Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. element=4 is nonzero. Give (element,one,inverse,found?).
G001 — Use the same already qualified complete finite commutative unital-ring tables on Fin(5)={0,…,4}, addition=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] and multiplication=[[0, 0, 0, 0, 0], [0, 3, 4, 2, 1], [0, 4, 3, 1, 2], [0, 2, 1, 4, 3], [0, 1, 2, 3, 4]] in carrier-label order. The actual zero and one are distinct and need not be labels 0 and 1. The inverse audit is additional to, not a replacement for, those ring axioms. Division by zero is never defined. A finite field has prime-power order, not necessarily prime order. element=4 is nonzero. Give (element,one,inverse,found?).
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.
Even where the raw inverse oracle has looser guards,independently qualify the complete commutative unital parent declared by the source English scope,including every additive/distributive axiom and distinct identities.Do not skip them by name or inverse search alone.
Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation.
Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation.
Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation.
Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation.
Search the whole carrier for both products equaling the actual one.Nonzero is judged against the actual zero,not label0;zero has no inverse obligation.
Existence follows from full field qualification;read back both products to one.The frozen inverse/division families contain no undefined output.Do not fabricate null for a missing inverse or invert zero.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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
(4, 4, 4, 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.