# Complete Finite Distance Tables and All Triangle Inequalities

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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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r06-metric-space-0100/)

Metric space · qf-str22-metric-full-metric-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r06-metric-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261022-r06-metric-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r06-metric-space-foundations/validation-report.json)