OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 30151: Ordered Integer Principal Segment Lengths

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

Return to catalogue page 151

← Question derivation index · Original question batch

R0059-30151 · q-20261008-advance-20261012-r06-euclidean-length-0026

Original question and givens

Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [0, 0], [0, 0]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.

  • G001 — Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P,Q,R are [[0, 0], [0, 0], [0, 0]] in that order; both segment squared norms are perfect squares. Give (d(P,Q), d(Q,R)) as nonnegative integers.

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-density12-length-integer-distance-pair, density12_length_integer_distance_pair, integer_euclidean_plane]

After

derived = [[nonnegative_distance_PQ, nonnegative_distance_QR]]

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

Length segment0 ORIGINAL Q minus P coordinate0 := sub(0, 0)

After

Length segment0 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, S001

S003 · calculation

Before

Length segment0 ORIGINAL Q minus P coordinate1 := sub(0, 0)

After

Length segment0 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, S002

S004 · calculation

Before

Length segment0 ORIGINAL square coordinate0 := mul(0, 0)

After

Length segment0 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, S003

S005 · calculation

Before

Length segment0 ORIGINAL square coordinate1 := mul(0, 0)

After

Length segment0 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, S004

S006 · calculation

Before

Length segment0 ORIGINAL squared norm sum := sum(0, 0)

After

Length segment0 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, S005

S007 · implies

Before

source roles = [Length segment0, [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, S006

S008 · calculation

Before

Length segment0 candidate nonnegative root squared := mul(0, 0)

After

Length segment0 candidate nonnegative root 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Length segment0 candidate next integer := sum(0, 1)

After

Length segment0 candidate next integer = 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, S008

S010 · calculation

Before

Length segment0 next integer squared := mul(1, 1)

After

Length segment0 next integer 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, S009

S011 · implies

Before

source roles = [Length segment0, 0]

After

derived = [0, 0, 1, Yes]

Conditions: none

The exact certificate r>=0,r²=N and(r+1)²>N identifies the UNIQUE nonnegative principal root.This slice requires an actual perfect square.Do not round a float or return both±r as distances;derived r is not SAFE-capped.

Step identifiers and dependencies

Rule: Length_EXACT_perfect_square_certificate_NONNEGATIVE_principal_root · Uses: G001, S010

S012 · calculation

Before

Length segment1 ORIGINAL Q minus P coordinate0 := sub(0, 0)

After

Length segment1 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, S011

S013 · calculation

Before

Length segment1 ORIGINAL Q minus P coordinate1 := sub(0, 0)

After

Length segment1 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

Length segment1 ORIGINAL square coordinate0 := mul(0, 0)

After

Length segment1 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, S013

S015 · calculation

Before

Length segment1 ORIGINAL square coordinate1 := mul(0, 0)

After

Length segment1 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

Length segment1 ORIGINAL squared norm sum := sum(0, 0)

After

Length segment1 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, S015

S017 · implies

Before

source roles = [Length segment1, [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, S016

S018 · calculation

Before

Length segment1 candidate nonnegative root squared := mul(0, 0)

After

Length segment1 candidate nonnegative root 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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Length segment1 candidate next integer := sum(0, 1)

After

Length segment1 candidate next integer = 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, S018

S020 · calculation

Before

Length segment1 next integer squared := mul(1, 1)

After

Length segment1 next integer 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, S019

S021 · implies

Before

source roles = [Length segment1, 0]

After

derived = [0, 0, 1, Yes]

Conditions: none

The exact certificate r>=0,r²=N and(r+1)²>N identifies the UNIQUE nonnegative principal root.This slice requires an actual perfect square.Do not round a float or return both±r as distances;derived r is not SAFE-capped.

Step identifiers and dependencies

Rule: Length_EXACT_perfect_square_certificate_NONNEGATIVE_principal_root · Uses: G001, S020

S022 · implies

Before

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

After

derived = [[0, 0]]

Conditions: none

Return the TWO segment distances(d(P,Q),d(Q,R)) in the ORIGINAL order.Coincident points and distance0 are legal;do not substitute the sum or d(P,R).

Step identifiers and dependencies

Rule: Length_TWO_original_segment_roots_IN_ORDER_not_sum_or_endpoint · Uses: G001, S021

Reverse checks

  • S022 · 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.
  • S021 · 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.
  • S020 · premise_and_result_audit — Recompute Length segment1 next integer squared with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Length segment1 candidate next integer with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Length segment1 candidate nonnegative root squared 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 Length segment1 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Length segment1 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Length segment1 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Length segment1 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Length segment1 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 Length segment0 next integer squared with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Length segment0 candidate next integer with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Length segment0 candidate nonnegative root squared with exact rational arithmetic, not rounded decimals.
  • S007 · 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.
  • S006 · premise_and_result_audit — Recompute Length segment0 ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Length segment0 ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Length segment0 ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Length segment0 ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Length segment0 ORIGINAL Q minus P coordinate0 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, 0)

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: 0ad7ede74a4912cbc8647cf4a13fdc344064bda42fdbc4ae674f36a449b72b8b
  • candidate record SHA-256: 167e422f657801163f7687ad5cb82cc9528ad51fe797f1a0668cf2720154625a
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON