Question 29634: The Coordinate Additive Inverse and Its Zero Sum

Original question and givens

Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-4, 1, -5]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

• G001 — Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. vector=[-4, 1, -5]. Give (vector,componentwise additive inverse,sum,all summed coordinates zero?).

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.

• mko-real-vector-space

Step-by-step derivation

source roles = [qf-str22-vectorspace-additive-inverse, vectorspace-additive-inverse]

derived = [[vector, inverse, sum, zero]]

Vector ORIGINAL negate coordinate0 := sub(0, -4)

Vector ORIGINAL negate coordinate0 = 4

Vector ORIGINAL negate coordinate1 := sub(0, 1)

Vector ORIGINAL negate coordinate1 = -1

Vector ORIGINAL negate coordinate2 := sub(0, -5)

Vector ORIGINAL negate coordinate2 = 5

Vector ORIGINAL inverse sum coordinate0 := sum(-4, 4)

Vector ORIGINAL inverse sum coordinate0 = 0

Vector ORIGINAL inverse sum coordinate1 := sum(1, -1)

Vector ORIGINAL inverse sum coordinate1 = 0

Vector ORIGINAL inverse sum coordinate2 := sum(-5, 5)

Vector ORIGINAL inverse sum coordinate2 = 0

source roles = [[-4, 1, -5]]

derived = [[4, -1, 5], [0, 0, 0], Yes]

Reverse checks

• S008 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.
• S007 · premise_and_result_audit — Recompute Vector ORIGINAL inverse sum coordinate2 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Vector ORIGINAL inverse sum coordinate1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Vector ORIGINAL inverse sum coordinate0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Vector ORIGINAL negate coordinate2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Vector ORIGINAL negate coordinate1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Vector ORIGINAL negate coordinate0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate.

Result

([-4, 1, -5], [4, -1, 5], [0, 0, 0], 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.