# Total Iff Maps, Directed Composition and Forward Decider Transfer

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-20261015-advance-20261115-r07-many-one-reduction-foundations · 2026-10-15 · 100 questions

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

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

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



## 1. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2], target carrier=[0, 1, 2], aligned mapping=[0, 1, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

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

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, null, No]], No, [2, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0001/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 2. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

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

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0002/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 3. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0003/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 4. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0004/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 5. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0005/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 6. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

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

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, null, No]], No, [7, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0006/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 7. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0007/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 8. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0008/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 9. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0009/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 10. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0010/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 11. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, null, No]], No, [12, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, null, No]], No, [12, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0011/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 12. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0012/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 13. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0013/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 14. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0014/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 15. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0015/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 16. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, null, No]], No, [17, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, null, No]], No, [17, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0016/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 17. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0017/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 18. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0018/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 19. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0019/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 20. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0020/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 21. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, null]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, null, No]], No, [22, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, null, No]], No, [22, null, No], many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0021/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 22. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0022/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 23. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes], [22, 2, Yes], [23, 0, Yes], [24, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes], [10, 2, Yes], [11, 0, Yes], [12, 1, Yes], [13, 2, Yes], [14, 0, Yes], [15, 1, Yes], [16, 2, Yes], [17, 0, Yes], [18, 1, Yes], [19, 2, Yes], [20, 0, Yes], [21, 1, Yes], [22, 2, Yes], [23, 0, Yes], [24, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0023/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 24. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1, 2], aligned mapping=[2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, 0, Yes], [23, 1, Yes], [24, 2, Yes], [25, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [0, 1, 2], [[0, 2, Yes], [1, 0, Yes], [2, 1, Yes], [3, 2, Yes], [4, 0, Yes], [5, 1, Yes], [6, 2, Yes], [7, 0, Yes], [8, 1, Yes], [9, 2, Yes], [10, 0, Yes], [11, 1, Yes], [12, 2, Yes], [13, 0, Yes], [14, 1, Yes], [15, 2, Yes], [16, 0, Yes], [17, 1, Yes], [18, 2, Yes], [19, 0, Yes], [20, 1, Yes], [21, 2, Yes], [22, 0, Yes], [23, 1, Yes], [24, 2, Yes], [25, 0, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0024/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 25. Complete Total Mapping Qualification Precedes Membership Equivalence

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1, 2], aligned mapping=[0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes], [24, 0, Yes], [25, 1, Yes], [26, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

For every domain value retain its image and whether it lies in the target. A null or outside image refutes totality, with the first invalid row retained. A total table by itself does not prove a reduction. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], [0, 1, 2], [[0, 0, Yes], [1, 1, Yes], [2, 2, Yes], [3, 0, Yes], [4, 1, Yes], [5, 2, Yes], [6, 0, Yes], [7, 1, Yes], [8, 2, Yes], [9, 0, Yes], [10, 1, Yes], [11, 2, Yes], [12, 0, Yes], [13, 1, Yes], [14, 2, Yes], [15, 0, Yes], [16, 1, Yes], [17, 2, Yes], [18, 0, Yes], [19, 1, Yes], [20, 2, Yes], [21, 0, Yes], [22, 1, Yes], [23, 2, Yes], [24, 0, Yes], [25, 1, Yes], [26, 2, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0025/)

Many-one reduction · qf-mod46-mreduction-total-map-qualification · candidate





## 26. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2], target carrier=[0, 1], aligned mapping=[0, 1, 0], source yes-set=[1], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

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

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0026/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 27. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

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

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0027/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 28. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

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

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0028/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 29. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

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

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0029/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 30. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

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

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0030/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 31. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

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

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0031/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 32. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

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

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0032/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 33. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0033/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 34. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0034/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 35. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0035/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 36. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0036/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 37. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0037/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 38. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0038/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 39. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0039/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 40. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0040/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 41. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0041/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 42. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0042/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 43. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0043/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 44. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0044/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 45. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0045/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 46. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0046/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 47. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0047/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 48. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0048/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 49. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0049/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 50. A Many-One Reduction Preserves Both Yes and No Membership

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

View answer and explanation([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

Require totality first. Retain each source predicate, image predicate and their equivalence, then compare the entire source yes-set with the inverse image of the target yes-set. Checking only yes instances is insufficient. This is finite-domain evidence, not an infinite computability proof. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0050/)

Many-one reduction · qf-mod46-mreduction-iff-audit · candidate





## 51. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1], [1], [11]], aligned maps A-to-B=[0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0051/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 52. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0052/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 53. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0053/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 54. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0054/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 55. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0055/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 56. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0056/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 57. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0057/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 58. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0058/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 59. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0059/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 60. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0060/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 61. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0061/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 62. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0062/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 63. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0063/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 64. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0064/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 65. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0065/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 66. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0066/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 67. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0067/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 68. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0068/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 69. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0069/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 70. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0070/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 71. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0071/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 72. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0072/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 73. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0073/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 74. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0074/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 75. Compose A-to-B then B-to-C without Reversing the Reduction

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], middle carrier=[0, 1], target carrier=[10, 11], yes-sets=[[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], [1], [11]], aligned maps A-to-B=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], B-to-C=[10, 11]. Give (first_membership_rows, second_membership_rows, composed_images, composed_membership_rows, composed_reduction, certificate).

View answer and explanation([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes], [26, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only)

Qualify both total reductions, then apply the second map to the first image for each source value. Retain both initial membership audits and the composed audit. A≤mB and B≤mC imply A≤mC only in that direction; no reverse decider transfer is inferred. The exact result, retaining the requested order, is ([[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes]], [10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10], [[0, No, 10, No, Yes], [1, Yes, 11, Yes, Yes], [2, No, 10, No, Yes], [3, Yes, 11, Yes, Yes], [4, No, 10, No, Yes], [5, Yes, 11, Yes, Yes], [6, No, 10, No, Yes], [7, Yes, 11, Yes, Yes], [8, No, 10, No, Yes], [9, Yes, 11, Yes, Yes], [10, No, 10, No, Yes], [11, Yes, 11, Yes, Yes], [12, No, 10, No, Yes], [13, Yes, 11, Yes, Yes], [14, No, 10, No, Yes], [15, Yes, 11, Yes, Yes], [16, No, 10, No, Yes], [17, Yes, 11, Yes, Yes], [18, No, 10, No, Yes], [19, Yes, 11, Yes, Yes], [20, No, 10, No, Yes], [21, Yes, 11, Yes, Yes], [22, No, 10, No, Yes], [23, Yes, 11, Yes, Yes], [24, No, 10, No, Yes], [25, Yes, 11, Yes, Yes], [26, No, 10, No, Yes]], Yes, A_le_m_B_and_B_le_m_C_imply_A_le_m_C_in_that_direction_only).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0075/)

Many-one reduction · qf-mod46-mreduction-composition-direction · candidate





## 76. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2], target carrier=[0, 1], aligned mapping=[0, 1, 0], source yes-set=[1], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

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

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0076/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 77. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

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

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0077/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 78. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, null], source yes-set=[1, 3], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

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

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0078/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 79. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

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

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0079/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 80. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

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

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0080/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 81. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

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

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0081/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 82. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

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

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0082/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 83. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0083/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 84. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0084/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 85. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0085/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 86. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0086/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 87. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0087/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 88. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0088/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 89. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0089/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 90. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0090/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 91. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0091/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 92. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0092/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 93. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0093/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 94. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0094/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 95. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0095/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 96. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0096/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 97. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=reverse_misuse. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (reverse_misuse, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes], Yes, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0097/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 98. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, null], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=partial_map. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (partial_map, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, null, null, No]], No, No, [No, Yes], Yes, null, No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0098/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 99. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], target carrier=[0, 1], aligned mapping=[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], source yes-set=[0], target yes-set=[0]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[true, false], aligned with the supplied target carrier. Mode=one_way_yes_only. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No], [22, No, 0, Yes, No], [23, No, 0, Yes, No], [24, No, 0, Yes, No], [25, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (one_way_yes_only, [[0, Yes, 0, Yes, Yes], [1, No, 0, Yes, No], [2, No, 0, Yes, No], [3, No, 0, Yes, No], [4, No, 0, Yes, No], [5, No, 0, Yes, No], [6, No, 0, Yes, No], [7, No, 0, Yes, No], [8, No, 0, Yes, No], [9, No, 0, Yes, No], [10, No, 0, Yes, No], [11, No, 0, Yes, No], [12, No, 0, Yes, No], [13, No, 0, Yes, No], [14, No, 0, Yes, No], [15, No, 0, Yes, No], [16, No, 0, Yes, No], [17, No, 0, Yes, No], [18, No, 0, Yes, No], [19, No, 0, Yes, No], [20, No, 0, Yes, No], [21, No, 0, Yes, No], [22, No, 0, Yes, No], [23, No, 0, Yes, No], [24, No, 0, Yes, No], [25, No, 0, Yes, No]], Yes, No, [Yes, No], Yes, [Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes, Yes], No, No, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0099/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate





## 100. Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Target decider truth table=[false, true], aligned with the supplied target carrier. Mode=valid_transfer. Give (mode, membership_rows, total, reduction, target_decider, target_decider_correct, constructed_source_decider_or_null, constructed_source_decider_correct, requested_forward_transfer_certificate, reverse_or_infinite_computability_claim, boundary).

All four modes retain11 fields. In reverse_misuse the constructed A decider may be correct while requested_forward_transfer_certificate is No because reverse use is rejected. A partial map gives null, not an empty constructed table. Finite tables do not prove infinite-domain computability.

View answer and explanation(valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability)

Retain reduction and target-decider correctness separately, constructing the source table only for a total map. valid_transfer certifies the forward use. reverse_misuse may still construct a correct source decider but rejects reverse use. partial_map gives null, and one_way_yes_only can preserve yes cases while failing no cases. Finite tables certify only these finite domains, not infinite computability or a Turing reduction. The exact result, retaining the requested order, is (valid_transfer, [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes], [20, No, 0, No, Yes], [21, Yes, 1, Yes, Yes], [22, No, 0, No, Yes], [23, Yes, 1, Yes, Yes], [24, No, 0, No, Yes], [25, Yes, 1, Yes, Yes], [26, No, 0, No, Yes]], Yes, Yes, [No, Yes], Yes, [No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No, Yes, No], Yes, Yes, No, A_le_m_B_plus_B_decidable_transfers_decidability_to_A_not_the_reverse_partial_maps_and_yes_only_checks_are_invalid_and_finite_tables_do_not_prove_infinite_computability).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261115-r07-many-one-reduction-0100/)

Many-one reduction · qf-mod46-mreduction-decider-transfer-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r07-many-one-reduction-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261015-advance-20261115-r07-many-one-reduction-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r07-many-one-reduction-foundations/validation-report.json)