Question 4010: Candidate Sign and Squared-Distance Defect

Original question and givens

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[9, 10], integer candidate c=-12. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

• G001 — In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[9, 10], integer candidate c=-12. Give (is c nonnegative?, c²−L2²). The first field tests only sign, not full distance eligibility.

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

Step-by-step derivation

integer vectors from a common origin;orthogonal equal-length units

L2(v)≥0 and L2(v)²=v₁²+v₂²

candidate vector coordinate1 squared := square(9)

candidate vector coordinate1 squared = 81

candidate vector coordinate2 squared := square(10)

candidate vector coordinate2 squared = 100

candidate vector L2 squared := sum(81, 100)

candidate vector L2 squared = 181

candidate squared := square(-12)

candidate squared = 144

signed square defect := sub(144, 181)

signed square defect = -37

c=-12;c≥0=No;c²−L2²=-37

report sign and defect separately

Reverse checks

• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute signed square defect with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute candidate squared with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute candidate vector L2 squared with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute candidate vector coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute candidate vector 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

(No, -37)

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.