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R0051-26151 · q-20261015-advance-20261115-r07-many-one-reduction-0026
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).
G001 — 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).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.
Before
source roles = [mreduction-iff-audit]
After
derived = [[source_yes, target_yes, membership_rows, reduction, certificate]]
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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [Reduction ORIGINAL map, [0, 1, 2], [0, 1], [0, 1, 0]]
After
derived = [[0, 1, 2], [0, 1], [0, 1, 0], [[0, 0, Yes], [1, 1, Yes], [2, 0, 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.
Rule: Reduction_original_domain_IMAGE_pairs_sorted_together_STRICT_types · Uses: G001, S001
Before
source roles = [Reduction ORIGINAL map, [1], [1], [0, 1, 2], [0, 1], [0, 1, 0]]
After
derived = [[[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, 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.
Rule: Reduction_EVERY_yes_AND_no_membership_IFF_not_yes_only · Uses: G001, S002
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.([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)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ff1f0002c4fb6888e2c895adecc4b84abe12216d29f202689bdfa3935a1ad2eccandidate record SHA-256: 77aa640f2bc546d94dfebe3609060966e0ebb2fb6f0996dde4b30f5e9514b143semantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED