# Question 30485: Squared Circle Distances, Residuals, and Boundary Membership [Return to catalogue page 153](https://ocme.evemisslab.com/en/questions/page/153/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r07-circle-foundations/#q-20261010-advance-20261014-r07-circle-0010) 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. R0059-30485 · q-20261010-advance-20261014-r07-circle-0010 ## 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, 0], r=9, ordered points=[[-1, 0], [8, 0], [9, 1]]. 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, 0], r=9, ordered points=[[-1, 0], [8, 0], [9, 1]]. 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](https://ocme.evemisslab.com/en/mko/mko-circle/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density14-circle-offset-residual-triple, density14_circle_offset_residual_triple, finite_integer_circle_boundary_protocol] After derived = [[ALL_squared_distances, ALL_signed_residuals, ALL_location_labels, ALL_boundary_flags]] Conditions: none 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. Step identifiers and dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · calculationBefore Circle ORIGINAL nonnegative radius squared := mul(9, 9) After Circle ORIGINAL nonnegative radius squared = 81 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore source roles = [9, orthogonal_equal_unit] After derived = [81] Conditions: none The circle is the BOUNDARY with squared L2 distance EQUAL to r²,not the disk defined by<=.The radius is nonnegative and0 is legal;at r=0 only the center is on the boundary.Derived coordinates/squares are not SAFE-capped. Step identifiers and dependenciesRule: Circle_BOUNDARY_not_disk_and_zero_radius_SINGLEcenter · Uses: G001, S002 S004 · calculationBefore Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 := sub(-1, -1) After Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 := sub(0, 0) After Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Circle ORIGINAL center to point0 ORIGINAL square coordinate0 := mul(0, 0) After Circle ORIGINAL center to point0 ORIGINAL square coordinate0 = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Circle ORIGINAL center to point0 ORIGINAL square coordinate1 := mul(0, 0) After Circle ORIGINAL center to point0 ORIGINAL square coordinate1 = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Circle ORIGINAL center to point0 ORIGINAL squared norm sum := sum(0, 0) After Circle ORIGINAL center to point0 ORIGINAL squared norm sum = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore source roles = [Circle ORIGINAL center to point0, [-1, 0], [-1, 0]] After derived = [[0, 0], 0] Conditions: none In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs. Step identifiers and dependenciesRule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S008 S010 · calculationBefore Circle ORIGINAL point0 square minus radius square := sub(0, 81) After Circle ORIGINAL point0 square minus radius square = -81 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore source roles = [[-1, 0], 9, [-1, 0], 0, -81] After derived = [inside, No] Conditions: none A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center. Step identifiers and dependenciesRule: Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership · Uses: G001, S010 S012 · calculationBefore Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 := sub(8, -1) After Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 = 9 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 := sub(0, 0) After Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Circle ORIGINAL center to point1 ORIGINAL square coordinate0 := mul(9, 9) After Circle ORIGINAL center to point1 ORIGINAL square coordinate0 = 81 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Circle ORIGINAL center to point1 ORIGINAL square coordinate1 := mul(0, 0) After Circle ORIGINAL center to point1 ORIGINAL square coordinate1 = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Circle ORIGINAL center to point1 ORIGINAL squared norm sum := sum(81, 0) After Circle ORIGINAL center to point1 ORIGINAL squared norm sum = 81 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [Circle ORIGINAL center to point1, [-1, 0], [8, 0]] After derived = [[9, 0], 81] Conditions: none In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs. Step identifiers and dependenciesRule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S016 S018 · calculationBefore Circle ORIGINAL point1 square minus radius square := sub(81, 81) After Circle ORIGINAL point1 square minus radius square = 0 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [[-1, 0], 9, [8, 0], 81, 0] After derived = [on the circle, Yes] Conditions: none A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center. Step identifiers and dependenciesRule: Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership · Uses: G001, S018 S020 · calculationBefore Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 := sub(9, -1) After Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 = 10 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 := sub(1, 0) After Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 = 1 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Circle ORIGINAL center to point2 ORIGINAL square coordinate0 := mul(10, 10) After Circle ORIGINAL center to point2 ORIGINAL square coordinate0 = 100 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Circle ORIGINAL center to point2 ORIGINAL square coordinate1 := mul(1, 1) After Circle ORIGINAL center to point2 ORIGINAL square coordinate1 = 1 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Circle ORIGINAL center to point2 ORIGINAL squared norm sum := sum(100, 1) After Circle ORIGINAL center to point2 ORIGINAL squared norm sum = 101 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore source roles = [Circle ORIGINAL center to point2, [-1, 0], [9, 1]] After derived = [[10, 1], 101] Conditions: none In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs. Step identifiers and dependenciesRule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S024 S026 · calculationBefore Circle ORIGINAL point2 square minus radius square := sub(101, 81) After Circle ORIGINAL point2 square minus radius square = 20 Conditions: none 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · impliesBefore source roles = [[-1, 0], 9, [9, 1], 101, 20] After derived = [outside, No] Conditions: none A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center. Step identifiers and dependenciesRule: Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership · Uses: G001, S026 ## 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, 81, 101], [-81, 0, 20], [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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 62a5bfe3fe77ab4e42e6ff6ab4bdeb101bc5e0b442b0c278a0d910c707f032e8 - candidate record SHA-256: 24ece88af1bc1ef306c0358af171d9db97bf339cc81d1049249d92dbf0074d92 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r07-circle-0010.json)