Explore related knowledge and questions →
← Question derivation index · Original question batch
R0051-26226 · q-20261015-advance-20261116-r01-contraction-mapping-theorem-0051
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).
G001 — 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).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [contract-fixed-point-bound]
After
derived = [[orbit, fixed_point, actual_error, geometric_bound, bound_holds, certificate]]
Conditions: none
List the full source field order without JavaScript.Original modes select contract6/4/6/6 and IVT8/7/8 roles,without padding or answer-shape inference.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[[0, 1, 3], [1, 0, 2], [3, 2, 0]]]
After
derived = [3, Yes, Yes, Yes, [[0, 0, 0, 0, 0, Yes], [0, 0, 1, 1, 1, Yes], [0, 0, 2, 3, 3, Yes], [0, 1, 0, 0, 2, Yes], [0, 1, 1, 1, 1, Yes], [0, 1, 2, 3, 3, Yes], [0, 2, 0, 0, 6, Yes], [0, 2, 1, 1, 5, Yes], [0, 2, 2, 3, 3, Yes], [1, 0, 0, 1, 1, Yes], [1, 0, 1, 0, 2, Yes], [1, 0, 2, 2, 4, Yes], [1, 1, 0, 1, 1, Yes], [1, 1, 1, 0, 0, Yes], [1, 1, 2, 2, 2, Yes], [1, 2, 0, 1, 5, Yes], [1, 2, 1, 0, 4, Yes], [1, 2, 2, 2, 2, Yes], [2, 0, 0, 3, 3, Yes], [2, 0, 1, 2, 4, Yes], [2, 0, 2, 0, 6, Yes], [2, 1, 0, 3, 3, Yes], [2, 1, 1, 2, 2, Yes], [2, 1, 2, 0, 4, Yes], [2, 2, 0, 3, 3, Yes], [2, 2, 1, 2, 2, Yes], [2, 2, 2, 0, 0, Yes]], Yes]
Conditions: none
Check every triangle triple,not sampled metric behavior.A nonempty finite metric is complete:trivial for a singleton;otherwise the minimum positive distance delta forces a Cauchy tail with distances<delta to be constant,with limit in X.
Rule: Contraction_COMPLETE_metric_axioms_ALL_triangles_finite_completeness · Uses: G001, S001
Before
Contraction ORIGINAL pair0,0 q times distance := mul(1/2, 0)
After
Contraction ORIGINAL pair0,0 q times distance = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
Contraction ORIGINAL pair0,1 q times distance := mul(1/2, 1)
After
Contraction ORIGINAL pair0,1 q times distance = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
Contraction ORIGINAL pair0,2 q times distance := mul(1/2, 3)
After
Contraction ORIGINAL pair0,2 q times distance = 3/2
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
Contraction ORIGINAL pair1,0 q times distance := mul(1/2, 1)
After
Contraction ORIGINAL pair1,0 q times distance = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
Contraction ORIGINAL pair1,1 q times distance := mul(1/2, 0)
After
Contraction ORIGINAL pair1,1 q times distance = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
Contraction ORIGINAL pair1,2 q times distance := mul(1/2, 2)
After
Contraction ORIGINAL pair1,2 q times distance = 1
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
Contraction ORIGINAL pair2,0 q times distance := mul(1/2, 3)
After
Contraction ORIGINAL pair2,0 q times distance = 3/2
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
Contraction ORIGINAL pair2,1 q times distance := mul(1/2, 2)
After
Contraction ORIGINAL pair2,1 q times distance = 1
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
Contraction ORIGINAL pair2,2 q times distance := mul(1/2, 0)
After
Contraction ORIGINAL pair2,2 q times distance = 0
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
source roles = [[0, 0, 1], 1/2, [[0, 1, 3], [1, 0, 2], [3, 2, 0]]]
After
derived = [Yes, 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, [0]]
Conditions: none
Separate self-map,0<=q<1 and all pair inequalities.Nonself maps have empty source pair rows;self maps retain actual pair results even when q-range fails.Array length or samples do not prove global contraction.
Rule: Contraction_selfmap_q_range_ALL_ordered_pairs_separate · Uses: G001, S011
Before
source roles = [0, 2, [0, 0, 1]]
After
derived = [1, 1]
Conditions: none
Apply the same complete map at every step,retaining the start and all iterates rather than guessing a fixed point from a short prefix.
Rule: Contraction_EACH_Picard_iterate_original_total_map · Uses: G001, S012
Before
source roles = [1, 1, [0, 0, 1]]
After
derived = [0, 0]
Conditions: none
Apply the same complete map at every step,retaining the start and all iterates rather than guessing a fixed point from a short prefix.
Rule: Contraction_EACH_Picard_iterate_original_total_map · Uses: G001, S013
Before
source roles = [1/2, [0], 2, [2, 1, 0]]
After
derived = [[3, 1, 0], d(x_(n+k),x_n)<=q^n*(1-q^k)/(1-q)*d(x1,x0), complete limit fixed;two fixed points obey(1-q)d<=0 hence coincide]
Conditions: none
Successive distances shrink by q;the triangle inequality sums the finite geometric tail,making Picard Cauchy.Completeness gives a limit and Lipschitz continuity makes it fixed.Two fixed points satisfy distance<=q*distance,so q<1 forces equality.This pedagogical argument does not label a finite orbit a general formal proof.
Rule: Contraction_GENERAL_geometric_tail_complete_limit_and_unique_fixed · Uses: G001, S014
Before
(1/2)^2
After
Contraction ORIGINAL q powerN = 1/4
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S015
Before
Contraction strict one minus q := sub(1, 1/2)
After
Contraction strict one minus q = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
Contraction geometric tail factor := div(1/4, 1/2)
After
Contraction geometric tail factor = 1/2
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
Contraction factor times ORIGINAL first-step distance := mul(1/2, 2)
After
Contraction factor times ORIGINAL first-step distance = 1
Conditions: none
Substitute original roles and established intermediates exactly,without rounding;extra teaching identities do not restrict the source domain.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
source roles = [2, 1/2, [2, 1, 0], 0, 2]
After
derived = [0, 1, Yes]
Conditions: none
Let tail length tend to infinity to obtain the error bound.At N0,q^0 is the empty product1,including q0.The source computes powers/division before multiplying first distance;zero distance does not bypass its numeric profile.
Rule: Contraction_actual_error_bound_N0_q0_profile_NOT_shortcircuited · Uses: G001, S019
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S019 · premise_and_result_audit — Recompute Contraction factor times ORIGINAL first-step distance with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute Contraction geometric tail factor with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute Contraction strict one minus q with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute Contraction ORIGINAL pair2,2 q times distance with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute Contraction ORIGINAL pair2,1 q times distance with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Recompute Contraction ORIGINAL pair2,0 q times distance with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Contraction ORIGINAL pair1,2 q times distance with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Contraction ORIGINAL pair1,1 q times distance with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Contraction ORIGINAL pair1,0 q times distance with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Contraction ORIGINAL pair0,2 q times distance with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute Contraction ORIGINAL pair0,1 q times distance with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute Contraction ORIGINAL pair0,0 q times distance with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([2, 1, 0], 0, 0, 1, Yes, d_xn_fixed_le_qn_over_one_minus_q_times_d_x1_x0)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 4a1cd08bb78ff270d8380551166b7abe358cadcfa775b82102d1682f9f750446candidate record SHA-256: d73d4007f1d501262eb621b5f8bf9250f00f7b73a44013589842a91f058aee9esemantic review receipt SHA-256: 90df1e889242e9afdf01e068788ee8500ef366ffcfc0496e30a6800de32631c3question-first witness SHA-256: ae1e849608590ca856c9f3b6911ba58e3bf53c608eb7369606192a770f48bd5cLean: NOT_RUN · human review: NOT_PERFORMED