# Question 30509: Radius-Parameter Scaling and Positivity [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-0059) 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. R0060-30509 · q-20261010-advance-20261014-r07-circle-0059 ## 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=8, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked. - 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=8, scale k=3. Give (kr, r², (kr)², 2kr, is kr>0?). Only parameter arithmetic is asked. ## 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-radius-scale-witness, density14_circle_radius_scale_witness, finite_integer_circle_boundary_protocol] After derived = [[scaled_radius, original_radius_squared, scaled_radius_squared, diameter, scaled_positive]] Conditions: none density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status. Step identifiers and dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · calculationBefore Circle scale ORIGINAL k times r := mul(3, 8) After Circle scale ORIGINAL k times r = 24 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Circle scale ORIGINAL r squared := mul(8, 8) After Circle scale ORIGINAL r squared = 64 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Circle scale ORIGINAL scaled radius squared := mul(24, 24) After Circle scale ORIGINAL scaled radius squared = 576 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Circle scale ORIGINAL twice scaled radius := mul(2, 24) After Circle scale ORIGINAL twice scaled radius = 48 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore source roles = [8, 3, orthogonal_equal_unit] After derived = [24, 64, 576, 48, Yes] Conditions: none Both r,k are nonnegative and0 is legal.Compute kr,r²,(kr)²,2kr separately,then test ONLY kr>0.Derived values may reach SAFE^4 without raw recapping.This parameter arithmetic is not a similarity theorem or formal Circle API bridge. Step identifiers and dependenciesRule: Circle_radius_parameter_arithmetic_ZEROlegal_NOT_similarity_theorem · Uses: G001, S005 ## Reverse checks - S006 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Recompute Circle scale ORIGINAL twice scaled radius with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Circle scale ORIGINAL scaled radius squared with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Circle scale ORIGINAL r squared with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Circle scale ORIGINAL k times r with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. ## Result (24, 64, 576, 48, Yes) 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: d455aa930eb10ac3d82be0321d3ee03a083b9de67cb0d82fa464cec577fd6cc4 - candidate record SHA-256: 59556deb35c46d3655ec2883be92a6173fd4e4156c3dc192621988853adc75b1 - semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf - question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4 - 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-0059.json)