First f, then g: Fin(0)=∅, middle=Fin(0)=∅, target=Fin(0)=∅. Complete f=[]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.
G001 — First f, then g: Fin(0)=∅, middle=Fin(0)=∅, target=Fin(0)=∅. Complete f=[]; g has original ordered dom=[] paired with g values=[]. Give ([n,m,k], sorted dom, allowed source indices, aligned outputs, rejected source indices); retain the original dom/g pairing.
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.
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.
source roles = [g ORIGINAL dom-slot 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.
Retain the COMPLETE allowed/rejected partition of ORIGINAL source inputs,aligning outputs to allowed indices.An empty dom may reject every input;g length is the dom length,NOT m.
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 — 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
([0, 0, 0], [], [], [], [])
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.