Question 26259: A Global Contraction Needs Every Metric and Mapping Qualification

Original question and givens

Use the complete exact metric matrix=[[0, 9, 27], [9, 0, 18], [27, 18, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).

• G001 — Use the complete exact metric matrix=[[0, 9, 27], [9, 0, 18], [27, 18, 0]], aligned mapping=[0, 0, 1] and q=1/2. Qualify nonnegativity, separation, symmetry and every triangle, the entire self-map and every ordered contraction pair. A finite metric space is complete. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).

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-contraction-mapping-theorem

Step-by-step derivation

source roles = [contract-metric-qualification]

derived = [[point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate]]

source roles = [[[0, 9, 27], [9, 0, 18], [27, 18, 0]]]

derived = [3, Yes, Yes, Yes, [[0, 0, 0, 0, 0, Yes], [0, 0, 1, 9, 9, Yes], [0, 0, 2, 27, 27, Yes], [0, 1, 0, 0, 18, Yes], [0, 1, 1, 9, 9, Yes], [0, 1, 2, 27, 27, Yes], [0, 2, 0, 0, 54, Yes], [0, 2, 1, 9, 45, Yes], [0, 2, 2, 27, 27, Yes], [1, 0, 0, 9, 9, Yes], [1, 0, 1, 0, 18, Yes], [1, 0, 2, 18, 36, Yes], [1, 1, 0, 9, 9, Yes], [1, 1, 1, 0, 0, Yes], [1, 1, 2, 18, 18, Yes], [1, 2, 0, 9, 45, Yes], [1, 2, 1, 0, 36, Yes], [1, 2, 2, 18, 18, Yes], [2, 0, 0, 27, 27, Yes], [2, 0, 1, 18, 36, Yes], [2, 0, 2, 0, 54, Yes], [2, 1, 0, 27, 27, Yes], [2, 1, 1, 18, 18, Yes], [2, 1, 2, 0, 36, Yes], [2, 2, 0, 27, 27, Yes], [2, 2, 1, 18, 18, Yes], [2, 2, 2, 0, 0, Yes]], Yes]

Contraction ORIGINAL pair0,0 q times distance := mul(1/2, 0)

Contraction ORIGINAL pair0,0 q times distance = 0

Contraction ORIGINAL pair0,1 q times distance := mul(1/2, 9)

Contraction ORIGINAL pair0,1 q times distance = 9/2

Contraction ORIGINAL pair0,2 q times distance := mul(1/2, 27)

Contraction ORIGINAL pair0,2 q times distance = 27/2

Contraction ORIGINAL pair1,0 q times distance := mul(1/2, 9)

Contraction ORIGINAL pair1,0 q times distance = 9/2

Contraction ORIGINAL pair1,1 q times distance := mul(1/2, 0)

Contraction ORIGINAL pair1,1 q times distance = 0

Contraction ORIGINAL pair1,2 q times distance := mul(1/2, 18)

Contraction ORIGINAL pair1,2 q times distance = 9

Contraction ORIGINAL pair2,0 q times distance := mul(1/2, 27)

Contraction ORIGINAL pair2,0 q times distance = 27/2

Contraction ORIGINAL pair2,1 q times distance := mul(1/2, 18)

Contraction ORIGINAL pair2,1 q times distance = 9

Contraction ORIGINAL pair2,2 q times distance := mul(1/2, 0)

Contraction ORIGINAL pair2,2 q times distance = 0

source roles = [[0, 0, 1], 1/2, [[0, 9, 27], [9, 0, 18], [27, 18, 0]]]

derived = [Yes, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9/2, Yes], [0, 2, 9, 27/2, Yes], [1, 0, 0, 9/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 9, 9, Yes], [2, 0, 9, 27/2, Yes], [2, 1, 9, 9, Yes], [2, 2, 0, 0, Yes]], Yes, [0]]

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute Contraction ORIGINAL pair2,2 q times distance with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Contraction ORIGINAL pair2,1 q times distance with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Contraction ORIGINAL pair2,0 q times distance with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Contraction ORIGINAL pair1,2 q times distance with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Contraction ORIGINAL pair1,1 q times distance with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Contraction ORIGINAL pair1,0 q times distance with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Contraction ORIGINAL pair0,2 q times distance with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Contraction ORIGINAL pair0,1 q times distance with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Contraction ORIGINAL pair0,0 q times distance with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9/2, Yes], [0, 2, 9, 27/2, Yes], [1, 0, 0, 9/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 9, 9, Yes], [2, 0, 9, 27/2, Yes], [2, 1, 9, 9, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)

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.