# Question 20311: A Complete Larger-Domain Map Has Two Distinct Sources with One Image [Return to catalogue page 102](https://ocme.evemisslab.com/en/questions/page/102/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261107-r05-pigeonhole-principle-foundations/#q-20261014-advance-20261107-r05-pigeonhole-principle-0011) 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. R0039-20311 · q-20261014-advance-20261107-r05-pigeonhole-principle-0011 ## Original question and givens Use the complete total mapping=[1, 0, 2, 1, 0, 2] from source indices Fin(6)={0,…,5} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate). - G001 — Use the complete total mapping=[1, 0, 2, 1, 0, 2] from source indices Fin(6)={0,…,5} into Fin(3)={0,…,2}. The codomain is nonempty; every source has exactly one in-range image. Compute full fibre loads and collisions, not a sample. The domain is strictly larger than the codomain. Give (domain_count, codomain_count, collision, has_collision, certificate). ## 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-pigeonhole-principle](https://ocme.evemisslab.com/en/mko/mko-pigeonhole-principle/) ## Step-by-step derivation S001 · impliesBefore source operation=pigeon-collision-witness;original snapshot and mathematics retained After explicit ordered source output fields=[domain_count, codomain_count, collision, has_collision, certificate] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL complete source-indexed total map=[1, 0, 2, 1, 0, 2];source Fin(6);ACTUAL target Fin(3) After every source has ONE in-range image;all target labels0..2 retained even if unused;empty source allowed Conditions: none Retain original source order and target labels rather than relabeling bins by first appearance.Complete total-map data support full fibres and collisions,including an empty source into a nonempty codomain. Step identifiers and dependenciesRule: pigeon_complete_total_map_not_sample_or_bin_relabeling · Uses: G001, S001 S003 · impliesBefore next LEX ordered original source pair=[0, 1] with lefttarget size=3;actual first collision=[0, 3, 1] After two DISTINCT original sources have the SAME original image;exact instance witness and finite size theorem agree Conditions: none The two source indices in the witness are distinct.This is an actual collision in a complete total map,consistent with the size argument for every map with m>n,not a global inference from partial samples. Step identifiers and dependenciesRule: pigeon_actual_distinct_source_collision_not_equal_indices · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (6, 3, [0, 3, 1], Yes, complete_total_map_collision) 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: 0c55d9ae62be28ed039cd14bc4c3756e4a6f95773f8632e224bc37dc0250e870 - candidate record SHA-256: 956b43bd62ebae76c7b3c7ce0173c52bc871961e16b120e455c8f33f71b3ba8a - semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034 - question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r05-pigeonhole-principle-0011.json)