Use the complete total map Fin(0)=∅→Fin(2) 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 ([n,m], images, n=m?, every fiber, is the given map bijective?).
G001 — Use the complete total map Fin(0)=∅→Fin(2) 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 ([n,m], images, n=m?, every fiber, is the given map bijective?).
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, 2, []]
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.
Bijection ORIGINAL reverse ALL fiber sizes through target0 := sum(0, 0)
After
Bijection ORIGINAL reverse ALL fiber sizes through target0 = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Bijection ORIGINAL reverse ALL fiber sizes through target1 := sum(0, 0)
After
Bijection ORIGINAL reverse ALL fiber sizes through target1 = 0
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
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.
n=m removes a cardinality obstruction,but does not guarantee THIS table lacks collisions/missing values.Do not pre-reject a legal nonbijective map;classify the GIVEN map by its complete fibers.
S008 · 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.
S007 · 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.
S006 · 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.
S005 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target1 with exact rational arithmetic, not rounded decimals.
S004 · 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.
S003 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target0 with exact rational arithmetic, not rounded decimals.
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
([0, 2], [], No, [[], []], No)
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.