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Question 30222: Axis Path Sum versus Endpoint Length

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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R0059-30222 · q-20261008-advance-20261012-r06-euclidean-length-0097

Original question and givens

Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-21,0), R=(21,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).

  • G001 — Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=(0,0), Q=(-21,0), R=(21,0). Give (P-to-R path length via Q, direct P-to-R length, path−direct defect, are they equal?).

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-triangle-defect, density12_length_triangle_defect, integer_euclidean_axis]

After

derived = [[path_via_middle, direct_endpoint_length, path_minus_direct, equality]]

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

Axis ORIGINAL y minus x := sub(-21, 0)

After

Axis ORIGINAL y minus x = -21

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

Axis ORIGINAL z minus y := sub(21, -21)

After

Axis ORIGINAL z minus y = 42

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

Axis ORIGINAL z minus x := sub(21, 0)

After

Axis ORIGINAL z minus x = 21

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

Axis PQ ORIGINAL negative branch negation := sub(0, -21)

After

Axis PQ ORIGINAL negative branch negation = 21

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 · implies

Before

source roles = [Axis PQ, -21]

After

derived = [No, 21]

Conditions: none

Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.

Step identifiers and dependencies

Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S005

S007 · implies

Before

source roles = [Axis QR, 42]

After

derived = [Yes, 42]

Conditions: none

Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.

Step identifiers and dependencies

Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S006

S008 · implies

Before

source roles = [Axis PR, 21]

After

derived = [Yes, 21]

Conditions: none

Choose the branch by the NORMALIZED value:x>=0 keeps x;for x<0 use0-x.Zero is in the nonnegative branch;do not misclassify a raw negative denominator.

Step identifiers and dependencies

Rule: Absolute_value_normalized_sign_branch_ZERO_is_nonnegative · Uses: G001, S007

S009 · calculation

Before

Axis ORIGINAL path via middle := sum(21, 42)

After

Axis ORIGINAL path via middle = 63

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

Axis ORIGINAL path minus direct := sub(63, 21)

After

Axis ORIGINAL path minus direct = 42

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, -21, 21]

After

derived = [63, 21, 42, No]

Conditions: none

Take the absolute lengths of BOTH legs before adding;the direct endpoint length uses only abs(z-x).The gap is0 when the middle lies in the CLOSED endpoint segment,including endpoints,and positive outside.This exact axis instance is not a general infinite-metric proof.

Step identifiers and dependencies

Rule: Length_axis_PATH_not_endpoint_distance_closed_segment_equality · Uses: G001, S010

Reverse checks

  • 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 Axis ORIGINAL path minus direct with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Axis ORIGINAL path via middle with exact rational arithmetic, not rounded decimals.
  • S008 · 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.
  • 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 — 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.
  • S005 · premise_and_result_audit — Recompute Axis PQ ORIGINAL negative branch negation with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Axis ORIGINAL z minus x with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Axis ORIGINAL z minus y with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Axis ORIGINAL y minus x 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

(63, 21, 42, 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: b2050ec5da4f07e86921b3ffcc2a01cc603f99aab10121971aa4dc92a4557d46
  • candidate record SHA-256: 0fc64824ab846fc976c2e3978a544ec8c68a1d7710a06e89a7c27bf3555b6673
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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