73], [2, -24]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (2, -24, [[0, 23], [1, 73], [2, -24]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0074/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 75. Keep Enumeration Direction and Original List Order Explicit

Complete nonempty duplicate-free finite integer enumeration elements=[24, 74, -25], zero-based index=0. Give (index,element,enumeration_pairs,finite_enumeration_complete). Enumeration pairs are [position,element], not the reversed injection pairs.

View answer and explanation(0, 24, [[0, 24], [1, 74], [2, -25]], finite_enumeration_complete)

Read the chosen position and retain the complete ordered enumeration. Finite enumeration supplies a finite-set certificate; it does not make the set countably infinite. The exact result, retaining the requested order, is (0, 24, [[0, 24], [1, 74], [2, -25]], finite_enumeration_complete).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0075/)

At-most-countable set · qf-mod31-countable-enumeration · candidate





## 76. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=1. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 1, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 1, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0076/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 77. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=2. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 2, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 2, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0077/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 78. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=3. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 3, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 3, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0078/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 79. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=4. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 4, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 4, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0079/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 80. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=5. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 5, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 5, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0080/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 81. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=6. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 6, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 6, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0081/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 82. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=7. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 7, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 7, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0082/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 83. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=8. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 8, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 8, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0083/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 84. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=9. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 9, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 9, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0084/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 85. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=10. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 10, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 10, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0085/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 86. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=11. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 11, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 11, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0086/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 87. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=12. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 12, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 12, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0087/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 88. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=13. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 13, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 13, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0088/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 89. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=14. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 14, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 14, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0089/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 90. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=15. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 15, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 15, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0090/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 91. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=16. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 16, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 16, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0091/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 92. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=17. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 17, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 17, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0092/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 93. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=18. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 18, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 18, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0093/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 94. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=19. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 19, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 19, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0094/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 95. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=20. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 20, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 20, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0095/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 96. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=21. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 21, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 21, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0096/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 97. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=22. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 22, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 22, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0097/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 98. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=universal_affine_injection, certificate_parameter=23. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(universal_affine_injection, 23, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (universal_affine_injection, 23, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0098/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 99. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=failed_candidate_encoding, certificate_parameter=24. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(failed_candidate_encoding, 24, No, no_classification_from_failed_witness)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (failed_candidate_encoding, 24, No, no_classification_from_failed_witness).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0099/)

At-most-countable set · qf-mod31-countable-classification · candidate





## 100. A Failed Encoding Does Not Prove Uncountability

Declared evidence_mode=complete_finite_list, certificate_parameter=25. The registered modes complete_finite_list and universal_affine_injection refer to their supplied complete or universal certificates; failed_candidate_encoding records only a failed witness. Give (evidence_mode,certificate_parameter,proved?,classification).

View answer and explanation(complete_finite_list, 25, Yes, at_most_countable)

This mode audit reports at_most_countable for a declared qualifying certificate and no_classification_from_failed_witness for the failed mode. A numeric certificate parameter alone is not a new construction or verification of an unspecified set. Failure of one encoding does not establish uncountability. The exact result, retaining the requested order, is (complete_finite_list, 25, Yes, at_most_countable).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261031-r01-at-most-countable-set-0100/)

At-most-countable set · qf-mod31-countable-classification · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261031-r01-at-most-countable-set-foundations/validation-report.json)