Use the complete total mapping=[1, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. 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, 3, 1, 3, 1] from source indices Fin(5)={0,…,4} into Fin(4)={0,…,3}. 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.
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.
ORIGINAL complete source-indexed total map=[1, 3, 1, 3, 1];source Fin(5);ACTUAL target Fin(4)
After
every source has ONE in-range image;all target labels0..3 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.
next LEX ordered original source pair=[0, 1] with left<right;original images=[1, 3]
After
same target image=No;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
next LEX ordered original source pair=[0, 2] with left<right;original images=[1, 1]
After
same target image=Yes;every earlier LEFT-then-RIGHT pair already checked unequal
Conditions: none
Check left first and then right;the first equal pair is the source's first lexicographic pair,which may differ from the earliest repeat encountered by increasing right endpoint.
ORIGINAL full domain size=5>target size=4;actual first collision=[0, 2, 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.
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.