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=13, scale k=2. 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=13, scale k=2. 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.
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.
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.
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.
Circle scale ORIGINAL scaled radius squared := mul(26, 26)
After
Circle scale ORIGINAL scaled radius squared = 676
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.
Circle scale ORIGINAL twice scaled radius := mul(2, 26)
After
Circle scale ORIGINAL twice scaled radius = 52
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.
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.
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
(26, 169, 676, 52, 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.