# Question 4004: Candidate Sign and Squared-Distance Defect

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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.

R0007-04004 · q-20261008-advance-20261010-r03-distance-formula-0029



## Original question and givens

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. Vector=[0, 0], integer candidate c=0. 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=[0, 0], integer candidate c=0. 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](https://ocme.evemisslab.com/en/mko/mko-distance-formula/)





## Step-by-step derivation

S001 · impliesBefore

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

After

L2(v)≥0 and L2(v)²=v₁²+v₂²

Conditions: none

A common origin,orthogonality and equal units support the sum-of-squares formula;coordinates may be negative but distances may not.

Step identifiers and dependenciesRule: euclidean_distance_frame · Uses: G001





S002 · calculationBefore

candidate vector coordinate1 squared := square(0)

After

candidate vector coordinate1 squared = 0

Conditions: none

Square the entire coordinate,including a negative sign inside the square.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

candidate vector coordinate2 squared := square(0)

After

candidate vector coordinate2 squared = 0

Conditions: none

Square the entire coordinate,including a negative sign inside the square.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

candidate vector L2 squared := sum(0, 0)

After

candidate vector L2 squared = 0

Conditions: none

Add the coordinate squares to obtain squared distance,not a sum of the coordinates.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

candidate squared := square(0)

After

candidate squared = 0

Conditions: none

Square the candidate before comparing with squared distance.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

signed square defect := sub(0, 0)

After

signed square defect = 0

Conditions: none

The signed defect is c²−L2²,not its absolute value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · impliesBefore

c=0;c≥0=Yes;c²−L2²=0

After

report sign and defect separately

Conditions: none

The first field asks only whether the candidate is nonnegative;full distance validity also requires defect0.A negative candidate is not a distance even if its square matches.

Step identifiers and dependenciesRule: distance_sign_and_defect · Uses: G001, S006









## 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

(Yes, 0)

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: e9eed0119889dba9e807f354af0876bf054ad128a84c6cc426ca0ca2dcc1886d
- candidate record SHA-256: 7f8b5428ddea3dd5b455512c5ccd3226f0898ce86b2b25570dde28417fdb4f5b
- semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a
- question-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r03-distance-formula-0029.json)