# Question 29311: All Metric Axioms on a Complete Finite Distance Table [Return to catalogue page 147](https://ocme.evemisslab.com/en/questions/page/147/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r06-metric-space-foundations/#q-20261013-advance-20261022-r06-metric-space-0086) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0057-29311 · q-20261013-advance-20261022-r06-metric-space-0086 ## Original question and givens The complete finite integer distance table=[[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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. - [mko-metric-space](https://ocme.evemisslab.com/en/mko/mko-metric-space/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001 S002 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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 dependenciesRule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S001 S003 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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 dependenciesRule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S002 S004 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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 dependenciesRule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S003 S005 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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 dependenciesRule: Metric_ALL_row_nonnegativity_AND_zero_IFF_samepoint · Uses: G001, S004 S006 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 0] After derived = [[0, 11, 23, 36], [0, 11, 23, 36], [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 dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S005 S007 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 1] After derived = [[11, 0, 12, 25], [11, 0, 12, 25], [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 dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S006 S008 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 2] After derived = [[23, 12, 0, 13], [23, 12, 0, 13], [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 dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S007 S009 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 3] After derived = [[36, 25, 13, 0], [36, 25, 13, 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 dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S008 S010 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Metric ORIGINAL via sum i0 j0 k1 := sum(0, 11) After Metric ORIGINAL via sum i0 j0 k1 = 11 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Metric ORIGINAL via sum i0 j0 k2 := sum(0, 23) After Metric ORIGINAL via sum i0 j0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Metric ORIGINAL via sum i0 j0 k3 := sum(0, 36) After Metric ORIGINAL via sum i0 j0 k3 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 0, 0] After derived = [[0, 11, 23, 36], [0, 11, 23, 36], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S013 S015 · calculationBefore Metric ORIGINAL via sum i0 j1 k0 := sum(11, 11) After Metric ORIGINAL via sum i0 j1 k0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Metric ORIGINAL via sum i0 j1 k1 := sum(11, 0) After Metric ORIGINAL via sum i0 j1 k1 = 11 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Metric ORIGINAL via sum i0 j1 k2 := sum(11, 12) After Metric ORIGINAL via sum i0 j1 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Metric ORIGINAL via sum i0 j1 k3 := sum(11, 25) After Metric ORIGINAL via sum i0 j1 k3 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 0, 1] After derived = [[0, 11, 23, 36], [22, 11, 23, 36], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S018 S020 · calculationBefore Metric ORIGINAL via sum i0 j2 k0 := sum(23, 23) After Metric ORIGINAL via sum i0 j2 k0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Metric ORIGINAL via sum i0 j2 k1 := sum(23, 12) After Metric ORIGINAL via sum i0 j2 k1 = 35 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Metric ORIGINAL via sum i0 j2 k2 := sum(23, 0) After Metric ORIGINAL via sum i0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Metric ORIGINAL via sum i0 j2 k3 := sum(23, 13) After Metric ORIGINAL via sum i0 j2 k3 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 0, 2] After derived = [[0, 11, 23, 36], [46, 35, 23, 36], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S023 S025 · calculationBefore Metric ORIGINAL via sum i0 j3 k0 := sum(36, 36) After Metric ORIGINAL via sum i0 j3 k0 = 72 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Metric ORIGINAL via sum i0 j3 k1 := sum(36, 25) After Metric ORIGINAL via sum i0 j3 k1 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Metric ORIGINAL via sum i0 j3 k2 := sum(36, 13) After Metric ORIGINAL via sum i0 j3 k2 = 49 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Metric ORIGINAL via sum i0 j3 k3 := sum(36, 0) After Metric ORIGINAL via sum i0 j3 k3 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 0, 3] After derived = [[0, 11, 23, 36], [72, 61, 49, 36], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S028 S030 · calculationBefore Metric ORIGINAL via sum i1 j0 k0 := sum(11, 0) After Metric ORIGINAL via sum i1 j0 k0 = 11 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Metric ORIGINAL via sum i1 j0 k1 := sum(11, 11) After Metric ORIGINAL via sum i1 j0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Metric ORIGINAL via sum i1 j0 k2 := sum(11, 23) After Metric ORIGINAL via sum i1 j0 k2 = 34 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Metric ORIGINAL via sum i1 j0 k3 := sum(11, 36) After Metric ORIGINAL via sum i1 j0 k3 = 47 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 1, 0] After derived = [[11, 0, 12, 25], [11, 22, 34, 47], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S033 S035 · calculationBefore Metric ORIGINAL via sum i1 j1 k0 := sum(0, 11) After Metric ORIGINAL via sum i1 j1 k0 = 11 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore Metric ORIGINAL via sum i1 j1 k2 := sum(0, 12) After Metric ORIGINAL via sum i1 j1 k2 = 12 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · calculationBefore Metric ORIGINAL via sum i1 j1 k3 := sum(0, 25) After Metric ORIGINAL via sum i1 j1 k3 = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 1, 1] After derived = [[11, 0, 12, 25], [11, 0, 12, 25], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S038 S040 · calculationBefore Metric ORIGINAL via sum i1 j2 k0 := sum(12, 23) After Metric ORIGINAL via sum i1 j2 k0 = 35 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore Metric ORIGINAL via sum i1 j2 k1 := sum(12, 12) After Metric ORIGINAL via sum i1 j2 k1 = 24 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore Metric ORIGINAL via sum i1 j2 k2 := sum(12, 0) After Metric ORIGINAL via sum i1 j2 k2 = 12 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore Metric ORIGINAL via sum i1 j2 k3 := sum(12, 13) After Metric ORIGINAL via sum i1 j2 k3 = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 1, 2] After derived = [[11, 0, 12, 25], [35, 24, 12, 25], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S043 S045 · calculationBefore Metric ORIGINAL via sum i1 j3 k0 := sum(25, 36) After Metric ORIGINAL via sum i1 j3 k0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore Metric ORIGINAL via sum i1 j3 k1 := sum(25, 25) After Metric ORIGINAL via sum i1 j3 k1 = 50 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · calculationBefore Metric ORIGINAL via sum i1 j3 k2 := sum(25, 13) After Metric ORIGINAL via sum i1 j3 k2 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S046 S048 · calculationBefore Metric ORIGINAL via sum i1 j3 k3 := sum(25, 0) After Metric ORIGINAL via sum i1 j3 k3 = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 1, 3] After derived = [[11, 0, 12, 25], [61, 50, 38, 25], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S048 S050 · calculationBefore Metric ORIGINAL via sum i2 j0 k0 := sum(23, 0) After Metric ORIGINAL via sum i2 j0 k0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore Metric ORIGINAL via sum i2 j0 k1 := sum(23, 11) After Metric ORIGINAL via sum i2 j0 k1 = 34 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore Metric ORIGINAL via sum i2 j0 k2 := sum(23, 23) After Metric ORIGINAL via sum i2 j0 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore Metric ORIGINAL via sum i2 j0 k3 := sum(23, 36) After Metric ORIGINAL via sum i2 j0 k3 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 2, 0] After derived = [[23, 12, 0, 13], [23, 34, 46, 59], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S053 S055 · calculationBefore Metric ORIGINAL via sum i2 j1 k0 := sum(12, 11) After Metric ORIGINAL via sum i2 j1 k0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · calculationBefore Metric ORIGINAL via sum i2 j1 k1 := sum(12, 0) After Metric ORIGINAL via sum i2 j1 k1 = 12 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S055 S057 · calculationBefore Metric ORIGINAL via sum i2 j1 k2 := sum(12, 12) After Metric ORIGINAL via sum i2 j1 k2 = 24 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S056 S058 · calculationBefore Metric ORIGINAL via sum i2 j1 k3 := sum(12, 25) After Metric ORIGINAL via sum i2 j1 k3 = 37 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S057 S059 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 2, 1] After derived = [[23, 12, 0, 13], [23, 12, 24, 37], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S058 S060 · calculationBefore Metric ORIGINAL via sum i2 j2 k0 := sum(0, 23) After Metric ORIGINAL via sum i2 j2 k0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S059 S061 · calculationBefore Metric ORIGINAL via sum i2 j2 k1 := sum(0, 12) After Metric ORIGINAL via sum i2 j2 k1 = 12 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S060 S062 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S061 S063 · calculationBefore Metric ORIGINAL via sum i2 j2 k3 := sum(0, 13) After Metric ORIGINAL via sum i2 j2 k3 = 13 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S062 S064 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 2, 2] After derived = [[23, 12, 0, 13], [23, 12, 0, 13], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S063 S065 · calculationBefore Metric ORIGINAL via sum i2 j3 k0 := sum(13, 36) After Metric ORIGINAL via sum i2 j3 k0 = 49 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S064 S066 · calculationBefore Metric ORIGINAL via sum i2 j3 k1 := sum(13, 25) After Metric ORIGINAL via sum i2 j3 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S065 S067 · calculationBefore Metric ORIGINAL via sum i2 j3 k2 := sum(13, 13) After Metric ORIGINAL via sum i2 j3 k2 = 26 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S066 S068 · calculationBefore Metric ORIGINAL via sum i2 j3 k3 := sum(13, 0) After Metric ORIGINAL via sum i2 j3 k3 = 13 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S067 S069 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 2, 3] After derived = [[23, 12, 0, 13], [49, 38, 26, 13], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S068 S070 · calculationBefore Metric ORIGINAL via sum i3 j0 k0 := sum(36, 0) After Metric ORIGINAL via sum i3 j0 k0 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S069 S071 · calculationBefore Metric ORIGINAL via sum i3 j0 k1 := sum(36, 11) After Metric ORIGINAL via sum i3 j0 k1 = 47 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S070 S072 · calculationBefore Metric ORIGINAL via sum i3 j0 k2 := sum(36, 23) After Metric ORIGINAL via sum i3 j0 k2 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S071 S073 · calculationBefore Metric ORIGINAL via sum i3 j0 k3 := sum(36, 36) After Metric ORIGINAL via sum i3 j0 k3 = 72 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S072 S074 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 3, 0] After derived = [[36, 25, 13, 0], [36, 47, 59, 72], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S073 S075 · calculationBefore Metric ORIGINAL via sum i3 j1 k0 := sum(25, 11) After Metric ORIGINAL via sum i3 j1 k0 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S074 S076 · calculationBefore Metric ORIGINAL via sum i3 j1 k1 := sum(25, 0) After Metric ORIGINAL via sum i3 j1 k1 = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S075 S077 · calculationBefore Metric ORIGINAL via sum i3 j1 k2 := sum(25, 12) After Metric ORIGINAL via sum i3 j1 k2 = 37 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S076 S078 · calculationBefore Metric ORIGINAL via sum i3 j1 k3 := sum(25, 25) After Metric ORIGINAL via sum i3 j1 k3 = 50 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S077 S079 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 3, 1] After derived = [[36, 25, 13, 0], [36, 25, 37, 50], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S078 S080 · calculationBefore Metric ORIGINAL via sum i3 j2 k0 := sum(13, 23) After Metric ORIGINAL via sum i3 j2 k0 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S079 S081 · calculationBefore Metric ORIGINAL via sum i3 j2 k1 := sum(13, 12) After Metric ORIGINAL via sum i3 j2 k1 = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S080 S082 · calculationBefore Metric ORIGINAL via sum i3 j2 k2 := sum(13, 0) After Metric ORIGINAL via sum i3 j2 k2 = 13 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S081 S083 · calculationBefore Metric ORIGINAL via sum i3 j2 k3 := sum(13, 13) After Metric ORIGINAL via sum i3 j2 k3 = 26 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S082 S084 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 3, 2] After derived = [[36, 25, 13, 0], [36, 25, 13, 26], []] 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S083 S085 · calculationBefore Metric ORIGINAL via sum i3 j3 k0 := sum(0, 36) After Metric ORIGINAL via sum i3 j3 k0 = 36 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S084 S086 · calculationBefore Metric ORIGINAL via sum i3 j3 k1 := sum(0, 25) After Metric ORIGINAL via sum i3 j3 k1 = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S085 S087 · calculationBefore Metric ORIGINAL via sum i3 j3 k2 := sum(0, 13) After Metric ORIGINAL via sum i3 j3 k2 = 13 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S086 S088 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S087 S089 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], 3, 3] After derived = [[36, 25, 13, 0], [36, 25, 13, 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 dependenciesRule: Metric_ALL_third_points_EACH_ordered_pair_triangle · Uses: G001, S088 S090 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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 dependenciesRule: Metric_ALL_bad_triangles_nested_order_not_only_first · Uses: G001, S089 S091 · impliesBefore source roles = [4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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 dependenciesRule: 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, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 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: e6fcbb6df8cd4826a782798e039affd3403ef0608c3f9a88e5ff2b52fd6775f3 - candidate record SHA-256: 0a94604c9939a5ee45e6f77e70ded296b2be7f360b0428caf148081f26f3dfeb - semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6 - question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r06-metric-space-0086.json)