Question 30462: Circle Squared Remainder at a Fixed Horizontal Offset

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=12, horizontal displacement dx=1, ordered candidate y values=[0, 12, 13] 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=12, horizontal displacement dx=1, ordered candidate y values=[0, 12, 13] 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

Step-by-step derivation

source roles = [qf-density14-circle-horizontal-section-candidates, density14_circle_horizontal_section_candidates, finite_integer_circle_boundary_protocol]

derived = [[required_vertical_square, ALL_squared_residuals, ALL_boundary_flags]]

Circle ORIGINAL nonnegative radius squared := mul(12, 12)

Circle ORIGINAL nonnegative radius squared = 144

source roles = [12, orthogonal_equal_unit]

derived = [144]

Circle ORIGINAL horizontal displacement squared := mul(1, 1)

Circle ORIGINAL horizontal displacement squared = 1

Circle ORIGINAL required vertical square := sub(144, 1)

Circle ORIGINAL required vertical square = 143

Circle ORIGINAL vertical candidate0 squared := mul(0, 0)

Circle ORIGINAL vertical candidate0 squared = 0

Circle ORIGINAL candidate0 square minus required := sub(0, 143)

Circle ORIGINAL candidate0 square minus required = -143

Circle ORIGINAL vertical candidate1 squared := mul(12, 12)

Circle ORIGINAL vertical candidate1 squared = 144

Circle ORIGINAL candidate1 square minus required := sub(144, 143)

Circle ORIGINAL candidate1 square minus required = 1

Circle ORIGINAL vertical candidate2 squared := mul(13, 13)

Circle ORIGINAL vertical candidate2 squared = 169

Circle ORIGINAL candidate2 square minus required := sub(169, 143)

Circle ORIGINAL candidate2 square minus required = 26

source roles = [12, 1, [0, 12, 13]]

derived = [143, [-143, 1, 26], [No, No, No]]

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

(143, [-143, 1, 26], [No, No, 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.