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Question 17445: Validity Precedes a Maximum-Matching Certificate

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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R0033-17445 · q-20261014-advance-20261103-r06-maximum-matching-0045

Original question and givens

Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).

  • G001 — Use the complete bounded simple graph on Fin(6)={0,…,5} with canonical ordered undirected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]. Edge indices are zero-based positions in this exact list. A matching contains distinct edge indices with no shared endpoint. Enumerate every edge subset to determine the global maximum size; include the empty matching. List indices within each matching increasingly and enumerate matchings in increasing edge-subset mask order. Maximum,inclusion-maximal,perfect and unique are distinct properties. Proposed edge-index collection=[0, 1]. Give (matching,valid?,size,maximum_size,is_maximum?,maximum_not_necessarily_unique_or_perfect).

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

complete original carrier Fin(6);ALREADY canonical original edge list=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]

After

loop-free unique original edges=5<=10;all edge subsets have mask0 through31

Conditions: none

This family's bounds are n≤12 and at most10 edges,permitting exhaustive enumeration.Do not reuse the earlier n30 guard.Input endpoints and edges must already be canonical,with indices bound to original edges and no loops or duplicates.

Step identifiers and dependencies

Rule: maxmatching_complete_bounded_original_graph_mask_domain · Uses: G001

S002 · implies

Before

original edge-subset integer mask=0;increasing original indices=[];actual selected edges=[]

After

used endpoint union=[];repeated endpoints in traversal order=[];valid matching=Yes;size=0

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S001

S003 · implies

Before

original edge-subset integer mask=1;increasing original indices=[0];actual selected edges=[[0, 1]]

After

used endpoint union=[0, 1];repeated endpoints in traversal order=[];valid matching=Yes;size=1

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S002

S004 · implies

Before

original edge-subset integer mask=2;increasing original indices=[1];actual selected edges=[[0, 2]]

After

used endpoint union=[0, 2];repeated endpoints in traversal order=[];valid matching=Yes;size=1

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S003

S005 · implies

Before

original edge-subset integer mask=3;increasing original indices=[0, 1];actual selected edges=[[0, 1], [0, 2]]

After

used endpoint union=[0, 1, 2];repeated endpoints in traversal order=[0];valid matching=No;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S004

S006 · implies

Before

original edge-subset integer mask=4;increasing original indices=[2];actual selected edges=[[0, 3]]

After

used endpoint union=[0, 3];repeated endpoints in traversal order=[];valid matching=Yes;size=1

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S005

S007 · implies

Before

original edge-subset integer mask=5;increasing original indices=[0, 2];actual selected edges=[[0, 1], [0, 3]]

After

used endpoint union=[0, 1, 3];repeated endpoints in traversal order=[0];valid matching=No;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S006

S008 · implies

Before

original edge-subset integer mask=6;increasing original indices=[1, 2];actual selected edges=[[0, 2], [0, 3]]

After

used endpoint union=[0, 2, 3];repeated endpoints in traversal order=[0];valid matching=No;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S007

S009 · implies

Before

original edge-subset integer mask=7;increasing original indices=[0, 1, 2];actual selected edges=[[0, 1], [0, 2], [0, 3]]

After

used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[0, 0];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S008

S010 · implies

Before

original edge-subset integer mask=8;increasing original indices=[3];actual selected edges=[[2, 3]]

After

used endpoint union=[2, 3];repeated endpoints in traversal order=[];valid matching=Yes;size=1

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S009

S011 · implies

Before

original edge-subset integer mask=9;increasing original indices=[0, 3];actual selected edges=[[0, 1], [2, 3]]

After

used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[];valid matching=Yes;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S010

S012 · implies

Before

original edge-subset integer mask=10;increasing original indices=[1, 3];actual selected edges=[[0, 2], [2, 3]]

After

used endpoint union=[0, 2, 3];repeated endpoints in traversal order=[2];valid matching=No;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S011

S013 · implies

Before

original edge-subset integer mask=11;increasing original indices=[0, 1, 3];actual selected edges=[[0, 1], [0, 2], [2, 3]]

After

used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[0, 2];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S012

S014 · implies

Before

original edge-subset integer mask=12;increasing original indices=[2, 3];actual selected edges=[[0, 3], [2, 3]]

After

used endpoint union=[0, 2, 3];repeated endpoints in traversal order=[3];valid matching=No;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S013

S015 · implies

Before

original edge-subset integer mask=13;increasing original indices=[0, 2, 3];actual selected edges=[[0, 1], [0, 3], [2, 3]]

After

used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[0, 3];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S014

S016 · implies

Before

original edge-subset integer mask=14;increasing original indices=[1, 2, 3];actual selected edges=[[0, 2], [0, 3], [2, 3]]

After

used endpoint union=[0, 2, 3];repeated endpoints in traversal order=[0, 2, 3];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S015

S017 · implies

Before

original edge-subset integer mask=15;increasing original indices=[0, 1, 2, 3];actual selected edges=[[0, 1], [0, 2], [0, 3], [2, 3]]

After

used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[0, 0, 2, 3];valid matching=No;size=4

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S016

S018 · implies

Before

original edge-subset integer mask=16;increasing original indices=[4];actual selected edges=[[4, 5]]

After

used endpoint union=[4, 5];repeated endpoints in traversal order=[];valid matching=Yes;size=1

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S017

S019 · implies

Before

original edge-subset integer mask=17;increasing original indices=[0, 4];actual selected edges=[[0, 1], [4, 5]]

After

used endpoint union=[0, 1, 4, 5];repeated endpoints in traversal order=[];valid matching=Yes;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S018

S020 · implies

Before

original edge-subset integer mask=18;increasing original indices=[1, 4];actual selected edges=[[0, 2], [4, 5]]

After

used endpoint union=[0, 2, 4, 5];repeated endpoints in traversal order=[];valid matching=Yes;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S019

S021 · implies

Before

original edge-subset integer mask=19;increasing original indices=[0, 1, 4];actual selected edges=[[0, 1], [0, 2], [4, 5]]

After

used endpoint union=[0, 1, 2, 4, 5];repeated endpoints in traversal order=[0];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S020

S022 · implies

Before

original edge-subset integer mask=20;increasing original indices=[2, 4];actual selected edges=[[0, 3], [4, 5]]

After

used endpoint union=[0, 3, 4, 5];repeated endpoints in traversal order=[];valid matching=Yes;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S021

S023 · implies

Before

original edge-subset integer mask=21;increasing original indices=[0, 2, 4];actual selected edges=[[0, 1], [0, 3], [4, 5]]

After

used endpoint union=[0, 1, 3, 4, 5];repeated endpoints in traversal order=[0];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S022

S024 · implies

Before

original edge-subset integer mask=22;increasing original indices=[1, 2, 4];actual selected edges=[[0, 2], [0, 3], [4, 5]]

After

used endpoint union=[0, 2, 3, 4, 5];repeated endpoints in traversal order=[0];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S023

S025 · implies

Before

original edge-subset integer mask=23;increasing original indices=[0, 1, 2, 4];actual selected edges=[[0, 1], [0, 2], [0, 3], [4, 5]]

After

used endpoint union=[0, 1, 2, 3, 4, 5];repeated endpoints in traversal order=[0, 0];valid matching=No;size=4

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S024

S026 · implies

Before

original edge-subset integer mask=24;increasing original indices=[3, 4];actual selected edges=[[2, 3], [4, 5]]

After

used endpoint union=[2, 3, 4, 5];repeated endpoints in traversal order=[];valid matching=Yes;size=2

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S025

S027 · implies

Before

original edge-subset integer mask=25;increasing original indices=[0, 3, 4];actual selected edges=[[0, 1], [2, 3], [4, 5]]

After

used endpoint union=[0, 1, 2, 3, 4, 5];repeated endpoints in traversal order=[];valid matching=Yes;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S026

S028 · implies

Before

original edge-subset integer mask=26;increasing original indices=[1, 3, 4];actual selected edges=[[0, 2], [2, 3], [4, 5]]

After

used endpoint union=[0, 2, 3, 4, 5];repeated endpoints in traversal order=[2];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S027

S029 · implies

Before

original edge-subset integer mask=27;increasing original indices=[0, 1, 3, 4];actual selected edges=[[0, 1], [0, 2], [2, 3], [4, 5]]

After

used endpoint union=[0, 1, 2, 3, 4, 5];repeated endpoints in traversal order=[0, 2];valid matching=No;size=4

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S028

S030 · implies

Before

original edge-subset integer mask=28;increasing original indices=[2, 3, 4];actual selected edges=[[0, 3], [2, 3], [4, 5]]

After

used endpoint union=[0, 2, 3, 4, 5];repeated endpoints in traversal order=[3];valid matching=No;size=3

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S029

S031 · implies

Before

original edge-subset integer mask=29;increasing original indices=[0, 2, 3, 4];actual selected edges=[[0, 1], [0, 3], [2, 3], [4, 5]]

After

used endpoint union=[0, 1, 2, 3, 4, 5];repeated endpoints in traversal order=[0, 3];valid matching=No;size=4

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S030

S032 · implies

Before

original edge-subset integer mask=30;increasing original indices=[1, 2, 3, 4];actual selected edges=[[0, 2], [0, 3], [2, 3], [4, 5]]

After

used endpoint union=[0, 2, 3, 4, 5];repeated endpoints in traversal order=[0, 2, 3];valid matching=No;size=4

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S031

S033 · implies

Before

original edge-subset integer mask=31;increasing original indices=[0, 1, 2, 3, 4];actual selected edges=[[0, 1], [0, 2], [0, 3], [2, 3], [4, 5]]

After

used endpoint union=[0, 1, 2, 3, 4, 5];repeated endpoints in traversal order=[0, 0, 2, 3];valid matching=No;size=5

Conditions: none

Enumerate integer masks in order,including the empty subset,and check endpoint conflicts first.Invalid collections cannot enter the optimum comparison.Every original edge subset has one mask,including alternatives that replace selected edges.

Step identifiers and dependencies

Rule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S032

S034 · calculation

Before

exhaustive matching validity indicator count := sum(1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0)

After

exhaustive matching validity indicator count = 12

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,coordinates and indices.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · implies

Before

all32 original masks examined;valid matching count=12;all valid sizes=[0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3]

After

global maximum size=3;ALL maxima in ORIGINAL MASK order=[[0, 3, 4]]

Conditions: none

Every valid candidate has size at most this maximum,and every attaining witness is retained.This supplies both a global upper bound and attainment.Keep mask order rather than lexicographic list order,and infer no general complexity theorem.

Step identifiers and dependencies

Rule: maxmatching_exhaustive_upper_bound_and_attaining_all_witnesses · Uses: G001, S034

S036 · implies

Before

original proposed indices=[0, 1];sorted duplicate-free collection=[0, 1];repeated endpoints=[0];numeric size=2;true global maximum=3

After

proposal valid=No;is maximum=valid AND size equals optimum=No

Conditions: none

Use the revised contract employed by the actual English projection.A conflicting-edge collection is not a maximum matching even if its numeric size equals the optimum.Keep its original size and false flags rather than changing or discarding the proposal.

Step identifiers and dependencies

Rule: maxmatching_revised_validity_conjunction_not_legacy_size_only · Uses: G001, S035

Reverse checks

  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Recompute exhaustive matching validity indicator count with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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], No, 2, 3, No, maximum_not_necessarily_unique_or_perfect)

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: 801c8e32c4b2e9e92dc7a9ab759a539d5fc1281613582f83a82e91eb53386cbc
  • candidate record SHA-256: 0ebc3d8b91e912df3980032415194f38f7e09515cbf9e20baf46dd0dcdf927b5
  • semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3
  • question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON