Global Contraction Qualifications, Picard Orbits and Exact Tail Bounds
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261116-r01-contraction-mapping-theorem-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 1, 3], [1, 0, 2], [3, 2, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1/2, Yes], [0, 2, 1, 3/2, Yes], [1, 0, 0, 1/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 1, 1, Yes], [2, 0, 1, 3/2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1/2, Yes], [0, 2, 1, 3/2, Yes], [1, 0, 0, 1/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 1, 1, Yes], [2, 0, 1, 3/2, Yes], [2, 1, 1, 1, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
2. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 2, 6], [2, 0, 4], [6, 4, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1, Yes], [0, 2, 2, 3, Yes], [1, 0, 0, 1, Yes], [1, 1, 0, 0, Yes], [1, 2, 2, 2, Yes], [2, 0, 2, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 1, Yes], [0, 2, 2, 3, Yes], [1, 0, 0, 1, Yes], [1, 1, 0, 0, Yes], [1, 2, 2, 2, Yes], [2, 0, 2, 3, Yes], [2, 1, 2, 2, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
3. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 3, 9], [3, 0, 6], [9, 6, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3/2, Yes], [0, 2, 3, 9/2, Yes], [1, 0, 0, 3/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 3, 3, Yes], [2, 0, 3, 9/2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3/2, Yes], [0, 2, 3, 9/2, Yes], [1, 0, 0, 3/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 3, 3, Yes], [2, 0, 3, 9/2, Yes], [2, 1, 3, 3, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
4. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 4, 12], [4, 0, 8], [12, 8, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 2, Yes], [0, 2, 4, 6, Yes], [1, 0, 0, 2, Yes], [1, 1, 0, 0, Yes], [1, 2, 4, 4, Yes], [2, 0, 4, 6, Yes], [2, 1, 4, 4, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 2, Yes], [0, 2, 4, 6, Yes], [1, 0, 0, 2, Yes], [1, 1, 0, 0, Yes], [1, 2, 4, 4, Yes], [2, 0, 4, 6, Yes], [2, 1, 4, 4, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
5. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 5, 15], [5, 0, 10], [15, 10, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5/2, Yes], [0, 2, 5, 15/2, Yes], [1, 0, 0, 5/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 5, 5, Yes], [2, 0, 5, 15/2, Yes], [2, 1, 5, 5, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5/2, Yes], [0, 2, 5, 15/2, Yes], [1, 0, 0, 5/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 5, 5, Yes], [2, 0, 5, 15/2, Yes], [2, 1, 5, 5, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
6. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 6, 18], [6, 0, 12], [18, 12, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3, Yes], [0, 2, 6, 9, Yes], [1, 0, 0, 3, Yes], [1, 1, 0, 0, Yes], [1, 2, 6, 6, Yes], [2, 0, 6, 9, Yes], [2, 1, 6, 6, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 3, Yes], [0, 2, 6, 9, Yes], [1, 0, 0, 3, Yes], [1, 1, 0, 0, Yes], [1, 2, 6, 6, Yes], [2, 0, 6, 9, Yes], [2, 1, 6, 6, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
7. A Global Contraction Needs Every Metric and Mapping Qualification
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. Give (point_count, self_map, q, q_in_range, pair_inequality_rows, global_contraction, certificate).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7/2, Yes], [0, 2, 7, 21/2, Yes], [1, 0, 0, 7/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 7, 7, Yes], [2, 0, 7, 21/2, Yes], [2, 1, 7, 7, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7/2, Yes], [0, 2, 7, 21/2, Yes], [1, 0, 0, 7/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 7, 7, Yes], [2, 0, 7, 21/2, Yes], [2, 1, 7, 7, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
8. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 8, 24], [8, 0, 16], [24, 16, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 4, Yes], [0, 2, 8, 12, Yes], [1, 0, 0, 4, Yes], [1, 1, 0, 0, Yes], [1, 2, 8, 8, Yes], [2, 0, 8, 12, Yes], [2, 1, 8, 8, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 4, Yes], [0, 2, 8, 12, Yes], [1, 0, 0, 4, Yes], [1, 1, 0, 0, Yes], [1, 2, 8, 8, Yes], [2, 0, 8, 12, Yes], [2, 1, 8, 8, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
9. A Global Contraction Needs Every Metric and Mapping Qualification
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).
View answer and explanation
(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)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (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).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
10. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5, Yes], [0, 2, 10, 15, Yes], [1, 0, 0, 5, Yes], [1, 1, 0, 0, Yes], [1, 2, 10, 10, Yes], [2, 0, 10, 15, Yes], [2, 1, 10, 10, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 5, Yes], [0, 2, 10, 15, Yes], [1, 0, 0, 5, Yes], [1, 1, 0, 0, Yes], [1, 2, 10, 10, Yes], [2, 0, 10, 15, Yes], [2, 1, 10, 10, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
11. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 11, 33], [11, 0, 22], [33, 22, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11/2, Yes], [0, 2, 11, 33/2, Yes], [1, 0, 0, 11/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 11, 11, Yes], [2, 0, 11, 33/2, Yes], [2, 1, 11, 11, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11/2, Yes], [0, 2, 11, 33/2, Yes], [1, 0, 0, 11/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 11, 11, Yes], [2, 0, 11, 33/2, Yes], [2, 1, 11, 11, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
12. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 12, 36], [12, 0, 24], [36, 24, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 6, Yes], [0, 2, 12, 18, Yes], [1, 0, 0, 6, Yes], [1, 1, 0, 0, Yes], [1, 2, 12, 12, Yes], [2, 0, 12, 18, Yes], [2, 1, 12, 12, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 6, Yes], [0, 2, 12, 18, Yes], [1, 0, 0, 6, Yes], [1, 1, 0, 0, Yes], [1, 2, 12, 12, Yes], [2, 0, 12, 18, Yes], [2, 1, 12, 12, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
13. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 13, 39], [13, 0, 26], [39, 26, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 13/2, Yes], [0, 2, 13, 39/2, Yes], [1, 0, 0, 13/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 13, 13, Yes], [2, 0, 13, 39/2, Yes], [2, 1, 13, 13, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 13/2, Yes], [0, 2, 13, 39/2, Yes], [1, 0, 0, 13/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 13, 13, Yes], [2, 0, 13, 39/2, Yes], [2, 1, 13, 13, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
14. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 14, 42], [14, 0, 28], [42, 28, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7, Yes], [0, 2, 14, 21, Yes], [1, 0, 0, 7, Yes], [1, 1, 0, 0, Yes], [1, 2, 14, 14, Yes], [2, 0, 14, 21, Yes], [2, 1, 14, 14, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 7, Yes], [0, 2, 14, 21, Yes], [1, 0, 0, 7, Yes], [1, 1, 0, 0, Yes], [1, 2, 14, 14, Yes], [2, 0, 14, 21, Yes], [2, 1, 14, 14, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
15. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 15, 45], [15, 0, 30], [45, 30, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 15/2, Yes], [0, 2, 15, 45/2, Yes], [1, 0, 0, 15/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 15, 15, Yes], [2, 0, 15, 45/2, Yes], [2, 1, 15, 15, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 15/2, Yes], [0, 2, 15, 45/2, Yes], [1, 0, 0, 15/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 15, 15, Yes], [2, 0, 15, 45/2, Yes], [2, 1, 15, 15, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
16. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 16, 48], [16, 0, 32], [48, 32, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 8, Yes], [0, 2, 16, 24, Yes], [1, 0, 0, 8, Yes], [1, 1, 0, 0, Yes], [1, 2, 16, 16, Yes], [2, 0, 16, 24, Yes], [2, 1, 16, 16, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 8, Yes], [0, 2, 16, 24, Yes], [1, 0, 0, 8, Yes], [1, 1, 0, 0, Yes], [1, 2, 16, 16, Yes], [2, 0, 16, 24, Yes], [2, 1, 16, 16, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
17. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 17, 51], [17, 0, 34], [51, 34, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 17/2, Yes], [0, 2, 17, 51/2, Yes], [1, 0, 0, 17/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 17, 17, Yes], [2, 0, 17, 51/2, Yes], [2, 1, 17, 17, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 17/2, Yes], [0, 2, 17, 51/2, Yes], [1, 0, 0, 17/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 17, 17, Yes], [2, 0, 17, 51/2, Yes], [2, 1, 17, 17, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
18. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 18, 54], [18, 0, 36], [54, 36, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9, Yes], [0, 2, 18, 27, Yes], [1, 0, 0, 9, Yes], [1, 1, 0, 0, Yes], [1, 2, 18, 18, Yes], [2, 0, 18, 27, Yes], [2, 1, 18, 18, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 9, Yes], [0, 2, 18, 27, Yes], [1, 0, 0, 9, Yes], [1, 1, 0, 0, Yes], [1, 2, 18, 18, Yes], [2, 0, 18, 27, Yes], [2, 1, 18, 18, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
19. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 19, 57], [19, 0, 38], [57, 38, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 19/2, Yes], [0, 2, 19, 57/2, Yes], [1, 0, 0, 19/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 19, 19, Yes], [2, 0, 19, 57/2, Yes], [2, 1, 19, 19, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 19/2, Yes], [0, 2, 19, 57/2, Yes], [1, 0, 0, 19/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 19, 19, Yes], [2, 0, 19, 57/2, Yes], [2, 1, 19, 19, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
20. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 20, 60], [20, 0, 40], [60, 40, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 10, Yes], [0, 2, 20, 30, Yes], [1, 0, 0, 10, Yes], [1, 1, 0, 0, Yes], [1, 2, 20, 20, Yes], [2, 0, 20, 30, Yes], [2, 1, 20, 20, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 10, Yes], [0, 2, 20, 30, Yes], [1, 0, 0, 10, Yes], [1, 1, 0, 0, Yes], [1, 2, 20, 20, Yes], [2, 0, 20, 30, Yes], [2, 1, 20, 20, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
21. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 21, 63], [21, 0, 42], [63, 42, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 21/2, Yes], [0, 2, 21, 63/2, Yes], [1, 0, 0, 21/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 21, 21, Yes], [2, 0, 21, 63/2, Yes], [2, 1, 21, 21, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 21/2, Yes], [0, 2, 21, 63/2, Yes], [1, 0, 0, 21/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 21, 21, Yes], [2, 0, 21, 63/2, Yes], [2, 1, 21, 21, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
22. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 22, 66], [22, 0, 44], [66, 44, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11, Yes], [0, 2, 22, 33, Yes], [1, 0, 0, 11, Yes], [1, 1, 0, 0, Yes], [1, 2, 22, 22, Yes], [2, 0, 22, 33, Yes], [2, 1, 22, 22, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 11, Yes], [0, 2, 22, 33, Yes], [1, 0, 0, 11, Yes], [1, 1, 0, 0, Yes], [1, 2, 22, 22, Yes], [2, 0, 22, 33, Yes], [2, 1, 22, 22, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
23. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 23, 69], [23, 0, 46], [69, 46, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 23/2, Yes], [0, 2, 23, 69/2, Yes], [1, 0, 0, 23/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 23, 23, Yes], [2, 0, 23, 69/2, Yes], [2, 1, 23, 23, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 23/2, Yes], [0, 2, 23, 69/2, Yes], [1, 0, 0, 23/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 23, 23, Yes], [2, 0, 23, 69/2, Yes], [2, 1, 23, 23, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
24. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 24, 72], [24, 0, 48], [72, 48, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 12, Yes], [0, 2, 24, 36, Yes], [1, 0, 0, 12, Yes], [1, 1, 0, 0, Yes], [1, 2, 24, 24, Yes], [2, 0, 24, 36, Yes], [2, 1, 24, 24, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 12, Yes], [0, 2, 24, 36, Yes], [1, 0, 0, 12, Yes], [1, 1, 0, 0, Yes], [1, 2, 24, 24, Yes], [2, 0, 24, 36, Yes], [2, 1, 24, 24, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
25. A Global Contraction Needs Every Metric and Mapping Qualification
Use the complete exact metric matrix=[[0, 25, 75], [25, 0, 50], [75, 50, 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).
View answer and explanation
(3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 25/2, Yes], [0, 2, 25, 75/2, Yes], [1, 0, 0, 25/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 25, 25, Yes], [2, 0, 25, 75/2, Yes], [2, 1, 25, 25, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required)
Retain point count, self-map qualification, q-range check and every exact pair inequality. Being a total array or passing sampled pairs is insufficient. Finite completeness does not replace the all-pairs bound or a general complete-space theorem proof. The exact result, retaining the requested order, is (3, Yes, 1/2, Yes, [[0, 0, 0, 0, Yes], [0, 1, 0, 25/2, Yes], [0, 2, 25, 75/2, Yes], [1, 0, 0, 25/2, Yes], [1, 1, 0, 0, Yes], [1, 2, 25, 25, Yes], [2, 0, 25, 75/2, Yes], [2, 1, 25, 25, Yes], [2, 2, 0, 0, Yes]], Yes, finite_metric_is_complete_but_full_pair_audit_is_still_required).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-metric-qualification · candidate
26. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 1, 3], [1, 0, 2], [3, 2, 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=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 2, [2, 1, 0], 0, [3, 1, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [3, 1, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
27. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 2, 6], [2, 0, 4], [6, 4, 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 (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 3, [2, 1, 0, 0], 0, [6, 2, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [6, 2, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
28. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 3, 9], [3, 0, 6], [9, 6, 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=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 4, [2, 1, 0, 0, 0], 0, [9, 3, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [9, 3, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
29. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 4, 12], [4, 0, 8], [12, 8, 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=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 5, [2, 1, 0, 0, 0, 0], 0, [12, 4, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [12, 4, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
30. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 5, 15], [5, 0, 10], [15, 10, 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=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [15, 5, 0, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [15, 5, 0, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
31. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 6, 18], [6, 0, 12], [18, 12, 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=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 2, [2, 1, 0], 0, [18, 6, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [18, 6, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
32. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
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 (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 3, [2, 1, 0, 0], 0, [21, 7, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [21, 7, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
33. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 8, 24], [8, 0, 16], [24, 16, 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=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 4, [2, 1, 0, 0, 0], 0, [24, 8, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [24, 8, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
34. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
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. Start=2, steps=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 5, [2, 1, 0, 0, 0, 0], 0, [27, 9, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [27, 9, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
35. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 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=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [30, 10, 0, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [30, 10, 0, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
36. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 11, 33], [11, 0, 22], [33, 22, 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=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 2, [2, 1, 0], 0, [33, 11, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [33, 11, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
37. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 12, 36], [12, 0, 24], [36, 24, 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 (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 3, [2, 1, 0, 0], 0, [36, 12, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [36, 12, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
38. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 13, 39], [13, 0, 26], [39, 26, 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=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 4, [2, 1, 0, 0, 0], 0, [39, 13, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [39, 13, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
39. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 14, 42], [14, 0, 28], [42, 28, 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=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 5, [2, 1, 0, 0, 0, 0], 0, [42, 14, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [42, 14, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
40. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 15, 45], [15, 0, 30], [45, 30, 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=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [45, 15, 0, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [45, 15, 0, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
41. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 16, 48], [16, 0, 32], [48, 32, 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=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 2, [2, 1, 0], 0, [48, 16, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [48, 16, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
42. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 17, 51], [17, 0, 34], [51, 34, 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 (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 3, [2, 1, 0, 0], 0, [51, 17, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [51, 17, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
43. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 18, 54], [18, 0, 36], [54, 36, 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=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 4, [2, 1, 0, 0, 0], 0, [54, 18, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [54, 18, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
44. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 19, 57], [19, 0, 38], [57, 38, 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=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 5, [2, 1, 0, 0, 0, 0], 0, [57, 19, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [57, 19, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
45. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 20, 60], [20, 0, 40], [60, 40, 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=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [60, 20, 0, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [60, 20, 0, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
46. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 21, 63], [21, 0, 42], [63, 42, 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=2. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 2, [2, 1, 0], 0, [63, 21, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 2, [2, 1, 0], 0, [63, 21, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
47. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 22, 66], [22, 0, 44], [66, 44, 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 (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 3, [2, 1, 0, 0], 0, [66, 22, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 3, [2, 1, 0, 0], 0, [66, 22, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
48. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 23, 69], [23, 0, 46], [69, 46, 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=4. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 4, [2, 1, 0, 0, 0], 0, [69, 23, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 4, [2, 1, 0, 0, 0], 0, [69, 23, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
49. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 24, 72], [24, 0, 48], [72, 48, 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=5. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 5, [2, 1, 0, 0, 0, 0], 0, [72, 24, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 5, [2, 1, 0, 0, 0, 0], 0, [72, 24, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
50. Generate the Complete Picard Orbit and Compare with the Unique Fixed Point
Use the complete exact metric matrix=[[0, 25, 75], [25, 0, 50], [75, 50, 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=6. Give (start, steps, orbit, fixed_point, distances_to_fixed_point, certificate).
View answer and explanation
(2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [75, 25, 0, 0, 0, 0, 0], exact_Picard_orbit)
Require a genuine contraction and exactly one fixed point. Apply the full map at every step, retain the start and every iterate, and report each exact distance to the fixed point. A finite observed orbit by itself is not the general convergence theorem. The exact result, retaining the requested order, is (2, 6, [2, 1, 0, 0, 0, 0, 0], 0, [75, 25, 0, 0, 0, 0, 0], exact_Picard_orbit).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-picard-orbit · candidate
51. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 1, 3], [1, 0, 2], [3, 2, 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=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
52. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 2, 6], [2, 0, 4], [6, 4, 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).
View answer and explanation
([2, 1, 0, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
53. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 3, 9], [3, 0, 6], [9, 6, 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=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0], 0, 0, 3/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 3/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
54. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 4, 12], [4, 0, 8], [12, 8, 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=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0], 0, 0, 1/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 1/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
55. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 5, 15], [5, 0, 10], [15, 10, 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=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
56. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 6, 18], [6, 0, 12], [18, 12, 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=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
57. The Geometric Tail Bound Controls the Exact Picard Error
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).
View answer and explanation
([2, 1, 0, 0], 0, 0, 7/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 7/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
58. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 8, 24], [8, 0, 16], [24, 16, 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=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0], 0, 0, 2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
59. The Geometric Tail Bound Controls the Exact Picard Error
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. Start=2, steps=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0], 0, 0, 9/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 9/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
60. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 10, 30], [10, 0, 20], [30, 20, 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=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
61. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 11, 33], [11, 0, 22], [33, 22, 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=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
62. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 12, 36], [12, 0, 24], [36, 24, 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).
View answer and explanation
([2, 1, 0, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 6, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
63. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 13, 39], [13, 0, 26], [39, 26, 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=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0], 0, 0, 13/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 13/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
64. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 14, 42], [14, 0, 28], [42, 28, 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=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0], 0, 0, 7/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 7/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
65. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 15, 45], [15, 0, 30], [45, 30, 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=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0, 0], 0, 0, 15/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 15/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
66. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 16, 48], [16, 0, 32], [48, 32, 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=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0], 0, 0, 16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
67. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 17, 51], [17, 0, 34], [51, 34, 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).
View answer and explanation
([2, 1, 0, 0], 0, 0, 17/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 17/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
68. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 18, 54], [18, 0, 36], [54, 36, 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=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0], 0, 0, 9/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 9/2, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
69. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 19, 57], [19, 0, 38], [57, 38, 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=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0], 0, 0, 19/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 19/8, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
70. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 20, 60], [20, 0, 40], [60, 40, 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=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 5/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
71. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 21, 63], [21, 0, 42], [63, 42, 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=2. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0], 0, 0, 21, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0], 0, 0, 21, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
72. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 22, 66], [22, 0, 44], [66, 44, 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).
View answer and explanation
([2, 1, 0, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0], 0, 0, 11, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
73. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 23, 69], [23, 0, 46], [69, 46, 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=4. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0], 0, 0, 23/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0], 0, 0, 23/4, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
74. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 24, 72], [24, 0, 48], [72, 48, 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=5. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0], 0, 0, 3, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0], 0, 0, 3, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
75. The Geometric Tail Bound Controls the Exact Picard Error
Use the complete exact metric matrix=[[0, 25, 75], [25, 0, 50], [75, 50, 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=6. Give (orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate).
View answer and explanation
([2, 1, 0, 0, 0, 0, 0], 0, 0, 25/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
Compare actual distance d(x_n,x*) with q^n/(1−q)*d(x_1,x_0). Keep n=0 and q=0 conventions explicit through exact finite arithmetic. The unique fixed point and full contraction premises precede this bound. The exact result, retaining the requested order, is ([2, 1, 0, 0, 0, 0, 0], 0, 0, 25/16, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-fixed-point-bound · candidate
76. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=q_equal_one and positive exact scale=1. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(q_equal_one, 1, two_point_metric, 1, identity_map_has_two_fixed_points, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 1, two_point_metric, 1, identity_map_has_two_fixed_points, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
77. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=non_self_map and positive exact scale=2. Give (mode, scale, mapping_description, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(non_self_map, 2, T_maps_one_point_outside_X, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 2, T_maps_one_point_outside_X, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
78. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=noncomplete_space and positive exact scale=3. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(noncomplete_space, X=(0,3), T(x)=(x+3)/2, fixed_point=3_outside_X, [1, 2], No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,3), T(x)=(x+3)/2, fixed_point=3_outside_X, [1, 2], No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
79. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=sampled_pairs and positive exact scale=4. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(sampled_pairs, 4, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 4, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
80. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=q_equal_one and positive exact scale=5. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(q_equal_one, 5, two_point_metric, 1, identity_map_has_two_fixed_points, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 5, two_point_metric, 1, identity_map_has_two_fixed_points, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
81. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=non_self_map and positive exact scale=6. Give (mode, scale, mapping_description, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(non_self_map, 6, T_maps_one_point_outside_X, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 6, T_maps_one_point_outside_X, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
82. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=noncomplete_space and positive exact scale=7. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(noncomplete_space, X=(0,7), T(x)=(x+7)/2, fixed_point=7_outside_X, [1, 2], No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,7), T(x)=(x+7)/2, fixed_point=7_outside_X, [1, 2], No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
83. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=sampled_pairs and positive exact scale=8. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(sampled_pairs, 8, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 8, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
84. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=q_equal_one and positive exact scale=9. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(q_equal_one, 9, two_point_metric, 1, identity_map_has_two_fixed_points, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 9, two_point_metric, 1, identity_map_has_two_fixed_points, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
85. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=non_self_map and positive exact scale=10. Give (mode, scale, mapping_description, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(non_self_map, 10, T_maps_one_point_outside_X, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 10, T_maps_one_point_outside_X, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
86. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=noncomplete_space and positive exact scale=11. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(noncomplete_space, X=(0,11), T(x)=(x+11)/2, fixed_point=11_outside_X, [1, 2], No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,11), T(x)=(x+11)/2, fixed_point=11_outside_X, [1, 2], No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
87. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=sampled_pairs and positive exact scale=12. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(sampled_pairs, 12, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 12, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
88. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=q_equal_one and positive exact scale=13. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(q_equal_one, 13, two_point_metric, 1, identity_map_has_two_fixed_points, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 13, two_point_metric, 1, identity_map_has_two_fixed_points, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
89. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=non_self_map and positive exact scale=14. Give (mode, scale, mapping_description, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(non_self_map, 14, T_maps_one_point_outside_X, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 14, T_maps_one_point_outside_X, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
90. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=noncomplete_space and positive exact scale=15. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,15), T(x)=(x+15)/2, fixed_point=15_outside_X, [1, 2], No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
91. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=sampled_pairs and positive exact scale=16. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(sampled_pairs, 16, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 16, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
92. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=q_equal_one and positive exact scale=17. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(q_equal_one, 17, two_point_metric, 1, identity_map_has_two_fixed_points, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 17, two_point_metric, 1, identity_map_has_two_fixed_points, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
93. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=non_self_map and positive exact scale=18. Give (mode, scale, mapping_description, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(non_self_map, 18, T_maps_one_point_outside_X, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 18, T_maps_one_point_outside_X, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
94. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=noncomplete_space and positive exact scale=19. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(noncomplete_space, X=(0,19), T(x)=(x+19)/2, fixed_point=19_outside_X, [1, 2], No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,19), T(x)=(x+19)/2, fixed_point=19_outside_X, [1, 2], No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
95. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=sampled_pairs and positive exact scale=20. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(sampled_pairs, 20, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 20, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
96. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=q_equal_one and positive exact scale=21. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(q_equal_one, 21, two_point_metric, 1, identity_map_has_two_fixed_points, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 21, two_point_metric, 1, identity_map_has_two_fixed_points, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
97. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=non_self_map and positive exact scale=22. Give (mode, scale, mapping_description, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(non_self_map, 22, T_maps_one_point_outside_X, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (non_self_map, 22, T_maps_one_point_outside_X, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
98. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=noncomplete_space and positive exact scale=23. Give (mode, space_description, mapping_description, outside_fixed_point, q_as_fraction, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(noncomplete_space, X=(0,23), T(x)=(x+23)/2, fixed_point=23_outside_X, [1, 2], No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (noncomplete_space, X=(0,23), T(x)=(x+23)/2, fixed_point=23_outside_X, [1, 2], No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
99. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=sampled_pairs and positive exact scale=24. Give (mode, scale, sampled_pair_result, unchecked_pair_statement, global_contraction_claim, boundary).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(sampled_pairs, 24, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (sampled_pairs, 24, checked_pairs_pass, one_unchecked_pair_violates_global_bound, No, sampled_pairs_do_not_prove_global_contraction).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
100. Failed Completeness, Self-Map or Global Strict-Contraction Premises Block the Theorem
Use boundary mode=q_equal_one and positive exact scale=25. Give (mode, scale, metric_description, q, fixed_point_counterexample, theorem_application_claim).
The q=1, non-self-map, noncomplete and sampled-pairs branches have6/4/6/6 fields with different roles. sampled_pairs retains the source scope description, not a supplied complete metric/map counterexample.
View answer and explanation
(q_equal_one, 25, two_point_metric, 1, identity_map_has_two_fixed_points, No)
For q_equal_one the two-point identity has two fixed points; non_self_map leaves the space. On (0,scale), T(x)=(x+scale)/2 has its only fixed point outside the space. sampled_pairs records only the source stated unchecked-pair failure scope, not a supplied complete metric/map witness. Do not infer global contraction or a general theorem from sampled checks. The exact result, retaining the requested order, is (q_equal_one, 25, two_point_metric, 1, identity_map_has_two_fixed_points, No).
Reviewed step-by-step derivation
Contraction mapping theorem · qf-mod47-contract-boundary · candidate
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