Question 4084: Distance Order with Explicit Tie Groups

Original question and givens

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[9, 0], [8, 1], [8, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

• G001 — In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Indices 1,2,3 correspond to [[9, 0], [8, 1], [8, 0]]. List index groups in increasing distance order, with indices increasing inside each tie group.

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-distance-formula

Step-by-step derivation

integer vectors from a common origin;orthogonal equal-length units

L2(v)≥0 and L2(v)²=v₁²+v₂²

vector1 coordinate1 squared := square(9)

vector1 coordinate1 squared = 81

vector1 coordinate2 squared := square(0)

vector1 coordinate2 squared = 0

vector1 L2 squared := sum(81, 0)

vector1 L2 squared = 81

vector2 coordinate1 squared := square(8)

vector2 coordinate1 squared = 64

vector2 coordinate2 squared := square(1)

vector2 coordinate2 squared = 1

vector2 L2 squared := sum(64, 1)

vector2 L2 squared = 65

vector3 coordinate1 squared := square(8)

vector3 coordinate1 squared = 64

vector3 coordinate2 squared := square(0)

vector3 coordinate2 squared = 0

vector3 L2 squared := sum(64, 0)

vector3 L2 squared = 64

indexed squares=[(1, 81), (2, 65), (3, 64)]

[[3], [2], [1]]

Reverse checks

• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute vector3 L2 squared with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute vector3 coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute vector3 coordinate1 squared with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute vector2 L2 squared with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute vector2 coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute vector2 coordinate1 squared with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute vector1 L2 squared with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute vector1 coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute vector1 coordinate1 squared with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

[[3], [2], [1]]

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.