from itertools import combinations

def covers(groups,target):
 allrows=[]
 for count in range(1,len(groups)+1):
  for selection in combinations(groups,count):
   union=set().union(*(set(g['indices']) for g in selection))
   if target<=union:allrows.append({'size':count,'groups':[g['id'] for g in selection],'indices':sorted(union)})
 return allrows

def run(load,check):
 seg=load('C302_DATA.json')['complete_segments'];pairs=[p for p in seg if p['profile']=='LXX_OFF']
 grouped={};geared={}
 for p in pairs:
  t=p['upper']['regular'];u=p['lower']['regular'];key=(t['tradition'],t['Cainan'],t['Terah60'],t['full430'],t['role'],u['role'])
  grouped.setdefault(key,set()).add(p['index']);geared.setdefault(key+(t['gear'],),set()).add(p['index'])
 groups=[{'id':'family'+str(j+1),'tradition':k[0],'Cainan':k[1],'Terah60':k[2],'full430':k[3],'upper_role':k[4],'lower_role':k[5],'indices':sorted(v)} for j,(k,v) in enumerate(sorted(grouped.items()))]
 geargroups=[{'id':'gear_family'+str(j+1),'key':list(k),'indices':sorted(v)} for j,(k,v) in enumerate(sorted(geared.items()))]
 available=set(p['index'] for p in pairs);allcovers=covers(groups,available);minimal=[c for c in allcovers if c['size']==min(x['size'] for x in allcovers)]
 gearcovers=covers(geargroups,available);gearminimum=min(c['size'] for c in gearcovers);mingear=[c for c in gearcovers if c['size']==gearminimum]
 check('tenLXXprofile labeledsegments sevenindices',len(pairs)==10 and sorted(available)==[0,2,3,4,5,6,7])
 check('three regularcontextfamilies',len(groups)==3)
 check('familyindexcounts3LXX2MT5SP',[(g['tradition'],len(g['indices'])) for g in groups]==[('LXX',3),('MT',2),('SP',5)])
 check('unique minimum twofamilycover',len(minimal)==1 and minimal[0]['size']==2)
 chosen=[g for g in groups if g['id'] in minimal[0]['groups']]
 check('minimum cover isMTbaselineandSPalternate',[g['tradition'] for g in chosen]==['MT','SP'] and [(g['Cainan'],g['Terah60'],g['full430']) for g in chosen]==[(0,0,1),(0,1,0)])
 check('endpointroleschange',[g['upper_role'] for g in chosen]==['Noah birth','Shem birth'] and [g['lower_role'] for g in chosen]==['Flood start','Shem death'])
 check('noonefamilycover',all(c['size']>1 for c in allcovers))
 check('full eightindexcover impossible',not covers(groups,set(range(8))))
 check('LXXcoordinatecover redundant',set(next(g['indices'] for g in groups if g['tradition']=='LXX'))<=set(next(g['indices'] for g in groups if g['tradition']=='SP')))
 check('gearretained minimumfivefamilies twochoices',gearminimum==5 and len(mingear)==2)
 check('contextshift separately60minus215',60-215==-155)
 core=load('C287_DATA.json');reflection=load('C295_DATA.json');audit=load('C309_DATA.json')
 check('corestatementfourcompletecoordinate rows',sum(r['complete'] for r in core['rows'])==4)
 check('finalsummaryretainsreflection incompatibility',reflection['counts']['both']==0)
 check('finalsummaryretainsrankandplacement',audit['fixedK_rank']==3 and audit['fixedK_translation_freedom']==1)
 anchors=load('C308_DATA.json')['anchor_cases']
 check('finalsummaryretainsanchorpointpathdistinction',anchors[0]['labeled_segments']==12 and anchors[1]['labeled_segments']==0)
 return {'LXX_OFF_target_indices':sorted(available),'missing_annual_index':[1],'context_family_definition':'Tradition,Cainan,Terah,sojourn and endpoint roles fixed; existing Gear and rail labels remain on each segment.','families':groups,'all_context_covers':allcovers,'minimum_context_covers':minimal,'gear_specific_families':geargroups,'minimum_gear_specific_covers':mingear,'net_regular_context_shift':-155,'coverage_status':'Unique minimum two-context cover, with a role change and one missing annual index.','full_single_context_chronology_claimed':False,'batch_end':'C310','next_unexecuted':'C311 source-rule audit of the MT-to-SP context and endpoint-role switch; no new endpoint search.'}