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, 1, 1, 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. Complete time-aligned process_values=[[0, 0, 0, 0, 0, 0], [1, 1, 1, 2, 2, 2], [2, 3, 3, 12, 12, 12]]. Give (filtration?,measurability_failures,adapted?,qualified_adapted_process?). Failure rows are [time,block_label,indices,distinct_values], traversing times and first-seen block labels in row order.
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, 1, 1, 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. Complete time-aligned process_values=[[0, 0, 0, 0, 0, 0], [1, 1, 1, 2, 2, 2], [2, 3, 3, 12, 12, 12]]. Give (filtration?,measurability_failures,adapted?,qualified_adapted_process?). Failure rows are [time,block_label,indices,distinct_values], traversing times and first-seen block labels in row order.
Step-by-step derivation
S001 · implies
Before
same carrier Fin(6);complete partitions in original time order=[[0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1], [0, 1, 1, 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 dependencies
Rule: filtration_same_carrier_time_and_label_roles · Uses: G001
S002 · implies
Before
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 dependencies
Rule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S001
S003 · implies
Before
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 dependencies
Rule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S002
S004 · implies
Before
time=2;later block label=0;indices=[0]
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 dependencies
Rule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S003
S005 · implies
Before
time=2;later block label=1;indices=[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 dependencies
Rule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S004
S006 · implies
Before
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 dependencies
Rule: filtration_later_refines_earlier_complete_blocks · Uses: G001, S005
S007 · implies
Before
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 dependencies
Rule: filtration_nesting_not_adaptedness · Uses: G001, S006
S008 · implies
Before
time=0;first-seen block label=0;indices=[0, 1, 2, 3, 4, 5];aligned process values=[0, 0, 0, 0, 0, 0]
After
distinct values in encounter order=[0];constant=Yes
Conditions: none
At each time the process must be constant on every information block.Order failures by time and first-seen block label,and values by first encounter rather than sorting.
Step identifiers and dependencies
Rule: adaptedness_every_current_block_not_nesting_alone · Uses: G001, S007
S009 · implies
Before
time=1;first-seen block label=0;indices=[0, 1, 2];aligned process values=[1, 1, 1]
After
distinct values in encounter order=[1];constant=Yes
Conditions: none
At each time the process must be constant on every information block.Order failures by time and first-seen block label,and values by first encounter rather than sorting.
Step identifiers and dependencies
Rule: adaptedness_every_current_block_not_nesting_alone · Uses: G001, S008
S010 · implies
Before
time=1;first-seen block label=1;indices=[3, 4, 5];aligned process values=[2, 2, 2]
After
distinct values in encounter order=[2];constant=Yes
Conditions: none
At each time the process must be constant on every information block.Order failures by time and first-seen block label,and values by first encounter rather than sorting.
Step identifiers and dependencies
Rule: adaptedness_every_current_block_not_nesting_alone · Uses: G001, S009
S011 · implies
Before
time=2;first-seen block label=0;indices=[0];aligned process values=[2]
After
distinct values in encounter order=[2];constant=Yes
Conditions: none
At each time the process must be constant on every information block.Order failures by time and first-seen block label,and values by first encounter rather than sorting.
Step identifiers and dependencies
Rule: adaptedness_every_current_block_not_nesting_alone · Uses: G001, S010
S012 · implies
Before
time=2;first-seen block label=1;indices=[1, 2];aligned process values=[3, 3]
After
distinct values in encounter order=[3];constant=Yes
Conditions: none
At each time the process must be constant on every information block.Order failures by time and first-seen block label,and values by first encounter rather than sorting.
Step identifiers and dependencies
Rule: adaptedness_every_current_block_not_nesting_alone · Uses: G001, S011
S013 · implies
Before
time=2;first-seen block label=10;indices=[3, 4, 5];aligned process values=[12, 12, 12]
After
distinct values in encounter order=[12];constant=Yes
Conditions: none
At each time the process must be constant on every information block.Order failures by time and first-seen block label,and values by first encounter rather than sorting.
Step identifiers and dependencies
Rule: adaptedness_every_current_block_not_nesting_alone · Uses: G001, S012
S014 · implies
Before
filtration=Yes;all per-time measurability failures=[]
After
source adapted checks pass=Yes;qualified adapted process=Yes
Conditions: none
Here the source adapted field records per-time measurability checks.The last field additionally needs filtration qualification;nesting and measurability do not replace one another or imply naturalness,independence or Markov behavior.
Step identifiers and dependencies
Rule: adapted_flag_vs_qualified_adapted_process · Uses: G001, S013