# Question 26127: Forward Decider Transfer Needs a Total Iff Reduction and a Correct Target Decider [Return to catalogue page 131](https://ocme.evemisslab.com/en/questions/page/131/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r07-many-one-reduction-foundations/#q-20261015-advance-20261115-r07-many-one-reduction-0077) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0051-26127 · q-20261015-advance-20261115-r07-many-one-reduction-0077 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-many-one-reduction](https://ocme.evemisslab.com/en/mko/mko-many-one-reduction/) ## Step-by-step derivation S001 · impliesBefore source roles = [mreduction-decider-transfer-boundary] After derived = [[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]] Conditions: none List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [Reduction ORIGINAL map, [0, 1, 2, 3], [0, 1], [0, 1, 0, 1]] After derived = [[0, 1, 2, 3], [0, 1], [0, 1, 0, 1], [[0, 0, Yes], [1, 1, Yes], [2, 0, Yes], [3, 1, Yes]], Yes, null] Conditions: none Sort source values numerically with their aligned images,and target values numerically.Null,outside and Boolean images are not valid integer target values;False is not0.Qualification retains actual raw images and the first invalid row,not an invented total map. Step identifiers and dependenciesRule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S001 S003 · impliesBefore source roles = [Reduction ORIGINAL map, [1, 3], [1], [0, 1, 2, 3], [0, 1], [0, 1, 0, 1]] After derived = [[[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]], Yes, Yes] Conditions: none For every x audit x in A iff f(x) in B,in both yes and no directions.Invalid images have null target predicates,notFalse.Totality and full membership equivalence jointly qualify this finite reduction. Step identifiers and dependenciesRule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S002 S004 · impliesBefore source roles = [reverse_misuse, [0, 1], [No, Yes], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes]]] After derived = [[0, 1], [No, Yes], Yes, [No, Yes, No, Yes], Yes, No, No] Conditions: none Align the exact Boolean decider with ORIGINAL target values before reordering.Construct A only for a total map;partial gives null.reverse_misuse may construct a correct A table while rejecting reverse use.Yes-only may be partial/vacuous;do not add a totality premise.Finite tables certify these domains,not infinite computability. Step identifiers and dependenciesRule: Reduction_original_TARGET_decider_alignment_FORWARD_transfer_mode_boundary · Uses: G001, S003 ## Reverse checks - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (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) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2b50de37e300751ffc54556bc35c7c67b09c5c3e5e01b31a5f94e909d3fb05d5 - candidate record SHA-256: d6e559e2981654940ccc164014a4048db22166c1388976c885898755f7e58d3c - semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3 - question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5c - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r07-many-one-reduction-0077.json)