Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(2,0), R=(4,0). 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=(2,0), R=(4,0). 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.
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.
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.
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.
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.
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.
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.
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.
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
(4, 4, 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.