# Complete Finite-Map Collisions and Original Source Labels

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

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

batch-20261012-advance-20261016-r06-injective-foundations · 2026-10-12 · 100 questions

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



## 1. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0001/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 2. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0002/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 3. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 1, 2]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0003/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 4. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0004/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 5. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0005/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 6. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0006/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 7. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], [[0, 1], [2, 3], [2, 4], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0007/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 8. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [[0, 1], [0, 3], [0, 4], [1, 3], [1, 4], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0008/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 9. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], [[0, 2], [1, 3], [1, 4], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0009/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 10. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], [[0, 3], [0, 4], [1, 2], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0010/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 11. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], [[0, 1], [0, 2], [1, 2], [3, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0011/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 12. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 1, 0], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0012/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 13. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 1, 0], [[0, 3], [1, 2], [1, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0013/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 14. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0], [[0, 2], [0, 4], [1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0014/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 15. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 1, 0], [[0, 1], [0, 3], [1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0015/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 16. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 1, 0], [[0, 1], [0, 4], [1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0016/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 17. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 1, 0], [[0, 2], [0, 3], [1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0017/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 18. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 1, 0], [[0, 4], [1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0018/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 19. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 1, 0], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0019/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 20. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0020/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 21. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [[0, 4], [1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0021/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 22. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [[0, 2], [0, 3], [1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0022/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 23. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [[0, 1], [0, 4], [1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0023/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 24. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [[0, 1], [0, 3], [1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0024/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 25. Every Distinct-Input Collision and Injectivity

Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.

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

Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [[0, 2], [0, 4], [1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0025/)

Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate





## 26. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], [], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0026/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 27. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [[0, 1]], [1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0027/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 28. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(4)={0,…,3}→Fin(2), complete images=[0, 0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 1], [[0, 1], [2, 3]], [1, 2, 3], [[2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0028/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 29. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], [1, 2, 3, 4], [[1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0029/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 30. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [[0, 4], [1, 2], [1, 3], [2, 3]], [0, 2, 3, 4], [[0, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0030/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 31. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [[1, 2], [1, 3], [2, 3]], [0, 1, 3, 4], [[1, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0031/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 32. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [[0, 2], [0, 3], [1, 4], [2, 3]], [0, 1, 2, 4], [[0, 2], [1, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0032/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 33. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [[0, 1], [0, 4], [1, 4], [2, 3]], [0, 1, 2, 3], [[0, 1], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0033/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 34. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [[1, 4], [2, 3]], [1, 2, 3, 4], [[1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0034/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 35. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [[0, 2], [0, 3], [2, 3]], [0, 2, 3, 4], [[0, 2], [0, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0035/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 36. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [[0, 4], [2, 3]], [0, 1, 3, 4], [[0, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0036/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 37. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [[0, 1], [2, 3]], [0, 1, 2, 4], [[0, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0037/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 38. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [[0, 1], [0, 3], [1, 3], [2, 4]], [0, 1, 2, 3], [[0, 1], [0, 3], [1, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0038/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 39. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [[0, 2], [0, 4], [1, 3], [2, 4]], [1, 2, 3, 4], [[1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0039/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 40. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [[1, 3], [2, 4]], [0, 2, 3, 4], [[2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0040/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 41. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [[0, 3], [1, 2], [1, 4], [2, 4]], [0, 1, 3, 4], [[0, 3], [1, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0041/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 42. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], [0, 1, 2, 4], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0042/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 43. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [[1, 2], [1, 4], [2, 4]], [0, 1, 2, 3], [[1, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0043/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 44. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [[0, 3], [2, 4]], [1, 2, 3, 4], [[2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0044/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 45. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [[0, 2], [0, 4], [2, 4]], [0, 2, 3, 4], [[0, 2], [0, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0045/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 46. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [[0, 1], [2, 4]], [0, 1, 3, 4], [[0, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0046/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 47. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [[0, 1], [0, 3], [1, 3]], [0, 1, 2, 4], [[0, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0047/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 48. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [[0, 4], [1, 3]], [0, 1, 2, 3], [[1, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0048/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 49. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [[0, 2], [1, 3]], [1, 2, 3, 4], [[1, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0049/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 50. Input Deletion with Original Collision Labels Retained

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.

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

Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [[0, 3], [1, 4]], [0, 2, 3, 4], [[0, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0050/)

Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate





## 51. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

View answer and explanation([], No, outside-codomain, not-evaluated, not-evaluated)

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], No, outside-codomain, not-evaluated, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0051/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 52. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(0)=∅→Fin(1), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], Yes, [0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0052/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 53. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], Yes, [0, 1, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0053/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 54. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(2)={0,…,1}→Fin(2), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], Yes, [0, 1, 0], [[0, 2]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0054/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 55. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(2)={0,…,1}→Fin(2), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

View answer and explanation([0, 1], No, outside-codomain, not-evaluated, not-evaluated)

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], No, outside-codomain, not-evaluated, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0055/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 56. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(8)={0,…,7}→Fin(8), complete images=[0, 1, 2, 3, 4, 5, 6, 7]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=7. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], Yes, [0, 1, 2, 3, 4, 5, 6, 7, 7], [[7, 8]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0056/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 57. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], Yes, [0, 0, 0, 0, 1, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [4, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0057/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 58. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], Yes, [1, 0, 0, 0, 1, 2], [[0, 4], [1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0058/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 59. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

View answer and explanation([2, 0, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0059/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 60. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], Yes, [0, 1, 0, 0, 1, 0], [[0, 2], [0, 3], [0, 5], [1, 4], [2, 3], [2, 5], [3, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0060/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 61. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], Yes, [1, 1, 0, 0, 1, 1], [[0, 1], [0, 4], [0, 5], [1, 4], [1, 5], [2, 3], [4, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0061/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 62. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], Yes, [2, 1, 0, 0, 1, 2], [[0, 5], [1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0062/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 63. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

View answer and explanation([0, 2, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0063/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 64. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], Yes, [1, 2, 0, 0, 1, 0], [[0, 4], [2, 3], [2, 5], [3, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0064/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 65. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], Yes, [2, 2, 0, 0, 1, 1], [[0, 1], [2, 3], [4, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0065/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 66. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], Yes, [0, 0, 1, 0, 1, 2], [[0, 1], [0, 3], [1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0066/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 67. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

View answer and explanation([1, 0, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0067/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 68. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], Yes, [2, 0, 1, 0, 1, 0], [[1, 3], [1, 5], [2, 4], [3, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0068/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 69. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], Yes, [0, 1, 1, 0, 1, 1], [[0, 3], [1, 2], [1, 4], [1, 5], [2, 4], [2, 5], [4, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0069/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 70. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], Yes, [1, 1, 1, 0, 1, 2], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0070/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 71. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

View answer and explanation([2, 1, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0071/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 72. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], Yes, [0, 2, 1, 0, 1, 0], [[0, 3], [0, 5], [2, 4], [3, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0072/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 73. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], Yes, [1, 2, 1, 0, 1, 1], [[0, 2], [0, 4], [0, 5], [2, 4], [2, 5], [4, 5]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0073/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 74. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], Yes, [2, 2, 1, 0, 1, 2], [[0, 1], [0, 5], [1, 5], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0074/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 75. Append Candidate Guard before New Table and Collisions

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

View answer and explanation([0, 0, 2, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)

Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0075/)

Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate





## 76. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0076/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 77. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[2, 0] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [0, 2], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0077/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 78. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[1, 0] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [0, 1], [[0, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0078/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 79. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0079/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 80. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0080/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 81. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 2], [1, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0081/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 82. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 3], [1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0082/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 83. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0083/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 84. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0084/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 85. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 3], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0085/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 86. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0086/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 87. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [2, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0087/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 88. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 3], [1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0088/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 89. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0089/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 90. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[1, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0090/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 91. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [1, 2], [1, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0091/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 92. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0092/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 93. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[1, 2], [1, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0093/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 94. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0094/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 95. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 4], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0095/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 96. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [2, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0096/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 97. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 3], [1, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0097/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 98. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [1, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0098/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 99. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [1, 3]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0099/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate





## 100. Typed Source Restriction and Complete Collision Witnesses

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).

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

Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [1, 4]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261012-advance-20261016-r06-injective-0100/)

Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r06-injective-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261012-advance-20261016-r06-injective-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261012-advance-20261016-r06-injective-foundations/validation-report.json)