# Total-Map Collisions, Ceiling Loads and Failed Size Converses

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

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

batch-20261014-advance-20261107-r05-pigeonhole-principle-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.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[0, 1, 2, 0] from source indices Fin(4)={0,…,3} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(4, 3, [0, 3, 0], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (4, 3, [0, 3, 0], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0001/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 2. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[1, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(5, 4, [0, 2, 1], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (5, 4, [0, 2, 1], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0002/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 3. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 0, 3, 1, 4, 2] from source indices Fin(6)={0,…,5} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(6, 5, [0, 5, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (6, 5, [0, 5, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0003/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 4. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[3, 4, 5, 0, 1, 2, 3] from source indices Fin(7)={0,…,6} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(7, 6, [0, 6, 3], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 6, [0, 6, 3], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0004/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 5. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(8)={0,…,7} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(8, 7, [0, 7, 4], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 7, [0, 7, 4], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0005/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 6. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 2, 2, 2, 2] from source indices Fin(5)={0,…,4} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(5, 3, [0, 1, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (5, 3, [0, 1, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0006/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 7. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 3, 0, 1, 2, 3] from source indices Fin(6)={0,…,5} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(6, 4, [0, 4, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (6, 4, [0, 4, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0007/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 8. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4] from source indices Fin(7)={0,…,6} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(7, 5, [0, 5, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 5, [0, 5, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0008/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 9. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(8)={0,…,7} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(8, 6, [0, 2, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 6, [0, 2, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0009/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 10. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 3, 4, 5, 6, 0, 1, 2, 3] from source indices Fin(9)={0,…,8} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(9, 7, [0, 7, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 7, [0, 7, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0010/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 11. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[1, 0, 2, 1, 0, 2] from source indices Fin(6)={0,…,5} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(6, 3, [0, 3, 1], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (6, 3, [0, 3, 1], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0011/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 12. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[3, 2, 1, 0, 3, 2, 1] from source indices Fin(7)={0,…,6} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(7, 4, [0, 4, 3], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 4, [0, 4, 3], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0012/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 13. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 3, 4, 0, 1, 2, 3, 4] from source indices Fin(8)={0,…,7} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(8, 5, [0, 5, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 5, [0, 5, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0013/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 14. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[1, 3, 5, 1, 3, 5, 1, 3, 5] from source indices Fin(9)={0,…,8} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(9, 6, [0, 3, 1], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 6, [0, 3, 1], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0014/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 15. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[0, 3, 6, 2, 5, 1, 4, 0, 3, 6] from source indices Fin(10)={0,…,9} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(10, 7, [0, 7, 0], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (10, 7, [0, 7, 0], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0015/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 16. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[0, 1, 2, 0, 1, 2, 0] from source indices Fin(7)={0,…,6} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(7, 3, [0, 3, 0], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (7, 3, [0, 3, 0], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0016/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 17. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[0, 2, 0, 2, 0, 2, 0, 2] from source indices Fin(8)={0,…,7} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(8, 4, [0, 2, 0], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 4, [0, 2, 0], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0017/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 18. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 0, 3, 1, 4, 2, 0, 3, 1] from source indices Fin(9)={0,…,8} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(9, 5, [0, 5, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 5, [0, 5, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0018/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 19. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[0, 1, 2, 3, 4, 5, 0, 1, 2, 3] from source indices Fin(10)={0,…,9} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(10, 6, [0, 6, 0], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (10, 6, [0, 6, 0], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0019/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 20. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(11)={0,…,10} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(11, 7, [0, 7, 5], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (11, 7, [0, 7, 5], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0020/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 21. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 2, 2, 2, 2, 2, 2, 2] from source indices Fin(8)={0,…,7} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(8, 3, [0, 1, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (8, 3, [0, 1, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0021/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 22. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[1, 2, 3, 0, 1, 2, 3, 0, 1] from source indices Fin(9)={0,…,8} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(9, 4, [0, 4, 1], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (9, 4, [0, 4, 1], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0022/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 23. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4, 1, 3, 0] from source indices Fin(10)={0,…,9} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(10, 5, [0, 5, 2], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (10, 5, [0, 5, 2], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0023/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 24. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(11)={0,…,10} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(11, 6, [0, 2, 5], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (11, 6, [0, 2, 5], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0024/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 25. A Complete Larger-Domain Map Has Two Distinct Sources with One Image

Use the complete total mapping=[3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0] from source indices Fin(12)={0,…,11} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate).

View answer and explanation(12, 7, [0, 7, 3], Yes, complete_total_map_collision)

Return the first collision [left_index,right_index,image] in increasing left/right order with left<right. The full finite map provides an exact instance witness, not a theorem derived from partial samples. The exact result, retaining the requested order, is (12, 7, [0, 7, 3], Yes, complete_total_map_collision).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0025/)

Pigeonhole principle · qf-mod38-pigeon-collision-witness · candidate





## 26. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[0, 1, 2, 0] from source indices Fin(4)={0,…,3} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0026/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 27. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[1, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([0, 3, 0, 2], 3, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 3, 0, 2], 3, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0027/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 28. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 0, 3, 1, 4, 2] from source indices Fin(6)={0,…,5} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0028/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 29. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[3, 4, 5, 0, 1, 2, 3] from source indices Fin(7)={0,…,6} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0029/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 30. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(8)={0,…,7} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 1, 1, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0030/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 31. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 2, 2, 2, 2] from source indices Fin(5)={0,…,4} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([0, 0, 5], 5, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 5], 5, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0031/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 32. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 3, 0, 1, 2, 3] from source indices Fin(6)={0,…,5} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 1, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0032/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 33. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4] from source indices Fin(7)={0,…,6} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 1, 2, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0033/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 34. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(8)={0,…,7} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([0, 0, 4, 0, 0, 4], 4, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 4, 0, 0, 4], 4, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0034/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 35. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 3, 4, 5, 6, 0, 1, 2, 3] from source indices Fin(9)={0,…,8} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 1, 2, 2, 1, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 2, 1, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0035/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 36. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[1, 0, 2, 1, 0, 2] from source indices Fin(6)={0,…,5} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0036/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 37. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[3, 2, 1, 0, 3, 2, 1] from source indices Fin(7)={0,…,6} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0037/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 38. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 3, 4, 0, 1, 2, 3, 4] from source indices Fin(8)={0,…,7} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([1, 1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([1, 1, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0038/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 39. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[1, 3, 5, 1, 3, 5, 1, 3, 5] from source indices Fin(9)={0,…,8} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([0, 3, 0, 3, 0, 3], 3, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 3, 0, 3, 0, 3], 3, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0039/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 40. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[0, 3, 6, 2, 5, 1, 4, 0, 3, 6] from source indices Fin(10)={0,…,9} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 1, 1, 2, 1, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 1, 2, 1, 1, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0040/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 41. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[0, 1, 2, 0, 1, 2, 0] from source indices Fin(7)={0,…,6} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0041/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 42. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[0, 2, 0, 2, 0, 2, 0, 2] from source indices Fin(8)={0,…,7} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([4, 0, 4, 0], 4, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([4, 0, 4, 0], 4, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0042/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 43. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 0, 3, 1, 4, 2, 0, 3, 1] from source indices Fin(9)={0,…,8} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 2, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0043/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 44. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[0, 1, 2, 3, 4, 5, 0, 1, 2, 3] from source indices Fin(10)={0,…,9} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 2, 2, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2, 2, 1, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0044/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 45. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[5, 0, 2, 4, 6, 1, 3, 5, 0, 2, 4] from source indices Fin(11)={0,…,10} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 1, 2, 1, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 2, 1, 2, 2, 1], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0045/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 46. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 2, 2, 2, 2, 2, 2, 2] from source indices Fin(8)={0,…,7} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([0, 0, 8], 8, 3, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 8], 8, 3, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0046/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 47. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[1, 2, 3, 0, 1, 2, 3, 0, 1] from source indices Fin(9)={0,…,8} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 3, 2, 2], 3, 3, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0047/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 48. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[2, 4, 1, 3, 0, 2, 4, 1, 3, 0] from source indices Fin(10)={0,…,9} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0048/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 49. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5] from source indices Fin(11)={0,…,10} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([0, 0, 5, 0, 0, 6], 6, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([0, 0, 5, 0, 0, 6], 6, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0049/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 50. The Largest Fibre Is at Least the Exact Ceiling Average

Use the complete total mapping=[3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0] from source indices Fin(12)={0,…,11} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Give (loads, maximum_load, ceiling_lower_bound, bound_holds, certificate).

View answer and explanation([2, 1, 1, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound)

List every codomain load in target order. Compare its maximum with ceil(domain_count/codomain_count); the lower bound is not generally equal to the maximum load. The empty domain gives all loads0 and lower bound0 on a nonempty codomain. The exact result, retaining the requested order, is ([2, 1, 1, 2, 2, 2, 2], 2, 2, Yes, generalized_pigeonhole_load_bound).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0050/)

Pigeonhole principle · qf-mod38-pigeon-generalized-load · candidate





## 51. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=5≥0 and codomain size=3≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(5, 3, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (5, 3, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0051/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 52. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=6≥0 and codomain size=4≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(6, 4, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (6, 4, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0052/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 53. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=7≥0 and codomain size=5≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(7, 5, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (7, 5, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0053/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 54. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=8≥0 and codomain size=6≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(8, 6, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (8, 6, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0054/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 55. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=9≥0 and codomain size=7≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(9, 7, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (9, 7, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0055/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 56. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=10≥0 and codomain size=8≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(10, 8, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (10, 8, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0056/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 57. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=11≥0 and codomain size=9≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(11, 9, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (11, 9, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0057/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 58. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=12≥0 and codomain size=10≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(12, 10, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (12, 10, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0058/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 59. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=13≥0 and codomain size=11≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(13, 11, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (13, 11, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0059/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 60. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=14≥0 and codomain size=12≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(14, 12, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (14, 12, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0060/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 61. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=15≥0 and codomain size=13≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(15, 13, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (15, 13, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0061/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 62. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=16≥0 and codomain size=14≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(16, 14, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (16, 14, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0062/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 63. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=17≥0 and codomain size=15≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(17, 15, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (17, 15, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0063/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 64. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=18≥0 and codomain size=16≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(18, 16, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (18, 16, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0064/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 65. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=19≥0 and codomain size=17≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(19, 17, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (19, 17, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0065/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 66. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=20≥0 and codomain size=18≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(20, 18, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (20, 18, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0066/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 67. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=21≥0 and codomain size=19≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(21, 19, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (21, 19, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0067/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 68. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=22≥0 and codomain size=20≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(22, 20, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (22, 20, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0068/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 69. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=23≥0 and codomain size=21≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(23, 21, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (23, 21, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0069/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 70. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=24≥0 and codomain size=22≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(24, 22, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (24, 22, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0070/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 71. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=25≥0 and codomain size=23≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(25, 23, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (25, 23, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0071/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 72. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=26≥0 and codomain size=24≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(26, 24, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (26, 24, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0072/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 73. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=27≥0 and codomain size=25≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(27, 25, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (27, 25, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0073/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 74. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=28≥0 and codomain size=26≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(28, 26, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (28, 26, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0074/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 75. The Size Premise Forces a Collision but Has No General Reverse Inference

Use complete finite domain size=29≥0 and codomain size=27≥1, with a total map understood but no particular map table supplied. Give (domain_count, codomain_count, domain_larger, collision_forced_by_size, general_reverse_implication_claim, boundary).

View answer and explanation(29, 27, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity)

Compare domain_count>codomain_count and retain the corresponding size-forced-collision flag. The final No withholds a general reverse implication; no particular map is classified as noninjective from its failure. The exact result, retaining the requested order, is (29, 27, Yes, Yes, No, premise_guarantees_collision_but_converse_does_not_guarantee_injectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0075/)

Pigeonhole principle · qf-mod38-pigeon-premise-audit · candidate





## 76. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(4)={0,…,3}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(3, 4, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 4, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0076/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 77. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(4, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0077/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 78. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(5, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0078/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 79. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(6, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0079/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 80. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(7, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0080/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 81. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(5)={0,…,4}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(3, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 5, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0081/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 82. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(4, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0082/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 83. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(5, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0083/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 84. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(6, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0084/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 85. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(7, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0085/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 86. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(6)={0,…,5}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(3, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 6, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0086/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 87. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(4, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0087/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 88. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(5, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0088/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 89. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(6, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0089/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 90. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(10)={0,…,9}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(7, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0090/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 91. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(7)={0,…,6}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(3, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 7, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0091/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 92. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(4, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0092/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 93. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(5, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0093/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 94. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(10)={0,…,9}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(6, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0094/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 95. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(11)={0,…,10}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(7, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0095/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 96. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2] from source indices Fin(3)={0,…,2} into Fin(8)={0,…,7}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(3, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (3, 8, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0096/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 97. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3] from source indices Fin(4)={0,…,3} into Fin(9)={0,…,8}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(4, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (4, 9, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0097/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 98. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4] from source indices Fin(5)={0,…,4} into Fin(10)={0,…,9}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(5, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (5, 10, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0098/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 99. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5] from source indices Fin(6)={0,…,5} into Fin(11)={0,…,10}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(6, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (6, 11, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0099/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate





## 100. A Smaller Domain Does Not Generally Decide the Given Map Is Injective

Use the complete total mapping=[0, 0, 2, 3, 4, 5, 6] from source indices Fin(7)={0,…,6} into Fin(12)={0,…,11}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. Domain size is at most codomain size. Give (domain_count, codomain_count, given_map_injective, general_reverse_implication_claim, boundary).

The final No withholds a general reverse implication; the preceding flag separately checks this map. Empty or singleton domains may be injective.

View answer and explanation(7, 12, No, No, size_condition_failure_does_not_imply_given_map_is_injective)

Check this complete map for injectivity separately, then retain No for the general reverse implication. Empty or singleton source maps are injective, while some larger sources still collide; the nonclaim is not a forced No injectivity flag. The exact result, retaining the requested order, is (7, 12, No, No, size_condition_failure_does_not_imply_given_map_is_injective).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0100/)

Pigeonhole principle · qf-mod38-pigeon-converse-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations/validation-report.json)