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Question 22002: Every Target Open Must Have an Open Source Preimage

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0043-22002 · q-20261014-advance-20261110-r03-continuous-image-of-compact-0027

Original question and givens

Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

  • G001 — Use complete source carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], target carrier Fin(3)={0,…,2} with topology=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]], total map=[1, 0, 0] and source subset K=[0, 1, 2]. The source/target topologies must qualify; continuity is tested by every target-open preimage. K is compact by finiteness, including empty K. Give (open_preimage_rows, continuous, certificate).

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.

Step-by-step derivation

S001 · implies

Before

source operation=compactimg-continuity-audit;original snapshot and mathematics retained

After

explicit ordered source output fields=[open_preimage_rows, continuous, certificate]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=0,U=[]

After

ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2], []], [4, [0, 2], []], [5, [1], []], [6, [1, 2], []], [7, [2], []]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S001

S003 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=1,U=[0]

After

ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2], [0]], [4, [0, 2], [0]], [5, [0, 1], []], [6, [0, 1, 2], []], [7, [0, 2], []]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S002

S004 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=2,U=[0, 1]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2], [0, 1]], [4, [0, 1, 2], [0]], [5, [0, 1], [1]], [6, [0, 1, 2], [1]], [7, [0, 1, 2], []]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S003

S005 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=3,U=[0, 1, 2]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2], []], [1, [0, 1, 2], [0]], [2, [0, 1, 2], [0, 1]], [3, [0, 1, 2], [0, 1, 2]], [4, [0, 1, 2], [0, 2]], [5, [0, 1, 2], [1]], [6, [0, 1, 2], [1, 2]], [7, [0, 1, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S004

S006 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=4,U=[0, 2]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 2], []], [1, [0, 2], [0]], [2, [0, 1, 2], [0]], [3, [0, 1, 2], [0, 2]], [4, [0, 2], [0, 2]], [5, [0, 1, 2], []], [6, [0, 1, 2], [2]], [7, [0, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S005

S007 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=5,U=[1]

After

ALL [other_open_index,union,intersection] rows=[[0, [1], []], [1, [0, 1], []], [2, [0, 1], [1]], [3, [0, 1, 2], [1]], [4, [0, 1, 2], []], [5, [1], [1]], [6, [1, 2], [1]], [7, [1, 2], []]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S006

S008 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=6,U=[1, 2]

After

ALL [other_open_index,union,intersection] rows=[[0, [1, 2], []], [1, [0, 1, 2], []], [2, [0, 1, 2], [1]], [3, [0, 1, 2], [1, 2]], [4, [0, 1, 2], [2]], [5, [1, 2], [1]], [6, [1, 2], [1, 2]], [7, [1, 2], [2]]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S007

S009 · implies

Before

SOURCE:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=7,U=[2]

After

ALL [other_open_index,union,intersection] rows=[[0, [2], []], [1, [0, 2], []], [2, [0, 1, 2], []], [3, [0, 1, 2], [2]], [4, [0, 2], [2]], [5, [1, 2], []], [6, [1, 2], [2]], [7, [2], [2]]];empty/full opens present;every result in SAME SOURCE topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S008

S010 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=0,U=[]

After

ALL [other_open_index,union,intersection] rows=[[0, [], []], [1, [0], []], [2, [0, 1], []], [3, [0, 1, 2], []], [4, [0, 2], []], [5, [1], []], [6, [1, 2], []], [7, [2], []]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S009

S011 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=1,U=[0]

After

ALL [other_open_index,union,intersection] rows=[[0, [0], []], [1, [0], [0]], [2, [0, 1], [0]], [3, [0, 1, 2], [0]], [4, [0, 2], [0]], [5, [0, 1], []], [6, [0, 1, 2], []], [7, [0, 2], []]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S010

S012 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=2,U=[0, 1]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1], []], [1, [0, 1], [0]], [2, [0, 1], [0, 1]], [3, [0, 1, 2], [0, 1]], [4, [0, 1, 2], [0]], [5, [0, 1], [1]], [6, [0, 1, 2], [1]], [7, [0, 1, 2], []]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S011

S013 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=3,U=[0, 1, 2]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 1, 2], []], [1, [0, 1, 2], [0]], [2, [0, 1, 2], [0, 1]], [3, [0, 1, 2], [0, 1, 2]], [4, [0, 1, 2], [0, 2]], [5, [0, 1, 2], [1]], [6, [0, 1, 2], [1, 2]], [7, [0, 1, 2], [2]]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S012

S014 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=4,U=[0, 2]

After

ALL [other_open_index,union,intersection] rows=[[0, [0, 2], []], [1, [0, 2], [0]], [2, [0, 1, 2], [0]], [3, [0, 1, 2], [0, 2]], [4, [0, 2], [0, 2]], [5, [0, 1, 2], []], [6, [0, 1, 2], [2]], [7, [0, 2], [2]]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S013

S015 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=5,U=[1]

After

ALL [other_open_index,union,intersection] rows=[[0, [1], []], [1, [0, 1], []], [2, [0, 1], [1]], [3, [0, 1, 2], [1]], [4, [0, 1, 2], []], [5, [1], [1]], [6, [1, 2], [1]], [7, [1, 2], []]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S014

S016 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=6,U=[1, 2]

After

ALL [other_open_index,union,intersection] rows=[[0, [1, 2], []], [1, [0, 1, 2], []], [2, [0, 1, 2], [1]], [3, [0, 1, 2], [1, 2]], [4, [0, 1, 2], [2]], [5, [1, 2], [1]], [6, [1, 2], [1, 2]], [7, [1, 2], [2]]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S015

S017 · implies

Before

TARGET:ORIGINAL carrier Fin(3);complete canonical opens=[[], [0], [0, 1], [0, 1, 2], [0, 2], [1], [1, 2], [2]];open index=7,U=[2]

After

ALL [other_open_index,union,intersection] rows=[[0, [2], []], [1, [0, 2], []], [2, [0, 1, 2], []], [3, [0, 1, 2], [2]], [4, [0, 2], [2]], [5, [1, 2], []], [6, [1, 2], [2]], [7, [2], [2]]];empty/full opens present;every result in SAME TARGET topology

Conditions: none

Keep source and target carriers distinct.Check all open pairs and empty/full membership.Any union in a finite family reduces to finitely many distinct opens;a finite image does not bypass topology or continuity obligations.

Step identifiers and dependencies

Rule: compactimg_qualified_complete_source_or_target_topology · Uses: G001, S016

S018 · implies

Before

ORIGINAL target-open index=0,O=[];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S017

S019 · implies

Before

ORIGINAL target-open index=1,O=[0];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[1, 2];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S018

S020 · implies

Before

ORIGINAL target-open index=2,O=[0, 1];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[0, 1, 2];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S019

S021 · implies

Before

ORIGINAL target-open index=3,O=[0, 1, 2];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[0, 1, 2];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S020

S022 · implies

Before

ORIGINAL target-open index=4,O=[0, 2];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[1, 2];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S021

S023 · implies

Before

ORIGINAL target-open index=5,O=[1];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[0];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S022

S024 · implies

Before

ORIGINAL target-open index=6,O=[1, 2];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[0];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S023

S025 · implies

Before

ORIGINAL target-open index=7,O=[2];total original map=[1, 0, 0];ENTIRE source carrier Fin(3)

After

full source preimage=[];source-open=Yes;NOT restricted to K

Conditions: none

Continuity checks every target-open preimage in the entire source,not just K or sampled opens.A failed source-open test remains false.

Step identifiers and dependencies

Rule: compactimg_EVERY_target_open_FULL_source_preimage · Uses: G001, S024

S026 · implies

Before

ORIGINAL source subset K=[0, 1, 2];ALL [x,map(x)] pairs=[[0, 1], [1, 0], [2, 0]];target carrier Fin(3)

After

exact sorted distinct image=[0, 1];K and image finite hence compact;continuous=Yes is the ALL-target-open conjunction

Conditions: none

Every image value has an original K witness and every K image is included,so the set is exactly f(K).A finite set has a finite subcover by choosing one cover member per point.This finite argument is separate from continuity and from the general pullback theorem.

Step identifiers and dependencies

Rule: compactimg_finite_image_and_continuity_separate · 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], [1, 2]], [[0, 1], [0, 1, 2]], [[0, 1, 2], [0, 1, 2]], [[0, 2], [1, 2]], [[1], [0]], [[1, 2], [0]], [[2], []]], Yes, every_target_open_has_source_open_preimage)

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: 0e34b03b345f278ef31971ddb836efb45a232757496b2c3b28a07e548e8f325a
  • candidate record SHA-256: aa73e70362ab1f475841b56938d2e876bbb86c49e630effb82d1e62bc28b1aa6
  • semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
  • question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON