# Question 26232: The Geometric Tail Bound Controls the Exact Picard Error [Return to catalogue page 132](https://ocme.evemisslab.com/en/questions/page/132/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations/#q-20261015-advance-20261116-r01-contraction-mapping-theorem-0057) 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. R0051-26232 · q-20261015-advance-20261116-r01-contraction-mapping-theorem-0057 ## Original question and givens Use the complete exact metric matrix=[[0, 7, 21], [7, 0, 14], [21, 14, 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. Start=2, steps=3. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate). - G001 — Use the complete exact metric matrix=[[0, 7, 21], [7, 0, 14], [21, 14, 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. Start=2, steps=3. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, 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](https://ocme.evemisslab.com/en/mko/mko-contraction-mapping-theorem/) ## Step-by-step derivation S001 · impliesBefore source roles = [contract-fixed-point-bound] After derived = [[orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate]] Conditions: none List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [[[0, 7, 21], [7, 0, 14], [21, 14, 0]]] After derived = [3, Yes, Yes, Yes, [[0, 0, 0, 0, 0, Yes], [0, 0, 1, 7, 7, Yes], [0, 0, 2, 21, 21, Yes], [0, 1, 0, 0, 14, Yes], [0, 1, 1, 7, 7, Yes], [0, 1, 2, 21, 21, Yes], [0, 2, 0, 0, 42, Yes], [0, 2, 1, 7, 35, Yes], [0, 2, 2, 21, 21, Yes], [1, 0, 0, 7, 7, Yes], [1, 0, 1, 0, 14, Yes], [1, 0, 2, 14, 28, Yes], [1, 1, 0, 7, 7, Yes], [1, 1, 1, 0, 0, Yes], [1, 1, 2, 14, 14, Yes], [1, 2, 0, 7, 35, Yes], [1, 2, 1, 0, 28, Yes], [1, 2, 2, 14, 14, Yes], [2, 0, 0, 21, 21, Yes], [2, 0, 1, 14, 28, Yes], [2, 0, 2, 0, 42, Yes], [2, 1, 0, 21, 21, Yes], [2, 1, 1, 14, 14, Yes], [2, 1, 2, 0, 28, Yes], [2, 2, 0, 21, 21, Yes], [2, 2, 1, 14, 14, Yes], [2, 2, 2, 0, 0, Yes]], Yes] Conditions: none Check every triangle triple,not sampled metric behavior.A nonempty finite metric is complete:trivial for a singleton;otherwise the minimum positive distance delta forces a Cauchy tail with distances