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Question 12896: An Explicit Infinite Tail Converges within Its Declared Space

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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R0024-12896 · q-20261014-advance-20261027-r03-complete-metric-space-0046

Original question and givens

Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 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=[0, 1, 2] then remains forever at tail=3. Give (prefix,tail,N,limit,cauchy?,converges_in_space?).

  • G001 — Use the complete qualified finite metric table=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 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=[0, 1, 2] then remains forever at tail=3. 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.

Step-by-step derivation

S001 · implies

Before

carrier Fin(4);complete nonnegative integer metric candidate=[[0, 7, 9, 10], [7, 0, 14, 15], [9, 14, 0, 17], [10, 15, 17, 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, 14, 18, 20];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=7;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[7, 7, 23, 25];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=9;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[9, 21, 9, 27];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=10;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[10, 22, 26, 10];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=7;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[7, 7, 23, 25];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=[14, 0, 28, 30];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=14;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[16, 14, 14, 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, S007

S009 · implies

Before

endpoints=[1, 3];direct distance=15;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[17, 15, 31, 15];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=9;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[9, 21, 9, 27];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=14;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[16, 14, 14, 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, S010

S012 · implies

Before

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

After

all two-leg distances=[18, 28, 0, 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, S011

S013 · implies

Before

endpoints=[2, 3];direct distance=17;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[19, 29, 17, 17];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=10;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[10, 22, 26, 10];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=15;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[17, 15, 31, 15];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=17;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[19, 29, 17, 17];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=[20, 30, 34, 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=[7, 9, 10, 7, 14, 15, 9, 14, 17, 10, 15, 17]

After

minimum positive distance delta=7>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=[0, 1, 2];all infinite later terms explicitly equal tail=3

After

N=3;limit=3 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

([0, 1, 2], 3, 3, 3, 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: eec08492c59d0460ea84da23c51c1861d6d76ff067d6ba700cad973689e76b2c
  • candidate record SHA-256: 544b01336946e873f14c80944ceda8ce807ba95968867cfa7034aace187f6cf1
  • semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
  • question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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