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
arbitrary Cauchy sequence in Fin(4);choose epsilon=delta=5>0
After
some N has every m,k>=N with d(x_m,x_k)<5;minimum separation forces x_m=x_k
Conditions: none
Apply the Cauchy definition at epsilon=delta to obtain N.Distinct points have distance at least delta,whereas every tail pair has distance strictly below delta.Thus all tail terms equal x_N.
Step identifiers and dependencies
Rule: every_Cauchy_sequence_eventually_constant · Uses: G001, S018
S020 · implies
Before
all tail terms equal x_N;typed sequence guarantees x_N in Fin(4)
After
for every positive eta,all k>=N have distance(x_k,x_N)=0<eta;limit belongs to same space
Conditions: none
The common tail value already belongs to the original space.For any positive eta,the same N gives zero distance.Hence every Cauchy sequence has an internal limit:completeness,not one successful sequence or a limit only in an ambient space.
Step identifiers and dependencies
Rule: finite_completeness_quantifier_and_internal_limit · Uses: G001, S019
S021 · implies
Before
qualified finite metric;delta=5;selected evidence mode=finite_minimum_separation
After
selected evidence sufficient=Yes;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, S020