# Question 30170: Ordered Integer Principal Segment Lengths [Return to catalogue page 151](https://ocme.evemisslab.com/en/questions/page/151/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r06-euclidean-length-foundations/#q-20261008-advance-20261012-r06-euclidean-length-0045) 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. R0059-30170 · q-20261008-advance-20261012-r06-euclidean-length-0045 ## Original question and givens Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [51, 68], [56, 80]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers. - G001 — Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [51, 68], [56, 80]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers. ## 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-euclidean-length](https://ocme.evemisslab.com/en/mko/mko-euclidean-length/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density12-length-integer-distance-pair, density12_length_integer_distance_pair, integer_euclidean_plane] After derived = [[nonnegative_distance_PQ, nonnegative_distance_QR]] Conditions: none Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources. Step identifiers and dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · calculationBefore Length segment0 ORIGINAL Q minus P coordinate0 := sub(51, 0) After Length segment0 ORIGINAL Q minus P coordinate0 = 51 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Length segment0 ORIGINAL Q minus P coordinate1 := sub(68, 0) After Length segment0 ORIGINAL Q minus P coordinate1 = 68 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Length segment0 ORIGINAL square coordinate0 := mul(51, 51) After Length segment0 ORIGINAL square coordinate0 = 2601 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Length segment0 ORIGINAL square coordinate1 := mul(68, 68) After Length segment0 ORIGINAL square coordinate1 = 4624 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Length segment0 ORIGINAL squared norm sum := sum(2601, 4624) After Length segment0 ORIGINAL squared norm sum = 7225 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Length segment0, [0, 0], [51, 68]] After derived = [[51, 68], 7225] Conditions: none In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs. Step identifiers and dependenciesRule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S006 S008 · calculationBefore Length segment0 candidate nonnegative root squared := mul(85, 85) After Length segment0 candidate nonnegative root squared = 7225 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Length segment0 candidate next integer := sum(85, 1) After Length segment0 candidate next integer = 86 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Length segment0 next integer squared := mul(86, 86) After Length segment0 next integer squared = 7396 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [Length segment0, 7225] After derived = [85, 7225, 7396, Yes] Conditions: none The exact certificate r>=0,r²=N and(r+1)²>N identifies the UNIQUE nonnegative principal root.This slice requires an actual perfect square.Do not round a float or return both±r as distances;derived r is not SAFE-capped. Step identifiers and dependenciesRule: Length_EXACT_perfect_square_certificate_NONNEGATIVE_principal_root · Uses: G001, S010 S012 · calculationBefore Length segment1 ORIGINAL Q minus P coordinate0 := sub(56, 51) After Length segment1 ORIGINAL Q minus P coordinate0 = 5 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Length segment1 ORIGINAL Q minus P coordinate1 := sub(80, 68) After Length segment1 ORIGINAL Q minus P coordinate1 = 12 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Length segment1 ORIGINAL square coordinate0 := mul(5, 5) After Length segment1 ORIGINAL square coordinate0 = 25 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Length segment1 ORIGINAL square coordinate1 := mul(12, 12) After Length segment1 ORIGINAL square coordinate1 = 144 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Length segment1 ORIGINAL squared norm sum := sum(25, 144) After Length segment1 ORIGINAL squared norm sum = 169 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [Length segment1, [51, 68], [56, 80]] After derived = [[5, 12], 169] Conditions: none In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs. Step identifiers and dependenciesRule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S016 S018 · calculationBefore Length segment1 candidate nonnegative root squared := mul(13, 13) After Length segment1 candidate nonnegative root squared = 169 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Length segment1 candidate next integer := sum(13, 1) After Length segment1 candidate next integer = 14 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Length segment1 next integer squared := mul(14, 14) After Length segment1 next integer squared = 196 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore source roles = [Length segment1, 169] After derived = [13, 169, 196, Yes] Conditions: none The exact certificate r>=0,r²=N and(r+1)²>N identifies the UNIQUE nonnegative principal root.This slice requires an actual perfect square.Do not round a float or return both±r as distances;derived r is not SAFE-capped. Step identifiers and dependenciesRule: Length_EXACT_perfect_square_certificate_NONNEGATIVE_principal_root · Uses: G001, S020 S022 · impliesBefore source roles = [[[0, 0], [51, 68], [56, 80]]] After derived = [[85, 13]] Conditions: none Return the TWO segment distances(d(P,Q),d(Q,R)) in the ORIGINAL order.Coincident points and distance0 are legal;do not substitute the sum or d(P,R). Step identifiers and dependenciesRule: Length_TWO_original_segment_roots_IN_ORDER_not_sum_or_endpoint · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S021 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S020 · premise_and_result_audit — Recompute Length segment1 next integer squared with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Length segment1 candidate next integer with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Length segment1 candidate nonnegative root squared with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S016 · premise_and_result_audit — Recompute Length segment1 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Length segment1 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Length segment1 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Length segment1 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Length segment1 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S010 · premise_and_result_audit — Recompute Length segment0 next integer squared with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Length segment0 candidate next integer with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Length segment0 candidate nonnegative root squared with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S006 · premise_and_result_audit — Recompute Length segment0 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Length segment0 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Length segment0 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Length segment0 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Length segment0 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. ## Result (85, 13) 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: 785d92daa566d88f04a99432929ce7904abd9fab880fd9d03ed58025b6239f79 - candidate record SHA-256: c514e6713273796408607b7c6efe75eb4e8a1d76e4b963d38713f017864e4f96 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r06-euclidean-length-0045.json)