Step-by-step derivation
S001 · implies
Before
complete original carrier Fin(6);ALREADY canonical original edge list=[[0, 1], [0, 3], [1, 3], [2, 3]]
After
loop-free unique original edges=4<=10;all edge subsets have mask0 through15
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, 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, S003
S005 · implies
Before
original edge-subset integer mask=3;increasing original indices=[0, 1];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, S004
S006 · implies
Before
original edge-subset integer mask=4;increasing original indices=[2];actual selected edges=[[1, 3]]
After
used endpoint union=[1, 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], [1, 3]]
After
used endpoint union=[0, 1, 3];repeated endpoints in traversal order=[1];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, 3], [1, 3]]
After
used endpoint union=[0, 1, 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, S007
S009 · implies
Before
original edge-subset integer mask=7;increasing original indices=[0, 1, 2];actual selected edges=[[0, 1], [0, 3], [1, 3]]
After
used endpoint union=[0, 1, 3];repeated endpoints in traversal order=[0, 1, 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, 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, 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, S011
S013 · implies
Before
original edge-subset integer mask=11;increasing original indices=[0, 1, 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, S012
S014 · implies
Before
original edge-subset integer mask=12;increasing original indices=[2, 3];actual selected edges=[[1, 3], [2, 3]]
After
used endpoint union=[1, 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], [1, 3], [2, 3]]
After
used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[1, 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, 3], [1, 3], [2, 3]]
After
used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[3, 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, 3], [1, 3], [2, 3]]
After
used endpoint union=[0, 1, 2, 3];repeated endpoints in traversal order=[0, 1, 3, 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 · calculation
Before
exhaustive matching validity indicator count := sum(1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0)
After
exhaustive matching validity indicator count = 6
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, S017
S019 · implies
Before
all16 original masks examined;valid matching count=6;all valid sizes=[0, 1, 1, 1, 1, 2]
After
global maximum size=2;ALL maxima in ORIGINAL MASK order=[[0, 3]]
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, S018
S020 · implies
Before
original proposed indices=[0, 1];sorted duplicate-free collection=[0, 1];repeated endpoints=[0];numeric size=2;true global maximum=2
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, S019