# Question 30438: Axis Antipodes, Squared Separation, and Diameter [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-0038) 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. R0059-30438 · q-20261010-advance-20261014-r07-circle-0038 ## 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, 0], r=12, 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, 0], r=12, 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](https://ocme.evemisslab.com/en/mko/mko-circle/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density14-circle-axis-antipodal-pair, density14_circle_axis_antipodal_pair, finite_integer_circle_boundary_protocol] After derived = [[PLUS_axis_endpoint, MINUS_axis_endpoint, SQUARED_separation, diameter]] 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(12, 12) After Circle ORIGINAL nonnegative radius squared = 144 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 = [12, orthogonal_equal_unit] After derived = [144] 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 positive-axis endpoint := sum(-2, 12) After Circle ORIGINAL positive-axis endpoint = 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, S003 S005 · calculationBefore Circle ORIGINAL negative-axis endpoint := sub(-2, 12) After Circle ORIGINAL negative-axis endpoint = -14 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 antipodal endpoints ORIGINAL Q minus P coordinate0 := sub(-14, 10) After Circle antipodal endpoints ORIGINAL Q minus P coordinate0 = -24 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 antipodal endpoints ORIGINAL Q minus P coordinate1 := sub(0, 0) After Circle antipodal endpoints 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, S006 S008 · calculationBefore Circle antipodal endpoints ORIGINAL square coordinate0 := mul(-24, -24) After Circle antipodal endpoints ORIGINAL square coordinate0 = 576 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 · calculationBefore Circle antipodal endpoints ORIGINAL square coordinate1 := mul(0, 0) After Circle antipodal endpoints 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, S008 S010 · calculationBefore Circle antipodal endpoints ORIGINAL squared norm sum := sum(576, 0) After Circle antipodal endpoints ORIGINAL squared norm sum = 576 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 = [Circle antipodal endpoints, [10, 0], [-14, 0]] After derived = [[-24, 0], 576] 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, S010 S012 · calculationBefore Circle ORIGINAL diameter twice radius := mul(2, 12) After Circle ORIGINAL diameter twice radius = 24 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 reverse diameter squared := mul(24, 24) After Circle reverse diameter squared = 576 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 · impliesBefore source roles = [[-2, 0], 12, x] After derived = [[10, 0], [-14, 0], 576, 24] Conditions: none Return the PLUS-axis endpoint THEN the minus endpoint,not coordinate-sorted points.The squared separation is(2r)²,but the diameter is the nonnegative2r,not4r².At r=0 retain BOTH coincident endpoint roles without inventing distinct points. Step identifiers and dependenciesRule: Circle_ORDERED_plus_minus_endpoints_SQUARED_separation_VS_diameter · Uses: G001, S013 ## 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 ([10, 0], [-14, 0], 576, 24) 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: c74c6f68dc24359f5415a7b04194dab867db1aed2a25c53b0e2d18709347e2ea - candidate record SHA-256: 5a92b736fa9309936fad27ce2e766a069c91cf21eef787d95002f03295bf27b1 - 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-0038.json)