# Question 17429: Validity Precedes a Maximum-Matching Certificate [Return to catalogue page 88](https://ocme.evemisslab.com/en/questions/page/88/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r06-maximum-matching-foundations/#q-20261014-advance-20261103-r06-maximum-matching-0029) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0033-17429 · q-20261014-advance-20261103-r06-maximum-matching-0029 ## 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]]. 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]]. 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. - [mko-maximum-matching](https://ocme.evemisslab.com/en/mko/mko-maximum-matching/) ## Step-by-step derivation S001 · impliesBefore complete original carrier Fin(6);ALREADY canonical original edge list=[[0, 1], [0, 2], [0, 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 dependenciesRule: maxmatching_complete_bounded_original_graph_mask_domain · Uses: G001 S002 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S001 S003 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S002 S004 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S003 S005 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S004 S006 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S005 S007 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S006 S008 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S007 S009 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S008 S010 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S009 S011 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S010 S012 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S011 S013 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S012 S014 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S013 S015 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S014 S016 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S015 S017 · impliesBefore 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 dependenciesRule: maxmatching_every_mask_validity_before_global_comparison · Uses: G001, S016 S018 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore 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 dependenciesRule: maxmatching_exhaustive_upper_bound_and_attaining_all_witnesses · Uses: G001, S018 S020 · impliesBefore 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 dependenciesRule: maxmatching_revised_validity_conjunction_not_legacy_size_only · Uses: G001, S019 ## Reverse checks - 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 — Recompute exhaustive matching validity indicator count with exact rational arithmetic, not rounded decimals. - 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, 2, 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: 239b0decda52301ba2499a0d518ff9d2663a574fdfba1c1de7d150a292b7491a - candidate record SHA-256: 5506436047fe3fd87dd410d553a562d0cd2b3a534e268c321783134efb975b77 - semantic review receipt SHA-256: 4a0957a15bb33fbe5f663e37a8babf46ab5a291eb5d1dfa3bf9045204cd43ec3 - question-first witness SHA-256: 22a8506bcfa34c503216083d1a8e813d61aad22dbc81589c0d41089c528a7f93 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261103-r06-maximum-matching-0029.json)