Question 12886: An Explicit Infinite Tail Converges within Its Declared Space

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0024-12886 · q-20261014-advance-20261027-r03-complete-metric-space-0036

Original question and givens

Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 13, 16], [10, 13, 0, 21], [13, 16, 21, 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 starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).

G001 — Use the complete qualified finite metric table=[[0, 5, 10, 13], [5, 0, 13, 16], [10, 13, 0, 21], [13, 16, 21, 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 starts with prefix=[2, 3, 0] then remains forever at tail=1. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).

Definition and scope

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

mko-complete-metric-space

Step-by-step derivation

S001 · implies

Before

carrier Fin(4);complete nonnegative integer metric candidate=[[0, 5, 10, 13], [5, 0, 13, 16], [10, 13, 0, 21], [13, 16, 21, 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, 10, 20, 26];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=5;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[5, 5, 23, 29];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=10;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[10, 18, 10, 34];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=13;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[13, 21, 31, 13];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=5;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[5, 5, 23, 29];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=[10, 0, 26, 32];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=13;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[15, 13, 13, 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=16;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[18, 16, 34, 16];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=10;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[10, 18, 10, 34];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=13;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[15, 13, 13, 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=[20, 26, 0, 42];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=21;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[23, 29, 21, 21];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=13;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[13, 21, 31, 13];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=16;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[18, 16, 34, 16];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=21;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[23, 29, 21, 21];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=[26, 32, 42, 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=[5, 10, 13, 5, 13, 16, 10, 13, 21, 13, 16, 21]

After

minimum positive distance delta=5>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 · implies

Before

typed prefix=[2, 3, 0];all infinite later terms explicitly equal tail=1

After

N=3;limit=1 in Fin(4);all tail pairs and tail-to-limit distances0

Conditions: none

Use the declared infinite tail,not a pattern guessed from finite data.For any positive epsilon,choose the prefix length N.All tail-pair and tail-to-limit distances are0,establishing both Cauchy behavior and convergence in this space.

Step identifiers and dependencies

Rule: declared_constant_tail_in_same_metric_space · Uses: G001, S018

Reverse checks

S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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, 0], 1, 3, 1, Yes, Yes)

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: 15be6233f2aad75fee511008432f6a4a02438e7bf5853c2e4bd47d218c408dd3
candidate record SHA-256: 523ed738f0b251ef806a77a1ec0486567e0a55167255039a80c52bac95556378
semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
Lean: NOT_RUN · human review: NOT_PERFORMED

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