# Question 14351: A Measurable Event Is a Union of Whole Current Blocks

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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.

R0027-14351 · q-20261014-advance-20261029-r05-finite-filtration-0076



## Original question and givens

Use the same finite carrier Fin(6)={0,…,5} at all times, with complete partitions=[[0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 10, 10, 10]] in time order starting at 0. Equal labels in one row mean the same information block; numerical label size and labels in other rows carry no extra meaning. A filtration requires every later partition to refine its predecessor. Event measurability and process adaptedness are separate checks; no natural-filtration, independence or Markov conclusion is inferred. time=0, event=[0]. Normalize the event as increasing distinct outcome indices. Give (time,event,measurable?,event_must_be_union_of_time_blocks).


- G001 — Use the same finite carrier Fin(6)={0,…,5} at all times, with complete partitions=[[0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 10, 10, 10]] in time order starting at 0. Equal labels in one row mean the same information block; numerical label size and labels in other rows carry no extra meaning. A filtration requires every later partition to refine its predecessor. Event measurability and process adaptedness are separate checks; no natural-filtration, independence or Markov conclusion is inferred. time=0, event=[0]. Normalize the event as increasing distinct outcome indices. Give (time,event,measurable?,event_must_be_union_of_time_blocks).





## 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-finite-filtration](https://ocme.evemisslab.com/en/mko/mko-finite-filtration/)





## Step-by-step derivation

S001 · impliesBefore

same carrier Fin(6);complete partitions in original time order=[[0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 0, 0, 10, 10, 10]]

After

equal labels mean same information block only within that row

Conditions: none

Labels encode equivalence blocks only within their time row.Label magnitude or reuse across rows carries no extra information;larger numbers do not mean more information.

Step identifiers and dependenciesRule: filtration_same_carrier_time_and_label_roles · Uses: G001





S002 · impliesBefore

time=1;later block label=0;indices=[0, 1, 2]

After

earlier block labels in encounter order=[0];contained in one earlier block=Yes

Conditions: none

Every later block must lie wholly in one earlier block.Merging previously distinguishable outcomes violates filtration;the refinement direction cannot be reversed.

Step identifiers and dependenciesRule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S001





S003 · impliesBefore

time=1;later block label=1;indices=[3, 4, 5]

After

earlier block labels in encounter order=[0];contained in one earlier block=Yes

Conditions: none

Every later block must lie wholly in one earlier block.Merging previously distinguishable outcomes violates filtration;the refinement direction cannot be reversed.

Step identifiers and dependenciesRule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S002





S004 · impliesBefore

time=2;later block label=0;indices=[0, 1, 2]

After

earlier block labels in encounter order=[0];contained in one earlier block=Yes

Conditions: none

Every later block must lie wholly in one earlier block.Merging previously distinguishable outcomes violates filtration;the refinement direction cannot be reversed.

Step identifiers and dependenciesRule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S003





S005 · impliesBefore

time=2;later block label=10;indices=[3, 4, 5]

After

earlier block labels in encounter order=[1];contained in one earlier block=Yes

Conditions: none

Every later block must lie wholly in one earlier block.Merging previously distinguishable outcomes violates filtration;the refinement direction cannot be reversed.

Step identifiers and dependenciesRule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S004





S006 · impliesBefore

all nesting failures in time,a,b order=[]

After

filtration=Yes;per-time measurability checks remain separate

Conditions: none

Audit refinement at every adjacent time.Even if nesting fails,the protocol may still look up a block or check measurability in one row.One passing row does not repair the whole information sequence.

Step identifiers and dependenciesRule: filtration_nesting_not_adaptedness · Uses: G001, S005





S007 · impliesBefore

time=0;block label=0;indices=[0, 1, 2, 3, 4, 5];normalized event=[0]

After

selected block indices=[0];whole-or-empty=No

Conditions: none

The event must include or exclude each block wholly.Partial inclusion fails;do not enlarge the event or omit outcomes without a probability premise.

Step identifiers and dependenciesRule: event_measurability_union_of_whole_current_blocks · Uses: G001, S006









## Reverse checks


- 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

(0, [0], No, event_must_be_union_of_time_blocks)

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: 15588282e7f7545ffaf6841e6f3a7504dfd010fe5af91a7ea42c9252eec93985
- candidate record SHA-256: a2e0ca2a52996d0abade196a8c10bfa786b7ec5206557e0c374c9d19a7688c1b
- semantic review receipt SHA-256: e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe
- question-first witness SHA-256: a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r05-finite-filtration-0076.json)