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=[a9, b9, c9];source canonical open family=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []];actual open index=0,U=[a9, b9, c9]
After
ALL [other_index,union,intersection] rows=[[0, [a9, b9, c9], [a9, b9, c9]], [1, [a9, b9, c9], [a9, b9]], [2, [a9, b9, c9], [a9, c9]], [3, [a9, b9, c9], [a9]], [4, [a9, b9, c9], [c9]], [5, [a9, b9, c9], []]];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=[a9, b9, c9];source canonical open family=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []];actual open index=1,U=[a9, b9]
After
ALL [other_index,union,intersection] rows=[[0, [a9, b9, c9], [a9, b9]], [1, [a9, b9], [a9, b9]], [2, [a9, b9, c9], [a9]], [3, [a9, b9], [a9]], [4, [a9, b9, c9], []], [5, [a9, b9], []]];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=[a9, b9, c9];source canonical open family=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []];actual open index=2,U=[a9, c9]
After
ALL [other_index,union,intersection] rows=[[0, [a9, b9, c9], [a9, c9]], [1, [a9, b9, c9], [a9]], [2, [a9, c9], [a9, c9]], [3, [a9, c9], [a9]], [4, [a9, c9], [c9]], [5, [a9, c9], []]];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=[a9, b9, c9];source canonical open family=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []];actual open index=3,U=[a9]
After
ALL [other_index,union,intersection] rows=[[0, [a9, b9, c9], [a9]], [1, [a9, b9], [a9]], [2, [a9, c9], [a9]], [3, [a9], [a9]], [4, [a9, c9], []], [5, [a9], []]];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=[a9, b9, c9];source canonical open family=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []];actual open index=4,U=[c9]
After
ALL [other_index,union,intersection] rows=[[0, [a9, b9, c9], [c9]], [1, [a9, b9, c9], []], [2, [a9, c9], [c9]], [3, [a9, c9], []], [4, [c9], [c9]], [5, [c9], []]];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=[a9, b9, c9];source canonical open family=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []];actual open index=5,U=[]
After
ALL [other_index,union,intersection] rows=[[0, [a9, b9, c9], []], [1, [a9, b9], []], [2, [a9, c9], []], [3, [a9], []], [4, [c9], []], [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=c9,y=a9;source open order=[[a9, b9, c9], [a9, b9], [a9, c9], [a9], [c9], []]
After
ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a9, b9, c9], No], [0, 1, [a9, b9], No], [0, 2, [a9, c9], No], [0, 3, [a9], No], [2, 0, [a9, c9], No], [2, 1, [a9], No], [2, 2, [a9, c9], No], [2, 3, [a9], No], [4, 0, [c9], No], [4, 1, [], Yes]];FIRST source-order witness=[[c9], [a9, b9]]
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=[[c9, a9, [[c9], [a9, b9]]]];sample positions count=1;ALL finite pairs=[[a9, b9], [a9, c9], [b9, c9]];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