Complete Finite Distance Tables and All Triangle Inequalities
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261022-r06-metric-space-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
2. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
3. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
4. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
5. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
6. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
7. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
8. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
9. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
10. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
11. Nonnegativity and Separation Are Independent Table Checks
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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
12. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
13. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
14. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
15. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
16. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
17. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
18. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
19. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
20. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
21. Nonnegativity and Separation Are Independent Table Checks
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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
22. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
23. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
24. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
25. Nonnegativity and Separation Are Independent Table Checks
The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 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,all entries nonnegative?,zero exactly on the diagonal?).
View answer and explanation
(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes)
Check every entry and both directions of the zero iff equal condition. Neither these two flags nor symmetry alone proves the full metric axioms. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-table-qualification · candidate
26. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
27. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
28. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
29. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
30. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
31. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
32. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
33. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
34. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
35. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
36. Compare Both Directions of Every Distance Pair
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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
37. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
38. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
39. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
40. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
41. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
42. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
43. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
44. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
45. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
46. Compare Both Directions of Every Distance Pair
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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
47. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
48. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
49. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
50. Compare Both Directions of Every Distance Pair
The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 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,d[i][j]=d[j][i] for every ordered pair?).
View answer and explanation
(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes)
Inspect corresponding transposed entries without assuming that nonnegativity, separation or triangle inequalities follow. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-symmetry-audit · candidate
51. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
52. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
53. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
54. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
55. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
56. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
57. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
58. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
59. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
60. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
61. Every Ordered Triangle Inequality in the Finite Table
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,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
62. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
63. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
64. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
65. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
66. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
67. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
68. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
69. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
70. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
71. Every Ordered Triangle Inequality in the Finite Table
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,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
72. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
73. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
74. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
75. Every Ordered Triangle Inequality in the Finite Table
The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,every [i,j,k] violating d[i][k]≤d[i][j]+d[j][k],all triangle inequalities hold?). Use nested increasing i,j,k order, including repeated points.
View answer and explanation
(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], [], Yes)
Test all ordered triples, not just unordered distinct triangles. A symmetry check does not replace these inequalities. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-triangle-audit · candidate
76. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 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?).
View answer and explanation
(4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 1, 3, 6], [1, 0, 2, 5], [3, 2, 0, 3], [6, 5, 3, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
77. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 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?).
View answer and explanation
(4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 2, 5, 9], [2, 0, 3, 7], [5, 3, 0, 4], [9, 7, 4, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
78. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 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?).
View answer and explanation
(4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 3, 7, 12], [3, 0, 4, 9], [7, 4, 0, 5], [12, 9, 5, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
79. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 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?).
View answer and explanation
(4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
80. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 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?).
View answer and explanation
(4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 5, 11, 18], [5, 0, 6, 13], [11, 6, 0, 7], [18, 13, 7, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
81. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 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?).
View answer and explanation
(4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 6, 13, 21], [6, 0, 7, 15], [13, 7, 0, 8], [21, 15, 8, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
82. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 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?).
View answer and explanation
(4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 7, 15, 24], [7, 0, 8, 17], [15, 8, 0, 9], [24, 17, 9, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
83. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 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?).
View answer and explanation
(4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 8, 17, 27], [8, 0, 9, 19], [17, 9, 0, 10], [27, 19, 10, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
84. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 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?).
View answer and explanation
(4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 9, 19, 30], [9, 0, 10, 21], [19, 10, 0, 11], [30, 21, 11, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
85. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 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?).
View answer and explanation
(4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 10, 21, 33], [10, 0, 11, 23], [21, 11, 0, 12], [33, 23, 12, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
86. All Metric Axioms on a Complete Finite Distance Table
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?).
View answer and explanation
(4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 11, 23, 36], [11, 0, 12, 25], [23, 12, 0, 13], [36, 25, 13, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
87. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 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?).
View answer and explanation
(4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 12, 25, 39], [12, 0, 13, 27], [25, 13, 0, 14], [39, 27, 14, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
88. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 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?).
View answer and explanation
(4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 13, 27, 42], [13, 0, 14, 29], [27, 14, 0, 15], [42, 29, 15, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
89. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 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?).
View answer and explanation
(4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 14, 29, 45], [14, 0, 15, 31], [29, 15, 0, 16], [45, 31, 16, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
90. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 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?).
View answer and explanation
(4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 15, 31, 48], [15, 0, 16, 33], [31, 16, 0, 17], [48, 33, 17, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
91. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 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?).
View answer and explanation
(4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 16, 33, 51], [16, 0, 17, 35], [33, 17, 0, 18], [51, 35, 18, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
92. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 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?).
View answer and explanation
(4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 17, 35, 54], [17, 0, 18, 37], [35, 18, 0, 19], [54, 37, 19, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
93. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 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?).
View answer and explanation
(4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 18, 37, 57], [18, 0, 19, 39], [37, 19, 0, 20], [57, 39, 20, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
94. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]] on Fin(4)={0,…,3} has entry d[i][j]. Retain every label and row/column order. A qualified metric requires nonnegativity, d[i][j]=0 iff i=j, symmetry and all triangle inequalities; finite-model verification does not establish a claim for arbitrary infinite spaces. Give (size,distances,nonnegative?,zero iff equal?,symmetric?,all bad [i,j,k] triangle triples in nested increasing order,is the table a metric?).
View answer and explanation
(4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 19, 39, 60], [19, 0, 20, 41], [39, 20, 0, 21], [60, 41, 21, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
95. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 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?).
View answer and explanation
(4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 20, 41, 63], [20, 0, 21, 43], [41, 21, 0, 22], [63, 43, 22, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
96. All Metric Axioms on a Complete Finite Distance Table
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?).
View answer and explanation
(4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 21, 43, 66], [21, 0, 22, 45], [43, 22, 0, 23], [66, 45, 23, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
97. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 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?).
View answer and explanation
(4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 22, 45, 69], [22, 0, 23, 47], [45, 23, 0, 24], [69, 47, 24, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
98. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 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?).
View answer and explanation
(4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 23, 47, 72], [23, 0, 24, 49], [47, 24, 0, 25], [72, 49, 25, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
99. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 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?).
View answer and explanation
(4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 24, 49, 75], [24, 0, 25, 51], [49, 25, 0, 26], [75, 51, 26, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
100. All Metric Axioms on a Complete Finite Distance Table
The complete finite integer distance table=[[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 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?).
View answer and explanation
(4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes, Yes, [], Yes)
Require all four metric obligations together. The exhaustive result concerns only this complete finite table. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (4, [[0, 25, 51, 78], [25, 0, 26, 53], [51, 26, 0, 27], [78, 53, 27, 0]], Yes, Yes, Yes, [], Yes).
Reviewed step-by-step derivation
Metric space · qf-str22-metric-full-metric-audit · candidate
Canonical question JSONL · English JSONL · Validation report