# Question 29329: Compare Both Directions of Every Distance Pair [Return to catalogue page 147](https://ocme.evemisslab.com/en/questions/page/147/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r06-metric-space-foundations/#q-20261013-advance-20261022-r06-metric-space-0029) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0057-29329 · q-20261013-advance-20261022-r06-metric-space-0029 ## Original question and givens 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?). - G001 — 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?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-metric-space](https://ocme.evemisslab.com/en/mko/mko-metric-space/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-str22-metric-symmetry-audit, metric-symmetry-audit] After derived = [[n, distances, symmetric]] Conditions: none Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning. Step identifiers and dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001 S002 · impliesBefore source roles = [4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], 0] After derived = [[0, 4, 9, 15], [0, 4, 9, 15], [Yes, Yes, Yes, Yes]] Conditions: none Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric. Step identifiers and dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S001 S003 · impliesBefore source roles = [4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], 1] After derived = [[4, 0, 5, 11], [4, 0, 5, 11], [Yes, Yes, Yes, Yes]] Conditions: none Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric. Step identifiers and dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S002 S004 · impliesBefore source roles = [4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], 2] After derived = [[9, 5, 0, 6], [9, 5, 0, 6], [Yes, Yes, Yes, Yes]] Conditions: none Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric. Step identifiers and dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S003 S005 · impliesBefore source roles = [4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], 3] After derived = [[15, 11, 6, 0], [15, 11, 6, 0], [Yes, Yes, Yes, Yes]] Conditions: none Compare d(i,j) with d(j,i) for EVERY original j,without assuming other axioms.Symmetry may hold even when the table is not a metric. Step identifiers and dependenciesRule: Metric_EVERY_transposed_pair_symmetry · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S004 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S003 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S002 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. ## Result (4, [[0, 4, 9, 15], [4, 0, 5, 11], [9, 5, 0, 6], [15, 11, 6, 0]], Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 72fe331ab4b6e32b8f339337fe8dede9d59c25d0167f68ce87b7c1eb9c29db2c - candidate record SHA-256: 429f7d35459eb49a092cd6f16b635fcb9d868d72f956ae95ef2588b4247bfb71 - semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6 - question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r06-metric-space-0029.json)