Question 30496: Squared Circle Distances, Residuals, and Boundary Membership

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. Center O=[1, 1], r=20, ordered points=[[1, 1], [21, 1], [22, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).

• 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. Center O=[1, 1], r=20, ordered points=[[1, 1], [21, 1], [22, 2]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).

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-offset-residual-triple, density14_circle_offset_residual_triple, finite_integer_circle_boundary_protocol]

derived = [[ALL_squared_distances, ALL_signed_residuals, ALL_location_labels, ALL_boundary_flags]]

Circle ORIGINAL nonnegative radius squared := mul(20, 20)

Circle ORIGINAL nonnegative radius squared = 400

source roles = [20, orthogonal_equal_unit]

derived = [400]

Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 := sub(1, 1)

Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 = 0

Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 := sub(1, 1)

Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 = 0

Circle ORIGINAL center to point0 ORIGINAL square coordinate0 := mul(0, 0)

Circle ORIGINAL center to point0 ORIGINAL square coordinate0 = 0

Circle ORIGINAL center to point0 ORIGINAL square coordinate1 := mul(0, 0)

Circle ORIGINAL center to point0 ORIGINAL square coordinate1 = 0

Circle ORIGINAL center to point0 ORIGINAL squared norm sum := sum(0, 0)

Circle ORIGINAL center to point0 ORIGINAL squared norm sum = 0

source roles = [Circle ORIGINAL center to point0, [1, 1], [1, 1]]

derived = [[0, 0], 0]

Circle ORIGINAL point0 square minus radius square := sub(0, 400)

Circle ORIGINAL point0 square minus radius square = -400

source roles = [[1, 1], 20, [1, 1], 0, -400]

derived = [inside, No]

Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 := sub(21, 1)

Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 = 20

Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 := sub(1, 1)

Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 = 0

Circle ORIGINAL center to point1 ORIGINAL square coordinate0 := mul(20, 20)

Circle ORIGINAL center to point1 ORIGINAL square coordinate0 = 400

Circle ORIGINAL center to point1 ORIGINAL square coordinate1 := mul(0, 0)

Circle ORIGINAL center to point1 ORIGINAL square coordinate1 = 0

Circle ORIGINAL center to point1 ORIGINAL squared norm sum := sum(400, 0)

Circle ORIGINAL center to point1 ORIGINAL squared norm sum = 400

source roles = [Circle ORIGINAL center to point1, [1, 1], [21, 1]]

derived = [[20, 0], 400]

Circle ORIGINAL point1 square minus radius square := sub(400, 400)

Circle ORIGINAL point1 square minus radius square = 0

source roles = [[1, 1], 20, [21, 1], 400, 0]

derived = [on the circle, Yes]

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 := sub(22, 1)

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 = 21

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 := sub(2, 1)

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 = 1

Circle ORIGINAL center to point2 ORIGINAL square coordinate0 := mul(21, 21)

Circle ORIGINAL center to point2 ORIGINAL square coordinate0 = 441

Circle ORIGINAL center to point2 ORIGINAL square coordinate1 := mul(1, 1)

Circle ORIGINAL center to point2 ORIGINAL square coordinate1 = 1

Circle ORIGINAL center to point2 ORIGINAL squared norm sum := sum(441, 1)

Circle ORIGINAL center to point2 ORIGINAL squared norm sum = 442

source roles = [Circle ORIGINAL center to point2, [1, 1], [22, 2]]

derived = [[21, 1], 442]

Circle ORIGINAL point2 square minus radius square := sub(442, 400)

Circle ORIGINAL point2 square minus radius square = 42

source roles = [[1, 1], 20, [22, 2], 442, 42]

derived = [outside, No]

Reverse checks

• S027 · 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.
• S026 · premise_and_result_audit — Recompute Circle ORIGINAL point2 square minus radius square with exact rational arithmetic, not rounded decimals.
• S025 · 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.
• S024 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals.
• S019 · 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.
• S018 · premise_and_result_audit — Recompute Circle ORIGINAL point1 square minus radius square with exact rational arithmetic, not rounded decimals.
• S017 · 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.
• S016 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 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 ORIGINAL point0 square minus radius square with exact rational arithmetic, not rounded decimals.
• S009 · 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.
• S008 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 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

([0, 400, 442], [-400, 0, 42], [inside, on-boundary, outside], [No, Yes, No])

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.