# Question 22976: Successful Pair Samples Do Not Certify Unchecked or Infinite Cases

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0044-22976 · q-20261014-advance-20261111-r03-hausdorff-space-0076



## Original question and givens

Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c0, a0]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).


- G001 — Use finite unique labeled points=[a0, b0, c0] and the complete canonical topology=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []]. Every neighborhood witness must be a declared open set containing its assigned point; the two points must be distinct and the witnesses disjoint. The complete topology must qualify before a Hausdorff audit. Supplied sampled_pairs=[[c0, a0]]. Give (sampled_pair_rows, all_sampled_separated, sample_count, total_pair_count, general_sample_certificate, boundary).





## 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.


- [mko-hausdorff-space](https://ocme.evemisslab.com/en/mko/mko-hausdorff-space/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL carrier order=[a0, b0, c0];source canonical open family=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []];actual open index=0,U=[a0, b0, c0]

After

ALL [other_index,union,intersection] rows=[[0, [a0, b0, c0], [a0, b0, c0]], [1, [a0, b0, c0], [a0, b0]], [2, [a0, b0, c0], [a0, c0]], [3, [a0, b0, c0], [a0]], [4, [a0, b0, c0], [c0]], [5, [a0, b0, c0], []]];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 dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S001





S003 · impliesBefore

ORIGINAL carrier order=[a0, b0, c0];source canonical open family=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []];actual open index=1,U=[a0, b0]

After

ALL [other_index,union,intersection] rows=[[0, [a0, b0, c0], [a0, b0]], [1, [a0, b0], [a0, b0]], [2, [a0, b0, c0], [a0]], [3, [a0, b0], [a0]], [4, [a0, b0, c0], []], [5, [a0, b0], []]];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 dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S002





S004 · impliesBefore

ORIGINAL carrier order=[a0, b0, c0];source canonical open family=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []];actual open index=2,U=[a0, c0]

After

ALL [other_index,union,intersection] rows=[[0, [a0, b0, c0], [a0, c0]], [1, [a0, b0, c0], [a0]], [2, [a0, c0], [a0, c0]], [3, [a0, c0], [a0]], [4, [a0, c0], [c0]], [5, [a0, c0], []]];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 dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S003





S005 · impliesBefore

ORIGINAL carrier order=[a0, b0, c0];source canonical open family=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []];actual open index=3,U=[a0]

After

ALL [other_index,union,intersection] rows=[[0, [a0, b0, c0], [a0]], [1, [a0, b0], [a0]], [2, [a0, c0], [a0]], [3, [a0], [a0]], [4, [a0, c0], []], [5, [a0], []]];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 dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S004





S006 · impliesBefore

ORIGINAL carrier order=[a0, b0, c0];source canonical open family=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []];actual open index=4,U=[c0]

After

ALL [other_index,union,intersection] rows=[[0, [a0, b0, c0], [c0]], [1, [a0, b0, c0], []], [2, [a0, c0], [c0]], [3, [a0, c0], []], [4, [c0], [c0]], [5, [c0], []]];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 dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S005





S007 · impliesBefore

ORIGINAL carrier order=[a0, b0, c0];source canonical open family=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []];actual open index=5,U=[]

After

ALL [other_index,union,intersection] rows=[[0, [a0, b0, c0], []], [1, [a0, b0], []], [2, [a0, c0], []], [3, [a0], []], [4, [c0], []], [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 dependenciesRule: hausdorff_complete_labeled_topology_and_source_order · Uses: G001, S006





S008 · impliesBefore

Hausdorff ORIGINAL sampled pair index0:ORIGINAL distinct carrier points x=c0,y=a0;source open order=[[a0, b0, c0], [a0, b0], [a0, c0], [a0], [c0], []]

After

ALL tested [U_index,V_index,intersection,disjoint] rows=[[0, 0, [a0, b0, c0], No], [0, 1, [a0, b0], No], [0, 2, [a0, c0], No], [0, 3, [a0], No], [2, 0, [a0, c0], No], [2, 1, [a0], No], [2, 2, [a0, c0], No], [2, 3, [a0], No], [4, 0, [c0], No], [4, 1, [], Yes]];FIRST source-order witness=[[c0], [a0, b0]]

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 dependenciesRule: hausdorff_FIRST_actual_open_pair_or_EXHAUSTIVE_failure · Uses: G001, S007





S009 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · impliesBefore

ORIGINAL sampled pair rows=[[c0, a0, [[c0], [a0, b0]]]];sample positions count=1;ALL finite pairs=[[a0, b0], [a0, c0], [b0, c0]];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 dependenciesRule: hausdorff_sample_No_not_separation_flag_or_actual_completeness_negation · Uses: G001, S010









## Reverse checks


- 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 — Recompute Hausdorff ALL unordered carrier pairs count with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute Hausdorff ORIGINAL sampled positions count with exact rational arithmetic, not rounded decimals.
- 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

([[c0, a0, [[c0], [a0, b0]]]], Yes, 1, 3, No, successful_sampled_pairs_do_not_certify_unchecked_pairs_or_infinite_spaces)

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: 811fdd77f2ea9fb46e707369d33648da48238d0d10d5c9349524a2e08497fec1
- candidate record SHA-256: abdef937cc6d29141d2eb790f9067137105da2fe58da5b5649dd9a2f43167838
- semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
- question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r03-hausdorff-space-0076.json)