Step-by-step derivation
S001 · implies
Before
source operation=hausdorff-sampled-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary]
Conditions: none
Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=0,U=[a1, b1, c1]
After
ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, b1, c1]], [1, [a1, b1, c1], [a1, b1]], [2, [a1, b1, c1], [a1, c1]], [3, [a1, b1, c1], [a1]], [4, [a1, b1, c1], [c1]], [5, [a1, b1, c1], []]];empty/full present;family qualification passed with exact source English collation
Conditions: none
Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].
Step identifiers and dependencies
Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S001
S003 · implies
Before
ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=1,U=[a1, b1]
After
ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, b1]], [1, [a1, b1], [a1, b1]], [2, [a1, b1, c1], [a1]], [3, [a1, b1], [a1]], [4, [a1, b1, c1], []], [5, [a1, b1], []]];empty/full present;family qualification passed with exact source English collation
Conditions: none
Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].
Step identifiers and dependencies
Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S002
S004 · implies
Before
ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=2,U=[a1, c1]
After
ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1, c1]], [1, [a1, b1, c1], [a1]], [2, [a1, c1], [a1, c1]], [3, [a1, c1], [a1]], [4, [a1, c1], [c1]], [5, [a1, c1], []]];empty/full present;family qualification passed with exact source English collation
Conditions: none
Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].
Step identifiers and dependencies
Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S003
S005 · implies
Before
ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=3,U=[a1]
After
ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [a1]], [1, [a1, b1], [a1]], [2, [a1, c1], [a1]], [3, [a1], [a1]], [4, [a1, c1], []], [5, [a1], []]];empty/full present;family qualification passed with exact source English collation
Conditions: none
Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].
Step identifiers and dependencies
Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S004
S006 · implies
Before
ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=4,U=[c1]
After
ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], [c1]], [1, [a1, b1, c1], []], [2, [a1, c1], [c1]], [3, [a1, c1], []], [4, [c1], [c1]], [5, [c1], []]];empty/full present;family qualification passed with exact source English collation
Conditions: none
Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].
Step identifiers and dependencies
Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S005
S007 · implies
Before
ORIGINAL carrier order=[a1, b1, c1];source canonical open family=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []];actual open index=5,U=[]
After
ALL [other_index,union,intersection] rows=[[0, [a1, b1, c1], []], [1, [a1, b1], []], [2, [a1, c1], []], [3, [a1], []], [4, [c1], []], [5, [], []]];empty/full present;family qualification passed with exact source English collation
Conditions: none
Retain labels and original carrier order.Sort elements and the family by the exact declared source conventions,not size or an alternate familiar convention.Check all open pairs;the empty carrier has topology[[]].
Step identifiers and dependencies
Rule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S006
S008 · implies
Before
Hausdorff ORIGINAL sampled pair index0:ORIGINAL distinct carrier points x=c1,y=a1;source open order=[[a1, b1, c1], [a1, b1], [a1, c1], [a1], [c1], []]
After
ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a1, b1, c1], No], [0, 1, [a1, b1], No], [0, 2, [a1, c1], No], [0, 3, [a1], No], [2, 0, [a1, c1], No], [2, 1, [a1], No], [2, 2, [a1, c1], No], [2, 3, [a1], No], [4, 0, [c1], No], [4, 1, [], Yes]];FIRST source-order witness=[[c1], [a1, b1]]
Conditions: none
Use actual opens U containing x and V containing y.Search original U then V order for the first disjoint pair.Null means no witness in this complete finite domain,not a failed sample.
Step identifiers and dependencies
Rule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S007
S009 · calculation
Before
Hausdorff ORIGINAL sampled positions count := sum(1)
After
Hausdorff ORIGINAL sampled positions count = 1
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
Hausdorff ALL unordered carrier pairs count := sum(1, 1, 1)
After
Hausdorff ALL unordered carrier pairs count = 3
Conditions: none
Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
ORIGINAL sampled pair rows=[[c1, a1, [[c1], [a1, b1]]]];sample positions count=1;ALL finite pairs=[[a1, b1], [a1, c1], [b1, c1]];total pair count=3
After
all_sampled_separated=Yes;general_sample_certificate fixed No;duplicates/full finite sample allowed;any actual completeness must be established separately
Conditions: none
Samples may be empty,repeat pairs or actually cover every finite pair.Position count is not distinct-pair count.The fixed No negates neither true separation results nor a finite conclusion with separately established completeness;finite sampling does not certify an infinite space.
Step identifiers and dependencies
Rule: hausdorff_sample_No_not_separation_flag_or_actual_completeness_negation · Uses: G001, S010