Complete Finite Certificates, Universal Integer Encodings and Failed-Witness Boundaries
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
At-most-countable set · qf-mod31-countable-classification · candidate
Canonical question JSONL · English JSONL · Validation report