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Question 12819: A Non-Cauchy Sequence Does Not Contradict Space Completeness

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0024-12819 · q-20261014-advance-20261027-r03-complete-metric-space-0069

Original question and givens

Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 9, 19], [8, 9, 0, 18], [18, 19, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=37. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.

  • G001 — Use the complete qualified finite metric table=[[0, 9, 8, 18], [9, 0, 9, 19], [8, 9, 0, 18], [18, 19, 18, 0]] on Fin(4)={0,…,3}, with at least two points for this minimum-positive-distance protocol. Completeness quantifies over every Cauchy sequence and requires a limit in this same space, not merely an ambient space. Testing finitely many sequences or terms is insufficient. The infinite sequence alternates forever between distinct left=2 and right=3, with arbitrary proposed cutoff N=37. Give [left,right,N,next_index,epsilon,distance,counterexample,cauchy] as a tuple. Choose next_index=N+1 and epsilon=d(left,right)>0. Retain the strict Cauchy condition on every pair of later terms.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

carrier Fin(4);complete nonnegative integer metric candidate=[[0, 9, 8, 18], [9, 0, 9, 19], [8, 9, 0, 18], [18, 19, 18, 0]]

After

zero iff equal=Yes;symmetric=Yes

Conditions: none

First qualify nonnegativity,diagonal/off-diagonal separation and symmetry on the entire table,not only queried cells.At least two points is a premise of this minimum-positive-distance protocol,not a claim that a singleton is incomplete.

Step identifiers and dependencies

Rule: complete_metric_separation_and_symmetry · Uses: G001

S002 · implies

Before

endpoints=[0, 0];direct distance=0;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[0, 18, 16, 36];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S001

S003 · implies

Before

endpoints=[0, 1];direct distance=9;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[9, 9, 17, 37];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S002

S004 · implies

Before

endpoints=[0, 2];direct distance=8;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[8, 18, 8, 36];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S003

S005 · implies

Before

endpoints=[0, 3];direct distance=18;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[18, 28, 26, 18];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S004

S006 · implies

Before

endpoints=[1, 0];direct distance=9;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[9, 9, 17, 37];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S005

S007 · implies

Before

endpoints=[1, 1];direct distance=0;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[18, 0, 18, 38];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S006

S008 · implies

Before

endpoints=[1, 2];direct distance=9;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[17, 9, 9, 37];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S007

S009 · implies

Before

endpoints=[1, 3];direct distance=19;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[27, 19, 27, 19];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S008

S010 · implies

Before

endpoints=[2, 0];direct distance=8;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[8, 18, 8, 36];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S009

S011 · implies

Before

endpoints=[2, 1];direct distance=9;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[17, 9, 9, 37];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S010

S012 · implies

Before

endpoints=[2, 2];direct distance=0;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[16, 18, 0, 36];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S011

S013 · implies

Before

endpoints=[2, 3];direct distance=18;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[26, 28, 18, 18];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S012

S014 · implies

Before

endpoints=[3, 0];direct distance=18;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[18, 28, 26, 18];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S013

S015 · implies

Before

endpoints=[3, 1];direct distance=19;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[27, 19, 27, 19];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S014

S016 · implies

Before

endpoints=[3, 2];direct distance=18;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[26, 28, 18, 18];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S015

S017 · implies

Before

endpoints=[3, 3];direct distance=0;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[36, 38, 36, 0];direct<=every two-leg=Yes

Conditions: none

For each ordered endpoint pair inspect every intermediate point.All triangle obligations must hold;this complete finite audit is not a few sampled triangles.

Step identifiers and dependencies

Rule: complete_metric_all_triangle_intermediates · Uses: G001, S016

S018 · implies

Before

all off-diagonal positive distances=[9, 8, 18, 9, 9, 19, 8, 9, 18, 18, 19, 18]

After

minimum positive distance delta=8>0

Conditions: none

A finite nonempty list of positive distances has a positive minimum.This is the space-wide scale for the universal Cauchy certificate,not a threshold guessed from a sequence.

Step identifiers and dependencies

Rule: finite_metric_positive_minimum_separation · Uses: G001, S017

S019 · calculation

Before

adjacent alternating tail index := sum(37, 1)

After

adjacent alternating tail index = 38

Conditions: none

Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

explicit infinite sequence alternates labels=[2, 3];arbitrary cutoff=37

After

indices=[37, 38];epsilon=d(left,right)=18>0;actual adjacent distance=18=epsilon

Conditions: none

Adjacent terms are these two points in one order,and symmetry gives the same distance.Fix positive epsilon=d.Equality with epsilon violates strict inequality,and this construction works for every cutoff N.

Step identifiers and dependencies

Rule: metric_alternation_fixed_epsilon_strict_failure · Uses: G001, S019

S021 · implies

Before

every cutoff defeated at fixed epsilon=18

After

counterexample=Yes;Cauchy=No;non-Cauchy sequence does not refute space completeness

Conditions: none

Completeness concerns only Cauchy sequences.A non-Cauchy sequence does not meet that premise.The final field uses the corrected source Cauchy label,not not_cauchy.

Step identifiers and dependencies

Rule: nonCauchy_not_counterexample_to_completeness · Uses: G001, S020

Reverse checks

  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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 adjacent alternating tail index with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(2, 3, 37, 38, 18, 18, Yes, No)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 230fa583f7118d11bd9fbc04f811e9e0502423ab58fc5e506f7bf9a10885d9cc
  • candidate record SHA-256: 5b3f0d23424d7755b5b7789a7f2670944bea91c7ac52d9621e6c695ecb2bcc8b
  • semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
  • question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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