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Question 29321: All Metric Axioms on a Complete Finite Distance Table

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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R0057-29321 · q-20261013-advance-20261022-r06-metric-space-0096

Original question and givens

The complete finite integer distance table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 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,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).

  • G001 — The complete finite integer distance table=[[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 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,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).

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-str22-metric-full-metric-audit, metric-full-metric-audit]

After

derived = [[n, distances, nonnegative, separation, symmetric, bad_triples, metric]]

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 0]

After

derived = [[Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes]]

Conditions: none

Check nonnegativity and BOTH directions of d(i,j)=0 IFF i=j at EVERY cell.Negative or nonseparating tables are legal audit counterexamples,not pre-rejected inputs.

Step identifiers and dependencies

Rule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S001

S003 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 1]

After

derived = [[Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes]]

Conditions: none

Check nonnegativity and BOTH directions of d(i,j)=0 IFF i=j at EVERY cell.Negative or nonseparating tables are legal audit counterexamples,not pre-rejected inputs.

Step identifiers and dependencies

Rule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S002

S004 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 2]

After

derived = [[Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes]]

Conditions: none

Check nonnegativity and BOTH directions of d(i,j)=0 IFF i=j at EVERY cell.Negative or nonseparating tables are legal audit counterexamples,not pre-rejected inputs.

Step identifiers and dependencies

Rule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S003

S005 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 3]

After

derived = [[Yes, Yes, Yes, Yes], [Yes, Yes, Yes, Yes]]

Conditions: none

Check nonnegativity and BOTH directions of d(i,j)=0 IFF i=j at EVERY cell.Negative or nonseparating tables are legal audit counterexamples,not pre-rejected inputs.

Step identifiers and dependencies

Rule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S004

S006 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 0]

After

derived = [[0, 21, 43, 66], [0, 21, 43, 66], [Yes, Yes, Yes, Yes]]

Conditions: none

Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric.

Step identifiers and dependencies

Rule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S005

S007 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 1]

After

derived = [[21, 0, 22, 45], [21, 0, 22, 45], [Yes, Yes, Yes, Yes]]

Conditions: none

Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric.

Step identifiers and dependencies

Rule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S006

S008 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 2]

After

derived = [[43, 22, 0, 23], [43, 22, 0, 23], [Yes, Yes, Yes, Yes]]

Conditions: none

Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric.

Step identifiers and dependencies

Rule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S007

S009 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 3]

After

derived = [[66, 45, 23, 0], [66, 45, 23, 0], [Yes, Yes, Yes, Yes]]

Conditions: none

Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric.

Step identifiers and dependencies

Rule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S008

S010 · 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, S009

S011 · calculation

Before

Metric ORIGINAL via sum i0 j0 k1 := sum(0, 21)

After

Metric ORIGINAL via sum i0 j0 k1 = 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, S010

S012 · calculation

Before

Metric ORIGINAL via sum i0 j0 k2 := sum(0, 43)

After

Metric ORIGINAL via sum i0 j0 k2 = 43

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 j0 k3 := sum(0, 66)

After

Metric ORIGINAL via sum i0 j0 k3 = 66

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 0, 0]

After

derived = [[0, 21, 43, 66], [0, 21, 43, 66], []]

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, S013

S015 · calculation

Before

Metric ORIGINAL via sum i0 j1 k0 := sum(21, 21)

After

Metric ORIGINAL via sum i0 j1 k0 = 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, S014

S016 · calculation

Before

Metric ORIGINAL via sum i0 j1 k1 := sum(21, 0)

After

Metric ORIGINAL via sum i0 j1 k1 = 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, S015

S017 · calculation

Before

Metric ORIGINAL via sum i0 j1 k2 := sum(21, 22)

After

Metric ORIGINAL via sum i0 j1 k2 = 43

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 j1 k3 := sum(21, 45)

After

Metric ORIGINAL via sum i0 j1 k3 = 66

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 0, 1]

After

derived = [[0, 21, 43, 66], [42, 21, 43, 66], []]

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, S018

S020 · calculation

Before

Metric ORIGINAL via sum i0 j2 k0 := sum(43, 43)

After

Metric ORIGINAL via sum i0 j2 k0 = 86

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 · calculation

Before

Metric ORIGINAL via sum i0 j2 k1 := sum(43, 22)

After

Metric ORIGINAL via sum i0 j2 k1 = 65

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, S020

S022 · calculation

Before

Metric ORIGINAL via sum i0 j2 k2 := sum(43, 0)

After

Metric ORIGINAL via sum i0 j2 k2 = 43

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 i0 j2 k3 := sum(43, 23)

After

Metric ORIGINAL via sum i0 j2 k3 = 66

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 0, 2]

After

derived = [[0, 21, 43, 66], [86, 65, 43, 66], []]

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, S023

S025 · calculation

Before

Metric ORIGINAL via sum i0 j3 k0 := sum(66, 66)

After

Metric ORIGINAL via sum i0 j3 k0 = 132

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 · calculation

Before

Metric ORIGINAL via sum i0 j3 k1 := sum(66, 45)

After

Metric ORIGINAL via sum i0 j3 k1 = 111

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, S025

S027 · calculation

Before

Metric ORIGINAL via sum i0 j3 k2 := sum(66, 23)

After

Metric ORIGINAL via sum i0 j3 k2 = 89

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 i0 j3 k3 := sum(66, 0)

After

Metric ORIGINAL via sum i0 j3 k3 = 66

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 0, 3]

After

derived = [[0, 21, 43, 66], [132, 111, 89, 66], []]

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, S028

S030 · calculation

Before

Metric ORIGINAL via sum i1 j0 k0 := sum(21, 0)

After

Metric ORIGINAL via sum i1 j0 k0 = 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, S029

S031 · calculation

Before

Metric ORIGINAL via sum i1 j0 k1 := sum(21, 21)

After

Metric ORIGINAL via sum i1 j0 k1 = 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, S030

S032 · calculation

Before

Metric ORIGINAL via sum i1 j0 k2 := sum(21, 43)

After

Metric ORIGINAL via sum i1 j0 k2 = 64

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 j0 k3 := sum(21, 66)

After

Metric ORIGINAL via sum i1 j0 k3 = 87

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 1, 0]

After

derived = [[21, 0, 22, 45], [21, 42, 64, 87], []]

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, S033

S035 · calculation

Before

Metric ORIGINAL via sum i1 j1 k0 := sum(0, 21)

After

Metric ORIGINAL via sum i1 j1 k0 = 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, S034

S036 · 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, S035

S037 · calculation

Before

Metric ORIGINAL via sum i1 j1 k2 := sum(0, 22)

After

Metric ORIGINAL via sum i1 j1 k2 = 22

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 j1 k3 := sum(0, 45)

After

Metric ORIGINAL via sum i1 j1 k3 = 45

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 1, 1]

After

derived = [[21, 0, 22, 45], [21, 0, 22, 45], []]

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, S038

S040 · calculation

Before

Metric ORIGINAL via sum i1 j2 k0 := sum(22, 43)

After

Metric ORIGINAL via sum i1 j2 k0 = 65

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 · calculation

Before

Metric ORIGINAL via sum i1 j2 k1 := sum(22, 22)

After

Metric ORIGINAL via sum i1 j2 k1 = 44

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, S040

S042 · calculation

Before

Metric ORIGINAL via sum i1 j2 k2 := sum(22, 0)

After

Metric ORIGINAL via sum i1 j2 k2 = 22

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 i1 j2 k3 := sum(22, 23)

After

Metric ORIGINAL via sum i1 j2 k3 = 45

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 1, 2]

After

derived = [[21, 0, 22, 45], [65, 44, 22, 45], []]

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, S043

S045 · calculation

Before

Metric ORIGINAL via sum i1 j3 k0 := sum(45, 66)

After

Metric ORIGINAL via sum i1 j3 k0 = 111

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 · calculation

Before

Metric ORIGINAL via sum i1 j3 k1 := sum(45, 45)

After

Metric ORIGINAL via sum i1 j3 k1 = 90

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, S045

S047 · calculation

Before

Metric ORIGINAL via sum i1 j3 k2 := sum(45, 23)

After

Metric ORIGINAL via sum i1 j3 k2 = 68

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 i1 j3 k3 := sum(45, 0)

After

Metric ORIGINAL via sum i1 j3 k3 = 45

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 1, 3]

After

derived = [[21, 0, 22, 45], [111, 90, 68, 45], []]

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, S048

S050 · calculation

Before

Metric ORIGINAL via sum i2 j0 k0 := sum(43, 0)

After

Metric ORIGINAL via sum i2 j0 k0 = 43

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 · calculation

Before

Metric ORIGINAL via sum i2 j0 k1 := sum(43, 21)

After

Metric ORIGINAL via sum i2 j0 k1 = 64

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, S050

S052 · calculation

Before

Metric ORIGINAL via sum i2 j0 k2 := sum(43, 43)

After

Metric ORIGINAL via sum i2 j0 k2 = 86

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 j0 k3 := sum(43, 66)

After

Metric ORIGINAL via sum i2 j0 k3 = 109

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 2, 0]

After

derived = [[43, 22, 0, 23], [43, 64, 86, 109], []]

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, S053

S055 · calculation

Before

Metric ORIGINAL via sum i2 j1 k0 := sum(22, 21)

After

Metric ORIGINAL via sum i2 j1 k0 = 43

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 · calculation

Before

Metric ORIGINAL via sum i2 j1 k1 := sum(22, 0)

After

Metric ORIGINAL via sum i2 j1 k1 = 22

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, S055

S057 · calculation

Before

Metric ORIGINAL via sum i2 j1 k2 := sum(22, 22)

After

Metric ORIGINAL via sum i2 j1 k2 = 44

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 j1 k3 := sum(22, 45)

After

Metric ORIGINAL via sum i2 j1 k3 = 67

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 2, 1]

After

derived = [[43, 22, 0, 23], [43, 22, 44, 67], []]

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, S058

S060 · calculation

Before

Metric ORIGINAL via sum i2 j2 k0 := sum(0, 43)

After

Metric ORIGINAL via sum i2 j2 k0 = 43

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 · calculation

Before

Metric ORIGINAL via sum i2 j2 k1 := sum(0, 22)

After

Metric ORIGINAL via sum i2 j2 k1 = 22

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, S060

S062 · 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, S061

S063 · calculation

Before

Metric ORIGINAL via sum i2 j2 k3 := sum(0, 23)

After

Metric ORIGINAL via sum i2 j2 k3 = 23

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 2, 2]

After

derived = [[43, 22, 0, 23], [43, 22, 0, 23], []]

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, S063

S065 · calculation

Before

Metric ORIGINAL via sum i2 j3 k0 := sum(23, 66)

After

Metric ORIGINAL via sum i2 j3 k0 = 89

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 · calculation

Before

Metric ORIGINAL via sum i2 j3 k1 := sum(23, 45)

After

Metric ORIGINAL via sum i2 j3 k1 = 68

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, S065

S067 · calculation

Before

Metric ORIGINAL via sum i2 j3 k2 := sum(23, 23)

After

Metric ORIGINAL via sum i2 j3 k2 = 46

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 i2 j3 k3 := sum(23, 0)

After

Metric ORIGINAL via sum i2 j3 k3 = 23

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 2, 3]

After

derived = [[43, 22, 0, 23], [89, 68, 46, 23], []]

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, S068

S070 · calculation

Before

Metric ORIGINAL via sum i3 j0 k0 := sum(66, 0)

After

Metric ORIGINAL via sum i3 j0 k0 = 66

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 · calculation

Before

Metric ORIGINAL via sum i3 j0 k1 := sum(66, 21)

After

Metric ORIGINAL via sum i3 j0 k1 = 87

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, S070

S072 · calculation

Before

Metric ORIGINAL via sum i3 j0 k2 := sum(66, 43)

After

Metric ORIGINAL via sum i3 j0 k2 = 109

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 j0 k3 := sum(66, 66)

After

Metric ORIGINAL via sum i3 j0 k3 = 132

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 3, 0]

After

derived = [[66, 45, 23, 0], [66, 87, 109, 132], []]

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, S073

S075 · calculation

Before

Metric ORIGINAL via sum i3 j1 k0 := sum(45, 21)

After

Metric ORIGINAL via sum i3 j1 k0 = 66

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 · calculation

Before

Metric ORIGINAL via sum i3 j1 k1 := sum(45, 0)

After

Metric ORIGINAL via sum i3 j1 k1 = 45

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, S075

S077 · calculation

Before

Metric ORIGINAL via sum i3 j1 k2 := sum(45, 22)

After

Metric ORIGINAL via sum i3 j1 k2 = 67

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 j1 k3 := sum(45, 45)

After

Metric ORIGINAL via sum i3 j1 k3 = 90

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 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 3, 1]

After

derived = [[66, 45, 23, 0], [66, 45, 67, 90], []]

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, S078

S080 · calculation

Before

Metric ORIGINAL via sum i3 j2 k0 := sum(23, 43)

After

Metric ORIGINAL via sum i3 j2 k0 = 66

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 · calculation

Before

Metric ORIGINAL via sum i3 j2 k1 := sum(23, 22)

After

Metric ORIGINAL via sum i3 j2 k1 = 45

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, S080

S082 · calculation

Before

Metric ORIGINAL via sum i3 j2 k2 := sum(23, 0)

After

Metric ORIGINAL via sum i3 j2 k2 = 23

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, S081

S083 · calculation

Before

Metric ORIGINAL via sum i3 j2 k3 := sum(23, 23)

After

Metric ORIGINAL via sum i3 j2 k3 = 46

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, S082

S084 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 3, 2]

After

derived = [[66, 45, 23, 0], [66, 45, 23, 46], []]

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, S083

S085 · calculation

Before

Metric ORIGINAL via sum i3 j3 k0 := sum(0, 66)

After

Metric ORIGINAL via sum i3 j3 k0 = 66

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, S084

S086 · calculation

Before

Metric ORIGINAL via sum i3 j3 k1 := sum(0, 45)

After

Metric ORIGINAL via sum i3 j3 k1 = 45

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, S085

S087 · calculation

Before

Metric ORIGINAL via sum i3 j3 k2 := sum(0, 23)

After

Metric ORIGINAL via sum i3 j3 k2 = 23

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, S086

S088 · 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, S087

S089 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], 3, 3]

After

derived = [[66, 45, 23, 0], [66, 45, 23, 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, S088

S090 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 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, S089

S091 · implies

Before

source roles = [4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]]]

After

derived = [Yes, Yes, Yes, [], Yes]

Conditions: none

ALL four obligations must hold for THIS complete finite table to be a metric.Legal counterexamples retain False and the complete bad list.This is not a general proof for arbitrary infinite spaces.

Step identifiers and dependencies

Rule: Metric_FULL_four_axioms_on_COMPLETE_finite_table · Uses: G001, S090

Reverse checks

  • S091 · 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.
  • S090 · 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.
  • S089 · 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.
  • S088 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k3 with exact rational arithmetic, not rounded decimals.
  • S087 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k2 with exact rational arithmetic, not rounded decimals.
  • S086 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k1 with exact rational arithmetic, not rounded decimals.
  • S085 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j3 k0 with exact rational arithmetic, not rounded decimals.
  • S084 · 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.
  • S083 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k3 with exact rational arithmetic, not rounded decimals.
  • S082 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k2 with exact rational arithmetic, not rounded decimals.
  • S081 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k1 with exact rational arithmetic, not rounded decimals.
  • S080 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j2 k0 with exact rational arithmetic, not rounded decimals.
  • S079 · 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.
  • S078 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k3 with exact rational arithmetic, not rounded decimals.
  • S077 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k2 with exact rational arithmetic, not rounded decimals.
  • S076 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k1 with exact rational arithmetic, not rounded decimals.
  • S075 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j1 k0 with exact rational arithmetic, not rounded decimals.
  • S074 · 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.
  • S073 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k3 with exact rational arithmetic, not rounded decimals.
  • S072 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k2 with exact rational arithmetic, not rounded decimals.
  • S071 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k1 with exact rational arithmetic, not rounded decimals.
  • S070 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i3 j0 k0 with exact rational arithmetic, not rounded decimals.
  • S069 · 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.
  • S068 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k3 with exact rational arithmetic, not rounded decimals.
  • S067 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k2 with exact rational arithmetic, not rounded decimals.
  • S066 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k1 with exact rational arithmetic, not rounded decimals.
  • S065 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j3 k0 with exact rational arithmetic, not rounded decimals.
  • S064 · 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.
  • S063 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k3 with exact rational arithmetic, not rounded decimals.
  • S062 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k2 with exact rational arithmetic, not rounded decimals.
  • S061 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k1 with exact rational arithmetic, not rounded decimals.
  • S060 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j2 k0 with exact rational arithmetic, not rounded decimals.
  • S059 · 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.
  • S058 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k3 with exact rational arithmetic, not rounded decimals.
  • S057 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k2 with exact rational arithmetic, not rounded decimals.
  • S056 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k1 with exact rational arithmetic, not rounded decimals.
  • S055 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j1 k0 with exact rational arithmetic, not rounded decimals.
  • S054 · 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.
  • S053 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k3 with exact rational arithmetic, not rounded decimals.
  • S052 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k2 with exact rational arithmetic, not rounded decimals.
  • S051 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k1 with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i2 j0 k0 with exact rational arithmetic, not rounded decimals.
  • S049 · 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.
  • S048 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k3 with exact rational arithmetic, not rounded decimals.
  • S047 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k2 with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k1 with exact rational arithmetic, not rounded decimals.
  • S045 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j3 k0 with exact rational arithmetic, not rounded decimals.
  • S044 · 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.
  • S043 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k3 with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k2 with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k1 with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j2 k0 with exact rational arithmetic, not rounded decimals.
  • S039 · 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.
  • S038 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k3 with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k2 with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k1 with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j1 k0 with exact rational arithmetic, not rounded decimals.
  • S034 · 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.
  • S033 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k3 with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k2 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k1 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i1 j0 k0 with exact rational arithmetic, not rounded decimals.
  • S029 · 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.
  • S028 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k3 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k2 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k1 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j3 k0 with exact rational arithmetic, not rounded decimals.
  • S024 · 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.
  • S023 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k3 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k2 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k1 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j2 k0 with exact rational arithmetic, not rounded decimals.
  • S019 · 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.
  • S018 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k3 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k2 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j1 k0 with exact rational arithmetic, not rounded decimals.
  • S014 · 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.
  • S013 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k3 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k2 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k1 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Metric ORIGINAL via sum i0 j0 k0 with exact rational arithmetic, not rounded decimals.
  • S009 · 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.
  • S008 · 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.
  • S007 · 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.
  • 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 — 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.
  • S004 · 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.
  • S003 · 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.
  • S002 · 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.
  • 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, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes, Yes, [], 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: a412fb1635697bd82c174ee6f012ec61169f0db48caa2f88f64a4774969b227e
  • candidate record SHA-256: f3b3d513a294c1a1f2480e637a91037f925a83ff227cd72731946b68f9d9146a
  • semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6
  • question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON