Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).
G001 — Use the complete total map Fin(0)=∅→Fin(0) with images=[] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (images, every codomain fiber, all lexicographically ordered distinct-input collision pairs, missing targets, injectivity, surjectivity, bijectivity).
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.
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
source roles = [Bijection ORIGINAL complete table, 0, 0, []]
After
derived = [[]]
Conditions: none
Require the WHOLE declared table length to equal its source size and EVERY cell to lie in Fin(m),including unused cells.Even an empty outer source cannot waive this complete table typing obligation.
A collision has x<y and equal images.List ALL pairs in GLOBAL numeric LEX order,not codomain-block order or first repeated output.No collision gives injectivity;no missing target gives surjectivity;BOTH give bijectivity.Empty->empty has all three;empty->nonempty only injectivity.
S003 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
Result
([], [], [], [], Yes, Yes, Yes)
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.