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Question 30477: Squared Circle Distances, Residuals, and Boundary Membership

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0059-30477 · q-20261010-advance-20261014-r07-circle-0002

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. Center O=[0, 0], r=1, ordered points=[[0, 0], [1, 0], [2, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).

  • 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. Center O=[0, 0], r=1, ordered points=[[0, 0], [1, 0], [2, 1]]. Give (three squared distances D, three D−r² values, inside/on-boundary/outside classifications, boundary-membership mask).

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-density14-circle-offset-residual-triple, density14_circle_offset_residual_triple, finite_integer_circle_boundary_protocol]

After

derived = [[ALL_squared_distances, ALL_signed_residuals, ALL_location_labels, 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 dependencies

Rule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001

S002 · calculation

Before

Circle ORIGINAL nonnegative radius squared := mul(1, 1)

After

Circle ORIGINAL nonnegative radius squared = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [1, orthogonal_equal_unit]

After

derived = [1]

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 dependencies

Rule: Circle_BOUNDARY_not_disk_and_zero_radius_SINGLEcenter · Uses: G001, S002

S004 · calculation

Before

Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 := sub(0, 0)

After

Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 := sub(0, 0)

After

Circle ORIGINAL center to point0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Circle ORIGINAL center to point0 ORIGINAL square coordinate0 := mul(0, 0)

After

Circle ORIGINAL center to point0 ORIGINAL square coordinate0 = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Circle ORIGINAL center to point0 ORIGINAL square coordinate1 := mul(0, 0)

After

Circle ORIGINAL center to point0 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Circle ORIGINAL center to point0 ORIGINAL squared norm sum := sum(0, 0)

After

Circle ORIGINAL center to point0 ORIGINAL squared norm sum = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [Circle ORIGINAL center to point0, [0, 0], [0, 0]]

After

derived = [[0, 0], 0]

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 dependencies

Rule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S008

S010 · calculation

Before

Circle ORIGINAL point0 square minus radius square := sub(0, 1)

After

Circle ORIGINAL point0 square minus radius square = -1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [[0, 0], 1, [0, 0], 0, -1]

After

derived = [inside, No]

Conditions: none

A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center.

Step identifiers and dependencies

Rule: Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership · Uses: G001, S010

S012 · calculation

Before

Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 := sub(1, 0)

After

Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 := sub(0, 0)

After

Circle ORIGINAL center to point1 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Circle ORIGINAL center to point1 ORIGINAL square coordinate0 := mul(1, 1)

After

Circle ORIGINAL center to point1 ORIGINAL square coordinate0 = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Circle ORIGINAL center to point1 ORIGINAL square coordinate1 := mul(0, 0)

After

Circle ORIGINAL center to point1 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Circle ORIGINAL center to point1 ORIGINAL squared norm sum := sum(1, 0)

After

Circle ORIGINAL center to point1 ORIGINAL squared norm sum = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [Circle ORIGINAL center to point1, [0, 0], [1, 0]]

After

derived = [[1, 0], 1]

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 dependencies

Rule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S016

S018 · calculation

Before

Circle ORIGINAL point1 square minus radius square := sub(1, 1)

After

Circle ORIGINAL point1 square minus radius square = 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

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

After

derived = [on the circle, Yes]

Conditions: none

A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center.

Step identifiers and dependencies

Rule: Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership · Uses: G001, S018

S020 · calculation

Before

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 := sub(2, 0)

After

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 = 2

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 := sub(1, 0)

After

Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Circle ORIGINAL center to point2 ORIGINAL square coordinate0 := mul(2, 2)

After

Circle ORIGINAL center to point2 ORIGINAL square coordinate0 = 4

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Circle ORIGINAL center to point2 ORIGINAL square coordinate1 := mul(1, 1)

After

Circle ORIGINAL center to point2 ORIGINAL square coordinate1 = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Circle ORIGINAL center to point2 ORIGINAL squared norm sum := sum(4, 1)

After

Circle ORIGINAL center to point2 ORIGINAL squared norm sum = 5

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

source roles = [Circle ORIGINAL center to point2, [0, 0], [2, 1]]

After

derived = [[2, 1], 5]

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 dependencies

Rule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S024

S026 · calculation

Before

Circle ORIGINAL point2 square minus radius square := sub(5, 1)

After

Circle ORIGINAL point2 square minus radius square = 4

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

source roles = [[0, 0], 1, [2, 1], 5, 4]

After

derived = [outside, No]

Conditions: none

A negative residual classifies inside,NOT boundary membership.ONLY zero is on-boundary with Yes;positive is outside with No.Check EVERY original point,including a zero-radius center.

Step identifiers and dependencies

Rule: Circle_EACH_signed_square_residual_location_AND_BOUNDARY_membership · Uses: G001, S026

Reverse checks

  • S027 · 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.
  • S026 · premise_and_result_audit — Recompute Circle ORIGINAL point2 square minus radius square with exact rational arithmetic, not rounded decimals.
  • S025 · 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.
  • S024 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Circle ORIGINAL center to point2 ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals.
  • S019 · 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.
  • S018 · premise_and_result_audit — Recompute Circle ORIGINAL point1 square minus radius square with exact rational arithmetic, not rounded decimals.
  • S017 · 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.
  • S016 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Circle ORIGINAL center to point1 ORIGINAL Q minus P coordinate0 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 ORIGINAL point0 square minus radius square with exact rational arithmetic, not rounded decimals.
  • S009 · 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.
  • S008 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Circle ORIGINAL center to point0 ORIGINAL Q minus P coordinate0 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

([0, 1, 5], [-1, 0, 4], [inside, on-boundary, outside], [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: bb85bf6d5f1cf2abff6cb8535fb46fe62cf5b0340c46cc75e776e62ca9e057d0
  • candidate record SHA-256: 3d0493939fbbd4d0f5fdcd8e6c870bae4802e152cbf3bace5078b9735820f173
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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