Question 29394: Every Ordered Triangle Inequality in the Finite Table

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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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R0057-29394 · q-20261013-advance-20261022-r06-metric-space-0069

Original question and givens

The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.

G001 — The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.

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-metric-space

Step-by-step derivation

S001 · implies

Before

source roles = [qf-str22-metric-triangle-audit, metric-triangle-audit]

After

derived = [[n, distances, bad_triples, triangle]]

Conditions: none

Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning.

Step identifiers and dependencies

Rule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001

S002 · calculation

Before

Metric ORIGINAL via sum i0 j0 k0 := sum(0, 0)

After

Metric ORIGINAL via sum i0 j0 k0 = 0

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

Metric ORIGINAL via sum i0 j0 k1 := sum(0, 19)

After

Metric ORIGINAL via sum i0 j0 k1 = 19

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Metric ORIGINAL via sum i0 j0 k2 := sum(0, 39)

After

Metric ORIGINAL via sum i0 j0 k2 = 39

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Metric ORIGINAL via sum i0 j0 k3 := sum(0, 60)

After

Metric ORIGINAL via sum i0 j0 k3 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 0, 0]

After

derived = [[0, 19, 39, 60], [0, 19, 39, 60], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S005

S007 · calculation

Before

Metric ORIGINAL via sum i0 j1 k0 := sum(19, 19)

After

Metric ORIGINAL via sum i0 j1 k0 = 38

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Metric ORIGINAL via sum i0 j1 k1 := sum(19, 0)

After

Metric ORIGINAL via sum i0 j1 k1 = 19

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Metric ORIGINAL via sum i0 j1 k2 := sum(19, 20)

After

Metric ORIGINAL via sum i0 j1 k2 = 39

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Metric ORIGINAL via sum i0 j1 k3 := sum(19, 41)

After

Metric ORIGINAL via sum i0 j1 k3 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 0, 1]

After

derived = [[0, 19, 39, 60], [38, 19, 39, 60], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S010

S012 · calculation

Before

Metric ORIGINAL via sum i0 j2 k0 := sum(39, 39)

After

Metric ORIGINAL via sum i0 j2 k0 = 78

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

Metric ORIGINAL via sum i0 j2 k1 := sum(39, 20)

After

Metric ORIGINAL via sum i0 j2 k1 = 59

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Metric ORIGINAL via sum i0 j2 k2 := sum(39, 0)

After

Metric ORIGINAL via sum i0 j2 k2 = 39

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Metric ORIGINAL via sum i0 j2 k3 := sum(39, 21)

After

Metric ORIGINAL via sum i0 j2 k3 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 0, 2]

After

derived = [[0, 19, 39, 60], [78, 59, 39, 60], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S015

S017 · calculation

Before

Metric ORIGINAL via sum i0 j3 k0 := sum(60, 60)

After

Metric ORIGINAL via sum i0 j3 k0 = 120

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Metric ORIGINAL via sum i0 j3 k1 := sum(60, 41)

After

Metric ORIGINAL via sum i0 j3 k1 = 101

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Metric ORIGINAL via sum i0 j3 k2 := sum(60, 21)

After

Metric ORIGINAL via sum i0 j3 k2 = 81

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Metric ORIGINAL via sum i0 j3 k3 := sum(60, 0)

After

Metric ORIGINAL via sum i0 j3 k3 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 0, 3]

After

derived = [[0, 19, 39, 60], [120, 101, 81, 60], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S020

S022 · calculation

Before

Metric ORIGINAL via sum i1 j0 k0 := sum(19, 0)

After

Metric ORIGINAL via sum i1 j0 k0 = 19

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

Metric ORIGINAL via sum i1 j0 k1 := sum(19, 19)

After

Metric ORIGINAL via sum i1 j0 k1 = 38

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

Metric ORIGINAL via sum i1 j0 k2 := sum(19, 39)

After

Metric ORIGINAL via sum i1 j0 k2 = 58

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

Metric ORIGINAL via sum i1 j0 k3 := sum(19, 60)

After

Metric ORIGINAL via sum i1 j0 k3 = 79

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 1, 0]

After

derived = [[19, 0, 20, 41], [19, 38, 58, 79], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S025

S027 · calculation

Before

Metric ORIGINAL via sum i1 j1 k0 := sum(0, 19)

After

Metric ORIGINAL via sum i1 j1 k0 = 19

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

Metric ORIGINAL via sum i1 j1 k1 := sum(0, 0)

After

Metric ORIGINAL via sum i1 j1 k1 = 0

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

Metric ORIGINAL via sum i1 j1 k2 := sum(0, 20)

After

Metric ORIGINAL via sum i1 j1 k2 = 20

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

Metric ORIGINAL via sum i1 j1 k3 := sum(0, 41)

After

Metric ORIGINAL via sum i1 j1 k3 = 41

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 1, 1]

After

derived = [[19, 0, 20, 41], [19, 0, 20, 41], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S030

S032 · calculation

Before

Metric ORIGINAL via sum i1 j2 k0 := sum(20, 39)

After

Metric ORIGINAL via sum i1 j2 k0 = 59

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Metric ORIGINAL via sum i1 j2 k1 := sum(20, 20)

After

Metric ORIGINAL via sum i1 j2 k1 = 40

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Metric ORIGINAL via sum i1 j2 k2 := sum(20, 0)

After

Metric ORIGINAL via sum i1 j2 k2 = 20

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Metric ORIGINAL via sum i1 j2 k3 := sum(20, 21)

After

Metric ORIGINAL via sum i1 j2 k3 = 41

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 1, 2]

After

derived = [[19, 0, 20, 41], [59, 40, 20, 41], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S035

S037 · calculation

Before

Metric ORIGINAL via sum i1 j3 k0 := sum(41, 60)

After

Metric ORIGINAL via sum i1 j3 k0 = 101

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Metric ORIGINAL via sum i1 j3 k1 := sum(41, 41)

After

Metric ORIGINAL via sum i1 j3 k1 = 82

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Metric ORIGINAL via sum i1 j3 k2 := sum(41, 21)

After

Metric ORIGINAL via sum i1 j3 k2 = 62

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Metric ORIGINAL via sum i1 j3 k3 := sum(41, 0)

After

Metric ORIGINAL via sum i1 j3 k3 = 41

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 1, 3]

After

derived = [[19, 0, 20, 41], [101, 82, 62, 41], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S040

S042 · calculation

Before

Metric ORIGINAL via sum i2 j0 k0 := sum(39, 0)

After

Metric ORIGINAL via sum i2 j0 k0 = 39

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Metric ORIGINAL via sum i2 j0 k1 := sum(39, 19)

After

Metric ORIGINAL via sum i2 j0 k1 = 58

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Metric ORIGINAL via sum i2 j0 k2 := sum(39, 39)

After

Metric ORIGINAL via sum i2 j0 k2 = 78

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

Metric ORIGINAL via sum i2 j0 k3 := sum(39, 60)

After

Metric ORIGINAL via sum i2 j0 k3 = 99

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 2, 0]

After

derived = [[39, 20, 0, 21], [39, 58, 78, 99], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S045

S047 · calculation

Before

Metric ORIGINAL via sum i2 j1 k0 := sum(20, 19)

After

Metric ORIGINAL via sum i2 j1 k0 = 39

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

Metric ORIGINAL via sum i2 j1 k1 := sum(20, 0)

After

Metric ORIGINAL via sum i2 j1 k1 = 20

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Metric ORIGINAL via sum i2 j1 k2 := sum(20, 20)

After

Metric ORIGINAL via sum i2 j1 k2 = 40

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

Metric ORIGINAL via sum i2 j1 k3 := sum(20, 41)

After

Metric ORIGINAL via sum i2 j1 k3 = 61

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 2, 1]

After

derived = [[39, 20, 0, 21], [39, 20, 40, 61], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S050

S052 · calculation

Before

Metric ORIGINAL via sum i2 j2 k0 := sum(0, 39)

After

Metric ORIGINAL via sum i2 j2 k0 = 39

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

Metric ORIGINAL via sum i2 j2 k1 := sum(0, 20)

After

Metric ORIGINAL via sum i2 j2 k1 = 20

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Metric ORIGINAL via sum i2 j2 k2 := sum(0, 0)

After

Metric ORIGINAL via sum i2 j2 k2 = 0

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

Metric ORIGINAL via sum i2 j2 k3 := sum(0, 21)

After

Metric ORIGINAL via sum i2 j2 k3 = 21

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 2, 2]

After

derived = [[39, 20, 0, 21], [39, 20, 0, 21], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S055

S057 · calculation

Before

Metric ORIGINAL via sum i2 j3 k0 := sum(21, 60)

After

Metric ORIGINAL via sum i2 j3 k0 = 81

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

Metric ORIGINAL via sum i2 j3 k1 := sum(21, 41)

After

Metric ORIGINAL via sum i2 j3 k1 = 62

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · calculation

Before

Metric ORIGINAL via sum i2 j3 k2 := sum(21, 21)

After

Metric ORIGINAL via sum i2 j3 k2 = 42

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S058

S060 · calculation

Before

Metric ORIGINAL via sum i2 j3 k3 := sum(21, 0)

After

Metric ORIGINAL via sum i2 j3 k3 = 21

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 2, 3]

After

derived = [[39, 20, 0, 21], [81, 62, 42, 21], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S060

S062 · calculation

Before

Metric ORIGINAL via sum i3 j0 k0 := sum(60, 0)

After

Metric ORIGINAL via sum i3 j0 k0 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

Metric ORIGINAL via sum i3 j0 k1 := sum(60, 19)

After

Metric ORIGINAL via sum i3 j0 k1 = 79

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

Metric ORIGINAL via sum i3 j0 k2 := sum(60, 39)

After

Metric ORIGINAL via sum i3 j0 k2 = 99

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

Metric ORIGINAL via sum i3 j0 k3 := sum(60, 60)

After

Metric ORIGINAL via sum i3 j0 k3 = 120

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 3, 0]

After

derived = [[60, 41, 21, 0], [60, 79, 99, 120], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S065

S067 · calculation

Before

Metric ORIGINAL via sum i3 j1 k0 := sum(41, 19)

After

Metric ORIGINAL via sum i3 j1 k0 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Metric ORIGINAL via sum i3 j1 k1 := sum(41, 0)

After

Metric ORIGINAL via sum i3 j1 k1 = 41

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Metric ORIGINAL via sum i3 j1 k2 := sum(41, 20)

After

Metric ORIGINAL via sum i3 j1 k2 = 61

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

Metric ORIGINAL via sum i3 j1 k3 := sum(41, 41)

After

Metric ORIGINAL via sum i3 j1 k3 = 82

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 3, 1]

After

derived = [[60, 41, 21, 0], [60, 41, 61, 82], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S070

S072 · calculation

Before

Metric ORIGINAL via sum i3 j2 k0 := sum(21, 39)

After

Metric ORIGINAL via sum i3 j2 k0 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

Metric ORIGINAL via sum i3 j2 k1 := sum(21, 20)

After

Metric ORIGINAL via sum i3 j2 k1 = 41

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

Metric ORIGINAL via sum i3 j2 k2 := sum(21, 0)

After

Metric ORIGINAL via sum i3 j2 k2 = 21

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

Metric ORIGINAL via sum i3 j2 k3 := sum(21, 21)

After

Metric ORIGINAL via sum i3 j2 k3 = 42

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 3, 2]

After

derived = [[60, 41, 21, 0], [60, 41, 21, 42], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S075

S077 · calculation

Before

Metric ORIGINAL via sum i3 j3 k0 := sum(0, 60)

After

Metric ORIGINAL via sum i3 j3 k0 = 60

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

Metric ORIGINAL via sum i3 j3 k1 := sum(0, 41)

After

Metric ORIGINAL via sum i3 j3 k1 = 41

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

Metric ORIGINAL via sum i3 j3 k2 := sum(0, 21)

After

Metric ORIGINAL via sum i3 j3 k2 = 21

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

Before

Metric ORIGINAL via sum i3 j3 k3 := sum(0, 0)

After

Metric ORIGINAL via sum i3 j3 k3 = 0

Conditions: none

Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], 3, 3]

After

derived = [[60, 41, 21, 0], [60, 41, 21, 0], []]

Conditions: none

For EACH(i,j),test d(i,k)<=d(i,j)+d(j,k) for EVERY k,including repeated points.Retain every failure,not only unordered distinct triangles.

Step identifiers and dependencies

Rule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S080

S082 · implies

Before

source roles = [4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]]]

After

derived = [[], Yes]

Conditions: none

List failures of ALL n³ original ordered triples in nested i,j,k order.No failures establish the triangle obligation for THIS table,not automatically nonnegativity,separation or symmetry.

Step identifiers and dependencies

Rule: Metric_ALL_bad_triangles_nested_order_not_only_first · Uses: G001, S081

Reverse checks

S082 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S081 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S080 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k3 with exact rational arithmetic, not rounded decimals.
S079 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k2 with exact rational arithmetic, not rounded decimals.
S078 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k1 with exact rational arithmetic, not rounded decimals.
S077 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k0 with exact rational arithmetic, not rounded decimals.
S076 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S075 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k3 with exact rational arithmetic, not rounded decimals.
S074 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k2 with exact rational arithmetic, not rounded decimals.
S073 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k1 with exact rational arithmetic, not rounded decimals.
S072 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k0 with exact rational arithmetic, not rounded decimals.
S071 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S070 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k3 with exact rational arithmetic, not rounded decimals.
S069 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k2 with exact rational arithmetic, not rounded decimals.
S068 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k1 with exact rational arithmetic, not rounded decimals.
S067 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k0 with exact rational arithmetic, not rounded decimals.
S066 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S065 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k3 with exact rational arithmetic, not rounded decimals.
S064 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k2 with exact rational arithmetic, not rounded decimals.
S063 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k1 with exact rational arithmetic, not rounded decimals.
S062 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k0 with exact rational arithmetic, not rounded decimals.
S061 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S060 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k3 with exact rational arithmetic, not rounded decimals.
S059 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k2 with exact rational arithmetic, not rounded decimals.
S058 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k1 with exact rational arithmetic, not rounded decimals.
S057 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k0 with exact rational arithmetic, not rounded decimals.
S056 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S055 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k3 with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k2 with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k1 with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k0 with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S050 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k3 with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k2 with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k1 with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k0 with exact rational arithmetic, not rounded decimals.
S046 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S045 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k3 with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k2 with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k1 with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k0 with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S040 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k3 with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k2 with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k1 with exact rational arithmetic, not rounded decimals.
S037 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k0 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S035 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k3 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k2 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k1 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k0 with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S030 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k3 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k2 with exact rational arithmetic, not rounded decimals.
S028 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k1 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k0 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S025 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k3 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k2 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k1 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k0 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S020 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k3 with exact rational arithmetic, not rounded decimals.
S019 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k2 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k1 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k0 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S015 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k3 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k2 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k1 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k0 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S010 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k3 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k2 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k1 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k0 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.
S005 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k3 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k2 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k1 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k0 with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct.

Result

(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], [], Yes)

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: 6c1aa3b2a889a100c8d48a417badf6e47086fcf20f1c935eed813d7374fd50f5
candidate record SHA-256: 9393ffdabeeee2005e3e62000d0289c1c455aa82da4b7db11837698a432baad7
semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON