# Question 30223: Axis Path Sum versus Endpoint Length [Return to catalogue page 152](https://ocme.evemisslab.com/en/questions/page/152/) [← 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-0098) 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. R0059-30223 · q-20261008-advance-20261012-r06-euclidean-length-0098 ## Original question and givens Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(11,0), R=(22,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?). - G001 — Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(11,0), R=(22,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?). ## 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-triangle-defect, density12_length_triangle_defect, integer_euclidean_axis] After derived = [[path_via_middle, direct_endpoint_length, path_minus_direct, equality]] 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 Axis ORIGINAL y minus x := sub(11, 0) After Axis ORIGINAL y minus x = 11 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 Axis ORIGINAL z minus y := sub(22, 11) After Axis ORIGINAL z minus y = 11 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 Axis ORIGINAL z minus x := sub(22, 0) After Axis ORIGINAL z minus x = 22 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 · impliesBefore source roles = [Axis PQ, 11] After derived = [Yes, 11] Conditions: none Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator. Step identifiers and dependenciesRule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S004 S006 · impliesBefore source roles = [Axis QR, 11] After derived = [Yes, 11] Conditions: none Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator. Step identifiers and dependenciesRule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S005 S007 · impliesBefore source roles = [Axis PR, 22] After derived = [Yes, 22] Conditions: none Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator. Step identifiers and dependenciesRule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S006 S008 · calculationBefore Axis ORIGINAL path via middle := sum(11, 11) After Axis ORIGINAL path via middle = 22 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 Axis ORIGINAL path minus direct := sub(22, 22) After Axis ORIGINAL path minus direct = 0 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 · impliesBefore source roles = [0, 11, 22] After derived = [22, 22, 0, Yes] Conditions: none Take the absolute lengths of BOTH legs before adding;the direct endpoint length uses only abs(z-x).The gap is0 when the middle lies in the CLOSED endpoint segment,including endpoints,and positive outside.This exact axis instance is not a general infinite-metric proof. Step identifiers and dependenciesRule: Length_axis_PATH_not_endpoint_distance_closed_segment_equality · Uses: G001, S009 ## Reverse checks - S010 · 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. - S009 · premise_and_result_audit — Recompute Axis ORIGINAL path minus direct with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Axis ORIGINAL path via middle 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 — 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. - S005 · 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. - S004 · premise_and_result_audit — Recompute Axis ORIGINAL z minus x with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Axis ORIGINAL z minus y with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Axis ORIGINAL y minus x 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 (22, 22, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: ce9c692a1e016f8c23542d8608cc04b3b506bd6652fe223a8fa2e302bf3cedee - candidate record SHA-256: f3863afaca7652d8bf6f3023deb21b0ba950771f80ba59e3882e7de8e6bb79db - 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-0098.json)