# Question 4096: Distance Order with Explicit Tie Groups [Return to catalogue page 21](https://ocme.evemisslab.com/en/questions/page/21/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r03-distance-formula-foundations/#q-20261008-advance-20261010-r03-distance-formula-0071) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0007-04096 · q-20261008-advance-20261010-r03-distance-formula-0071 ## 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 [[21, 0], [20, 1], [20, 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 [[21, 0], [20, 1], [20, 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](https://ocme.evemisslab.com/en/mko/mko-distance-formula/) ## Step-by-step derivation S001 · impliesBefore integer vectors from a common origin;orthogonal equal-length units After L2(v)≥0 and L2(v)²=v₁²+v₂² Conditions: none A common origin,orthogonality and equal units support the sum-of-squares formula;coordinates may be negative but distances may not. Step identifiers and dependenciesRule: euclidean_distance_frame · Uses: G001 S002 · calculationBefore vector1 coordinate1 squared := square(21) After vector1 coordinate1 squared = 441 Conditions: none Square the entire coordinate,including a negative sign inside the square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore vector1 coordinate2 squared := square(0) After vector1 coordinate2 squared = 0 Conditions: none Square the entire coordinate,including a negative sign inside the square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore vector1 L2 squared := sum(441, 0) After vector1 L2 squared = 441 Conditions: none Add the coordinate squares to obtain squared distance,not a sum of the coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore vector2 coordinate1 squared := square(20) After vector2 coordinate1 squared = 400 Conditions: none Square the entire coordinate,including a negative sign inside the square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore vector2 coordinate2 squared := square(1) After vector2 coordinate2 squared = 1 Conditions: none Square the entire coordinate,including a negative sign inside the square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore vector2 L2 squared := sum(400, 1) After vector2 L2 squared = 401 Conditions: none Add the coordinate squares to obtain squared distance,not a sum of the coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore vector3 coordinate1 squared := square(20) After vector3 coordinate1 squared = 400 Conditions: none Square the entire coordinate,including a negative sign inside the square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore vector3 coordinate2 squared := square(0) After vector3 coordinate2 squared = 0 Conditions: none Square the entire coordinate,including a negative sign inside the square. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore vector3 L2 squared := sum(400, 0) After vector3 L2 squared = 400 Conditions: none Add the coordinate squares to obtain squared distance,not a sum of the coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore indexed squares=[(1, 441), (2, 401), (3, 400)] After [[3], [2], [1]] Conditions: none Nonnegative distances have the same order as their squares.Group equal squares and retain increasing original1-based indices within ties,not coordinate lexicographic order. Step identifiers and dependenciesRule: distance_square_order_and_ties · Uses: G001, S010 ## 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 274114eeb8a133ff3138b9f6dcb5d6fe60ef98de2aa621883c7265cdefb1cc2b - candidate record SHA-256: 4200588143212cdc6e105a316ef33a466dd38d6182a3fefe2311899194e73f93 - semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a - question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r03-distance-formula-0071.json)