# Question 30473: Circle Squared Remainder at a Fixed Horizontal Offset [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-0098) 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. R0059-30473 · q-20261010-advance-20261014-r07-circle-0098 ## 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. r=23, horizontal displacement dx=0, ordered candidate y values=[0, 23, 24] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values). - 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. r=23, horizontal displacement dx=0, ordered candidate y values=[0, 23, 24] for points (dx,y). Give (required=r²−dx², three y²−required residuals, three boundary-eligibility values). ## 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-horizontal-section-candidates, density14_circle_horizontal_section_candidates, finite_integer_circle_boundary_protocol] After derived = [[required_vertical_square, ALL_squared_residuals, 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(23, 23) After Circle ORIGINAL nonnegative radius squared = 529 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 = [23, orthogonal_equal_unit] After derived = [529] 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 horizontal displacement squared := mul(0, 0) After Circle ORIGINAL horizontal displacement squared = 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 required vertical square := sub(529, 0) After Circle ORIGINAL required vertical square = 529 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 vertical candidate0 squared := mul(0, 0) After Circle ORIGINAL vertical candidate0 squared = 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 candidate0 square minus required := sub(0, 529) After Circle ORIGINAL candidate0 square minus required = -529 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 vertical candidate1 squared := mul(23, 23) After Circle ORIGINAL vertical candidate1 squared = 529 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 ORIGINAL candidate1 square minus required := sub(529, 529) After Circle ORIGINAL candidate1 square minus required = 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 ORIGINAL vertical candidate2 squared := mul(24, 24) After Circle ORIGINAL vertical candidate2 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, S009 S011 · calculationBefore Circle ORIGINAL candidate2 square minus required := sub(576, 529) After Circle ORIGINAL candidate2 square minus required = 47 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, S010 S012 · impliesBefore source roles = [23, 0, [0, 23, 24]] After derived = [529, [-529, 0, 47], [No, Yes, No]] Conditions: none The FIXED quantity is horizontal OFFSET dx;candidate points(dx,y) satisfy dx²+y²=r² IFF y²-required=0.A negative required value is LEGAL and gives no real intersection.STILL retain all three signed residuals/positions;do not take a negative square root,clamp0 or add an unspecified center translation. Step identifiers and dependenciesRule: Circle_FIXED_horizontal_OFFSET_ALL_vertical_candidates_NEGATIVE_required_LEGAL · Uses: G001, S011 ## Reverse checks - S012 · 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. - S011 · premise_and_result_audit — Recompute Circle ORIGINAL candidate2 square minus required with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Circle ORIGINAL vertical candidate2 squared with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Circle ORIGINAL candidate1 square minus required with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Circle ORIGINAL vertical candidate1 squared with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Circle ORIGINAL candidate0 square minus required with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Circle ORIGINAL vertical candidate0 squared with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Circle ORIGINAL required vertical square with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Circle ORIGINAL horizontal displacement squared 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 (529, [-529, 0, 47], [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: e5753a6d6e0cd193ff823b52cc4abb72d1b16e05b7314466adcdbe2ef62c0b94 - candidate record SHA-256: cceb26963583003ddc3e64e40dfc29b49343767a30b5f9b697d3adb8fb82ad9c - 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-0098.json)