Question 30320: Three Rational Distances and Path Defect

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0059-30320 · q-20261010-advance-20261014-r06-absolute-value-0045

Original question and givens

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 19/2, 20/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

G001 — Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 19/2, 20/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for 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.

mko-absolute-value

Step-by-step derivation

S001 · implies

Before

source roles = [qf-density14-absolute-three-point-distances, density14_absolute_three_point_distances, finite_rational_absolute_protocol]

After

derived = [[distance_01, distance_12, distance_02, path_minus_direct]]

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 · implies

Before

source roles = [Absolute original point0, [0, 1]]

After

derived = [0/1, 0, 1]

Conditions: none

Require a RAW SAFE pair with nonzero denominator.This bank ALLOWS negative denominators;normalize sign/gcd BEFORE deciding the value sign.Do not import a primitive positive-denominator rule or a STR22 component cap.

Step identifiers and dependencies

Rule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S001

S003 · implies

Before

source roles = [Absolute original point1, [19, 2]]

After

derived = [19/2, 19, 2]

Conditions: none

Step identifiers and dependencies

Rule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S002

S004 · implies

Before

source roles = [Absolute original point2, [20, 2]]

After

derived = [10/1, 10, 1]

Conditions: none

Step identifiers and dependencies

Rule: Absolute_RAW_SAFE_pair_NONZERO_den_normalize_SIGN_and_gcd · Uses: G001, S003

S005 · calculation

Before

Absolute ORIGINAL point0 minus point1 := sub(0, 19/2)

After

Absolute ORIGINAL point0 minus point1 = -19/2

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, S004

S006 · calculation

Before

Absolute segment01 ORIGINAL negative branch negation := sub(0, -19/2)

After

Absolute segment01 ORIGINAL negative branch negation = 19/2

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, S005

S007 · implies

Before

source roles = [Absolute segment01, -19/2]

After

derived = [No, 19/2]

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 · calculation

Before

Absolute ORIGINAL point1 minus point2 := sub(19/2, 10)

After

Absolute ORIGINAL point1 minus point2 = -1/2

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, S007

S009 · calculation

Before

Absolute segment12 ORIGINAL negative branch negation := sub(0, -1/2)

After

Absolute segment12 ORIGINAL negative branch negation = 1/2

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, S008

S010 · implies

Before

source roles = [Absolute segment12, -1/2]

After

derived = [No, 1/2]

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, S009

S011 · calculation

Before

Absolute ORIGINAL point0 minus point2 := sub(0, 10)

After

Absolute ORIGINAL point0 minus point2 = -10

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, S010

S012 · calculation

Before

Absolute segment02 ORIGINAL negative branch negation := sub(0, -10)

After

Absolute segment02 ORIGINAL negative branch negation = 10

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 · implies

Before

source roles = [Absolute segment02, -10]

After

derived = [No, 10]

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, S012

S014 · calculation

Before

Absolute ORIGINAL two-segment path sum := sum(19/2, 1/2)

After

Absolute ORIGINAL two-segment path sum = 10

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

Absolute ORIGINAL path minus endpoint distance := sub(10, 10)

After

Absolute ORIGINAL path minus endpoint distance = 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 · implies

Before

source roles = [[0/1, 19/2, 10/1]]

After

derived = [[19/2, 1/2, 10/1], 0/1]

Conditions: none

Take the absolute value for EACH separate segment.The first-two-leg sum differs from the endpoint distance;retain the ORIGINAL path order for the gap.This complete finite rational instance is not a formal proof of the general triangle inequality.

Step identifiers and dependencies

Rule: Absolute_THREE_original_distances_AND_signed_path_defect · Uses: G001, S015

Reverse checks

S016 · 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.
S015 · premise_and_result_audit — Recompute Absolute ORIGINAL path minus endpoint distance with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute Absolute ORIGINAL two-segment path sum with exact rational arithmetic, not rounded decimals.
S013 · 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.
S012 · premise_and_result_audit — Recompute Absolute segment02 ORIGINAL negative branch negation with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Recompute Absolute ORIGINAL point0 minus point2 with exact rational arithmetic, not rounded decimals.
S010 · 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.
S009 · premise_and_result_audit — Recompute Absolute segment12 ORIGINAL negative branch negation with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute Absolute ORIGINAL point1 minus point2 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 Absolute segment01 ORIGINAL negative branch negation with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Absolute ORIGINAL point0 minus point1 with exact rational arithmetic, not rounded decimals.
S004 · 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.
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 — 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.
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

(19/2, 1/2, 10/1, 0/1)

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: 5d279993f7afbf4f09d0412c164adb4637b5d5689d07d97e4f4ddf2755f282dd
candidate record SHA-256: e273d5cfb5548aa3a2ce7404b4271be678b01e5285fd66377e9f9ceee8278558
semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
Lean: NOT_RUN · human review: NOT_PERFORMED

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