Question 30430: Axis Antipodes, Squared Separation, and Diameter

Original question and givens

Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, -8], r=4, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.

• G001 — Use a circle boundary, not a disk, in one equal-unit orthogonal Euclidean L2 frame with nonnegative integer radius. For r=0 only the center is on the circle. O=[-2, -8], r=4, axis=x. Give (positive-axis endpoint, negative-axis endpoint, squared endpoint separation, diameter). For zero radius the endpoints coincide.

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-circle

Step-by-step derivation

source roles = [qf-density14-circle-axis-antipodal-pair, density14_circle_axis_antipodal_pair, finite_integer_circle_boundary_protocol]

derived = [[PLUS_axis_endpoint, MINUS_axis_endpoint, SQUARED_separation, diameter]]

Circle ORIGINAL nonnegative radius squared := mul(4, 4)

Circle ORIGINAL nonnegative radius squared = 16

source roles = [4, orthogonal_equal_unit]

derived = [16]

Circle ORIGINAL positive-axis endpoint := sum(-2, 4)

Circle ORIGINAL positive-axis endpoint = 2

Circle ORIGINAL negative-axis endpoint := sub(-2, 4)

Circle ORIGINAL negative-axis endpoint = -6

Circle antipodal endpoints ORIGINAL Q minus P coordinate0 := sub(-6, 2)

Circle antipodal endpoints ORIGINAL Q minus P coordinate0 = -8

Circle antipodal endpoints ORIGINAL Q minus P coordinate1 := sub(-8, -8)

Circle antipodal endpoints ORIGINAL Q minus P coordinate1 = 0

Circle antipodal endpoints ORIGINAL square coordinate0 := mul(-8, -8)

Circle antipodal endpoints ORIGINAL square coordinate0 = 64

Circle antipodal endpoints ORIGINAL square coordinate1 := mul(0, 0)

Circle antipodal endpoints ORIGINAL square coordinate1 = 0

Circle antipodal endpoints ORIGINAL squared norm sum := sum(64, 0)

Circle antipodal endpoints ORIGINAL squared norm sum = 64

source roles = [Circle antipodal endpoints, [2, -8], [-6, -8]]

derived = [[-8, 0], 64]

Circle ORIGINAL diameter twice radius := mul(2, 4)

Circle ORIGINAL diameter twice radius = 8

Circle reverse diameter squared := mul(8, 8)

Circle reverse diameter squared = 64

source roles = [[-2, -8], 4, x]

derived = [[2, -8], [-6, -8], 64, 8]

Reverse checks

• S014 · 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.
• S013 · premise_and_result_audit — Recompute Circle reverse diameter squared with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Circle ORIGINAL diameter twice radius 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 Circle antipodal endpoints ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Circle antipodal endpoints ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Circle antipodal endpoints ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Circle antipodal endpoints ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Circle antipodal endpoints ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Circle ORIGINAL negative-axis endpoint with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Circle ORIGINAL positive-axis endpoint with exact rational arithmetic, not rounded decimals.
• S003 · 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.
• S002 · premise_and_result_audit — Recompute Circle ORIGINAL nonnegative radius squared 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

([2, -8], [-6, -8], 64, 8)

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.