# Question 31417: Squared Euclidean Distance [Return to catalogue page 158](https://ocme.evemisslab.com/en/questions/page/158/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261005-advance-20261006-r04-distance-formula-foundations/#q-20261005-advance-20261006-r04-distance-formula-0017) 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. R0061-31417 · q-20261005-advance-20261006-r04-distance-formula-0017 ## Original question and givens Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 5). Find d(P,Q)² for the Euclidean L2 distance. - G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(4, -11) and Q=(4, 5). Find d(P,Q)² for the Euclidean L2 distance. ## 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 source roles = [qf-distance-squared, squared_distance, frame, euclidean_orthonormal_equal_units] After derived = [[integer_squared_norm]] Conditions: none Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status. Step identifiers and dependenciesRule: R61_SOURCE_exact_keys_operation_domain_OR_frame_and_output_roles · Uses: G001 S002 · calculationBefore Distance PQ ORIGINAL Q minus P axis0 := sub(4, 4) After Distance PQ ORIGINAL Q minus P axis0 = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Distance PQ ORIGINAL Q minus P axis1 := sub(5, -11) After Distance PQ ORIGINAL Q minus P axis1 = 16 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Distance PQ ORIGINAL signed delta squared axis0 := mul(0, 0) After Distance PQ ORIGINAL signed delta squared axis0 = 0 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Distance PQ ORIGINAL signed delta squared axis1 := mul(16, 16) After Distance PQ ORIGINAL signed delta squared axis1 = 256 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Distance PQ ORIGINAL sum of two squares := sum(0, 256) After Distance PQ ORIGINAL sum of two squares = 256 Conditions: none Retain ORIGINAL operands and order for exact integer arithmetic.Only raw inputs are SAFE-bounded;derived integers are not recapped or approximated. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Distance PQ, [4, -11], [4, 5]] After derived = [[0, 16], [0, 256], 256] Conditions: none In the DECLARED orthonormal equal-unit frame,distance squared is(Δx)²+(Δy)²≥0,including coincident points.Keep signed differences;do not mistake a signed component for distance. Step identifiers and dependenciesRule: Distance_ORTHONORMAL_equal_units_two_squares_NONNEGATIVE · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. - S006 · premise_and_result_audit — Recompute Distance PQ ORIGINAL sum of two squares with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Distance PQ ORIGINAL signed delta squared axis1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Distance PQ ORIGINAL signed delta squared axis0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Distance PQ ORIGINAL Q minus P axis1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Distance PQ ORIGINAL Q minus P axis0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof. ## Result 256 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: f6488bf9cc912825668f3f5c172717bb2e43faee7db7525a221727eefb236582 - candidate record SHA-256: 795c4d044c266631a8d61e5dad4d51c52f737f2aabcfb6bcd3d2306e989ed42f - semantic review receipt SHA-256: 5403317467ff7c6b74c18ae7211530d0156ebc67d63fc00a3e07b81a201d3d69 - question-first witness SHA-256: c2b5022aa77f803cd3e1ba9ce162a707da0281aa5bb5289111a97de51fe8c495 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r04-distance-formula-0017.json)