Question 5281: Finite Principal-Filter Direction Obstructions

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0009-05281 · q-20261008-advance-20261011-r07-tends-to-relation-0031

Original question and givens

The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

G001 — The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.

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-tends-to-relation

Step-by-step derivation

S001 · implies

Before

source index=0∈Fin(8)

After

image=1∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001

S002 · implies

Before

source index=1∈Fin(8)

After

image=0∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001, S001

S003 · implies

Before

source index=2∈Fin(8)

After

image=1∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001, S002

S004 · implies

Before

source index=3∈Fin(8)

After

image=0∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001, S003

S005 · implies

Before

source index=4∈Fin(8)

After

image=0∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001, S004

S006 · implies

Before

source index=5∈Fin(8)

After

image=0∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001, S005

S007 · implies

Before

source index=6∈Fin(8)

After

image=0∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001, S006

S008 · implies

Before

source index=7∈Fin(8)

After

image=0∈Fin(2)

Conditions: none

Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing.

Step identifiers and dependencies

Rule: complete_principal_map_row · Uses: G001, S007

S009 · implies

Before

Fin(8)→Fin(2);S=[0, 1, 2, 3, 4, 5, 6, 7];full map=[1, 0, 1, 0, 0, 0, 0, 0]

After

principal(S)={A:S⊆A};empty generators allowed,no NeBot

Conditions: none

The declared convention allows empty S;its family of supersets then contains the empty set,and no proper-filter or NeBot condition is added.This is a finite principal-generator model,not a general sequence or point limit.

Step identifiers and dependencies

Rule: finite_principal_generator_convention · Uses: G001, S008

S010 · implies

Before

source generator x=0∈S

After

f(x)=1

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S009

S011 · implies

Before

source generator x=1∈S

After

f(x)=0

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S010

S012 · implies

Before

source generator x=2∈S

After

f(x)=1

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S011

S013 · implies

Before

source generator x=3∈S

After

f(x)=0

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S012

S014 · implies

Before

source generator x=4∈S

After

f(x)=0

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S013

S015 · implies

Before

source generator x=5∈S

After

f(x)=0

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S014

S016 · implies

Before

source generator x=6∈S

After

f(x)=0

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S015

S017 · implies

Before

source generator x=7∈S

After

f(x)=0

Conditions: none

Map only source generators in S;do not confuse the whole codomain with f(S).

Step identifiers and dependencies

Rule: source_generator_image_row · Uses: G001, S016

S018 · implies

Before

x=0;f(x)=1;T=[0]

After

source obstruction=Yes

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S017

S019 · implies

Before

x=1;f(x)=0;T=[0]

After

source obstruction=No

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S018

S020 · implies

Before

x=2;f(x)=1;T=[0]

After

source obstruction=Yes

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S019

S021 · implies

Before

x=3;f(x)=0;T=[0]

After

source obstruction=No

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S020

S022 · implies

Before

x=4;f(x)=0;T=[0]

After

source obstruction=No

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S021

S023 · implies

Before

x=5;f(x)=0;T=[0]

After

source obstruction=No

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S022

S024 · implies

Before

x=6;f(x)=0;T=[0]

After

source obstruction=No

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S023

S025 · implies

Before

x=7;f(x)=0;T=[0]

After

source obstruction=No

Conditions: none

The obstruction is the source index x,not its target value;include it only when f(x) is outside T.

Step identifiers and dependencies

Rule: principal_target_obstruction_row · Uses: G001, S024

S026 · implies

Before

f(S)=[0, 1];T=[0];obstructions=[0, 2]

After

f(S)⊆T=No

Conditions: none

The direction is f(S)⊆T,from image generator to target generator;empty S makes it vacuously true.Do not reverse it to T⊆f(S).

Step identifiers and dependencies

Rule: principal_inclusion_direction · Uses: G001, S025

Reverse checks

S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

[0, 2]

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: 8661b4005c8029ffeaeff3e8081d00d324002ce8d7c64b6dfccc17057060b356
candidate record SHA-256: fbc5d404f6f0b6e4825e53cebd4192bc68b3854f1a0f59b9db9306298aeba854
semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
Lean: NOT_RUN · human review: NOT_PERFORMED

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