# Finite Principal-Filter Images and Direction Guards

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

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

batch-20261008-advance-20261011-r07-tends-to-relation-foundations · 2026-10-08 · 100 questions

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



## 1. Finite Principal-Filter Image Generator

The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0001/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 2. Finite Principal-Filter Image Generator

The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0002/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 3. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[1]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0003/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 4. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 1]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0004/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 5. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[2]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0005/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 6. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 2]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0006/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 7. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[1, 2]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0007/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 8. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 1, 2]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0008/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 9. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0009/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 10. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0010/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 11. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[1, 3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0011/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 12. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 1, 3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0012/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 13. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[2, 3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0013/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 14. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 2, 3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0014/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 15. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[1, 2, 3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0015/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 16. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 1, 2, 3]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0016/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 17. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0017/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 18. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0018/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 19. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[1, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0019/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 20. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 1, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0020/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 21. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[2, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0021/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 22. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 2, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0022/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 23. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[1, 2, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0023/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 24. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[0, 1, 2, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0024/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 25. Finite Principal-Filter Image Generator

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

View answer and explanation[3, 4]

Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0025/)

Tends-to relation · qf-density11-tends-principal-image · candidate





## 26. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0026/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 27. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0027/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 28. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[1]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0028/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 29. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 1]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0029/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 30. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[2]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0030/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 31. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 2]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0031/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 32. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[1, 2]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0032/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 33. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 1, 2]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0033/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 34. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0034/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 35. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0035/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 36. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[1, 3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0036/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 37. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 1, 3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0037/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 38. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[2, 3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0038/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 39. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 2, 3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0039/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 40. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[1, 2, 3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0040/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 41. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 1, 2, 3]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0041/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 42. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0042/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 43. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0043/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 44. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[1, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0044/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 45. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 1, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0045/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 46. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[2, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0046/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 47. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 2, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0047/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 48. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[1, 2, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0048/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 49. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[0, 1, 2, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0049/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 50. Finite Principal-Filter Direction Obstructions

The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 1, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

View answer and explanation[3, 4]

An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0050/)

Tends-to relation · qf-density11-tends-principal-obstructions · candidate





## 51. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[], ordered candidates=[-1, 0, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0051/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 52. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 4, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0052/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 53. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0053/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 54. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[], ordered candidates=[0, 1, 2]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[Yes, Yes, Yes]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, Yes].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0054/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 55. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 4, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0055/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 56. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0056/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 57. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 2, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[Yes, Yes, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0057/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 58. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 4, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0058/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 59. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 0, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0059/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 60. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 2, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, Yes, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0060/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 61. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 101, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[Yes, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0061/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 62. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0062/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 63. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 2, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, Yes, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0063/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 64. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 101, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, Yes]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, Yes].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0064/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 65. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 4, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[Yes, No, Yes]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, Yes].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0065/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 66. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 2, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, Yes, Yes]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, Yes].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0066/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 67. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 101, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0067/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 68. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 4, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0068/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 69. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 0, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[Yes, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0069/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 70. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 101, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0070/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 71. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0071/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 72. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 2, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[Yes, Yes, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0072/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 73. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 4, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0073/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 74. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, No, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0074/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 75. Typed Principal-Filter Obstruction Candidates

The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 2, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.

View answer and explanation[No, Yes, No]

Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0075/)

Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate





## 76. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(0)→Fin(0) has after=[]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([], [])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0076/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 77. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(4)→Fin(3) has images=[0, 0, 1, 2] indexed from 0; Fin(4)={0,…,3}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(3)→Fin(8) has after=[0, 0, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0077/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 78. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([1], [1])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0078/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 79. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0079/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 80. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2], [2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0080/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 81. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2], [0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0081/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 82. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2], [1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0082/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 83. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0083/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 84. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([3], [3])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([3], [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0084/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 85. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([0, 3], [0, 3])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 3], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0085/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 86. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([1, 3], [1, 3])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 3], [1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0086/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 87. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0087/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 88. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2, 3], [2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0088/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 89. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2, 3], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0089/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 90. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([1, 2, 3], [1, 2, 3])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2, 3], [1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0090/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 91. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0091/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 92. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([4], [4])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([4], [4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0092/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 93. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([0, 4], [0, 4])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 4], [0, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0093/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 94. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([1, 4], [1, 4])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 4], [1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0094/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 95. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 4], [0, 1, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0095/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 96. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([2, 4], [2, 4])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2, 4], [2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0096/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 97. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2, 4], [0, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0097/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 98. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([1, 2, 4], [1, 2, 4])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2, 4], [1, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0098/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 99. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

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

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2, 4], [0, 1, 2, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0099/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate





## 100. Two Finite Principal-Filter Composition Images

The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

View answer and explanation([3, 4], [3, 4])

Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([3, 4], [3, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r07-tends-to-relation-0100/)

Tends-to relation · qf-density11-tends-principal-composition-images · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r07-tends-to-relation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261011-r07-tends-to-relation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r07-tends-to-relation-foundations/validation-report.json)