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Question 12850: A Space-Wide Certificate Is Not a List of Successful Sequences

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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R0024-12850 · q-20261014-advance-20261027-r03-complete-metric-space-0100

Original question and givens

Use the complete qualified finite metric table=[[0, 11, 14, 15], [11, 0, 15, 16], [14, 15, 0, 19], [15, 16, 19, 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. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.

  • G001 — Use the complete qualified finite metric table=[[0, 11, 14, 15], [11, 0, 15, 16], [14, 15, 0, 19], [15, 16, 19, 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. evidence_mode=sampled_sequences. Give (n,min_positive_distance,evidence_mode,sufficient?,boundary). finite_minimum_separation yields universal_finite_certificate; sampled_sequences yields sampled_sequences_insufficient.

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, 11, 14, 15], [11, 0, 15, 16], [14, 15, 0, 19], [15, 16, 19, 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, 22, 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, S001

S003 · implies

Before

endpoints=[0, 1];direct distance=11;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[11, 11, 29, 31];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=14;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[14, 26, 14, 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=15;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[15, 27, 33, 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, S004

S006 · implies

Before

endpoints=[1, 0];direct distance=11;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[11, 11, 29, 31];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=[22, 0, 30, 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=15;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[25, 15, 15, 35];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=[26, 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=14;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[14, 26, 14, 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=15;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[25, 15, 15, 35];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=[28, 30, 0, 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, S011

S013 · implies

Before

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

After

all two-leg distances=[29, 31, 19, 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, S012

S014 · implies

Before

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

After

all two-leg distances=[15, 27, 33, 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, S013

S015 · implies

Before

endpoints=[3, 1];direct distance=16;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[26, 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=19;all intermediate labels=[0, 1, 2, 3]

After

all two-leg distances=[29, 31, 19, 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, S015

S017 · implies

Before

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

After

all two-leg distances=[30, 32, 38, 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=[11, 14, 15, 11, 15, 16, 14, 15, 19, 15, 16, 19]

After

minimum positive distance delta=11>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

qualified finite metric;delta=11;selected evidence mode=sampled_sequences

After

selected evidence sufficient=No;space completeness is not denied by sample insufficiency

Conditions: none

Minimum separation covers every Cauchy sequence;finite samples do not provide that quantifier coverage.No in sample mode means insufficient evidence,not that this qualified finite space is incomplete.

Step identifiers and dependencies

Rule: complete_evidence_mode_not_truth_of_incompleteness · 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

(4, 11, sampled_sequences, No, sampled_sequences_insufficient)

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: caf43723e65cbbf7a01b5aba36bbb9bd756b20390365210e5d9532db11d55b66
  • candidate record SHA-256: c823a2ab85b4754b742685de735aec4ad311ebadbd6388fd518ff01e2f51ba3a
  • semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
  • question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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