from collections import Counter,defaultdict
from itertools import product,combinations
from fractions import Fraction
import hashlib

def run(load,read,check):
    prior=[445,446,448,449,450,451,452,454,455,457,459,461,463,464,465,466]
    for n in prior:
        check(f'certificate_C{n}',hashlib.sha256(read(f'C{n}_DATA.json').encode()).hexdigest()==load(f'C{n}_CERTIFICATE.json')['data_sha256'])
    raw=load('C286_DATA.json')['target_register']
    bases={'Noah birth':2841,'Noah death':1891,'Shem birth':2341,'Shem death':1741,'Flood start':2241,'Flood close':2240}
    generated=[]
    for t,role in product(['MT','SP','LXX'],bases):
        rails=[None] if t!='SP' or role.startswith('Flood') else ['primary','companion']
        for rail,g,C,T,F in product(rails,range(1,4),range(2),range(2),range(2)):
            x=bases[role]+650*(t!='MT')+130*C+60*T+215*F+2*(g-1)-(rail=='companion')
            generated.append((t,role,rail,g,C,T,F,x))
    def key(r):return(r['tradition'],r['role'],r.get('rail'),r['gear'],r['Cainan'],r['Terah60'],r['full430'],r['center'])
    check('independent_source_grid',Counter(generated)==Counter(map(key,raw)))
    check('source_base_excerpt','3491/3490' in read('File_18.md') and '1743/1741' in read('File_02.md'))
    X=sorted({r[-1] for r in generated});S=set(X);C=[3188,3190,3248,3250];P=sorted({c+s*432 for c in C for s in [-1,1]})
    R=set();E=set()
    for x,y in product(X,repeat=2):
        if (x+y)%2==0 and (x+y)//2 in C:R.add((x,y,(x+y)//2))
        hh=25*x-23*y
        if hh%2==0 and hh//2 in P:E.add((x,y,hh//2))
    check('inverse_pivot_E_arrows',E=={(a['input'],a['output'],a['pivot']) for a in load('C450_DATA.json')['arrows']})
    expectedR=set()
    for a in load('C446_DATA.json')['edges']:
        expectedR|={(a['lower'],a['upper'],a['center']),(a['upper'],a['lower'],a['center'])}
    check('sum_enumeration_R_arrows',R==expectedR)
    compR={(a,b,x,m,y) for x,m,a in R for m2,y,b in R if m==m2}
    check('reflection_compositions',compR=={(p['first_center'],p['second_center'],p['input'],p['intermediate'],p['output']) for p in load('C445_DATA.json')['complete_paths']})
    compE={(a,b,x,m,y) for x,m,a in E for m2,y,b in E if m==m2}
    check('E_compositions',compE=={(p['first_pivot'],p['second_pivot'],*p['coordinates']) for p in load('C455_DATA.json')['complete_paths']})
    kh=[]
    for a,b,x,m,y in sorted(compE):
        H=Fraction(25*a+23*b,48)
        check(f'displacement_{a}_{b}_{x}',H-x==Fraction(529*(x-y),96))
        if H.denominator==1 and int(H) in S and H!=x and (H-x)%529==0:kh.append([a,b,x,m,y,int(H)])
    check('unique_sourced_K_ladder',kh==[[3620,2756,2148,2020,1956,3206]])
    buckets=defaultdict(list)
    for x in X:buckets[x%529].append(x)
    Kpairs={p for bucket in buckets.values() for p in combinations(bucket,2)}
    check('residue_bucket_K_pairs',Kpairs=={(p['lower'],p['upper']) for p in load('C454_DATA.json')['K_pairs']})
    index={x:i for i,x in enumerate(X)};bits=[1<<i for i in range(len(X))]
    for x,y,h in R|E:bits[index[x]]|=1<<index[y]
    for k in range(len(X)):
        bit=1<<k;row=bits[k]
        for i in range(len(X)):
            if bits[i]&bit:bits[i]|=row
    reached={(x,y) for x,y in product(X,repeat=2) if x!=y and bits[index[x]]&(1<<index[y])}
    check('bitset_full_reachability',reached=={(p['input'],p['output']) for p in load('C451_DATA.json')['reachable_pairs']})
    check('bitset_K_exclusion',all((y-x)%529 for x,y in reached))
    scc={tuple(y for y in X if (y==x or ((x,y) in reached and (y,x) in reached))) for x in X}
    check('bitset_strong_components',scc=={tuple(x) for x in load('C451_DATA.json')['strong_components']})
    maps=defaultdict(list)
    for op,ar in [('R',R),('E',E)]:
        for x,y,h in ar:maps[(op,h)].append((x,y))
    pair_relations=set()
    for (op,h),aa in maps.items():
        for (x,y),(z,w) in combinations(sorted(aa),2):
            pair_relations.add((tuple(sorted([x,z])),tuple(sorted([y,w])),op,h))
    pair_reach=set(Kpairs)
    while True:
        new={q for p,q,op,h in pair_relations if p in pair_reach};oldlen=len(pair_reach);pair_reach|=new
        if len(pair_reach)==oldlen:break
    admitted={r for r in pair_relations if r[0] in pair_reach}
    check('pair_relation_closure',pair_reach=={tuple(p['coordinates']) for p in load('C457_DATA.json')['all_reached_pairs']})
    check('pair_relation_edges',admitted=={(tuple(p['input_pair']),tuple(p['output_pair']),p['operator'],p['anchor']) for p in load('C457_DATA.json')['admitted_edges']})
    cum=load('C365_DATA.json')['cumulative_register'];Z=sorted({z['coordinate'] for z in cum});ZS=set(Z);mirror=set()
    for A in [-2639,-2635]:
        for u,v in combinations(Z,2):
            x,y=A-v,A-u
            if (x,y) in Kpairs:mirror.add((A,x,y,u,v))
    check('reverse_Mirror_enumeration',mirror=={(p['exchange_constant'],*p['regular_pair'],*p['cumulative_pair']) for p in load('C461_DATA.json')['complete_images']})
    qualified=set()
    for A,op in product([-2639,-2635],['R','E']):
        source=R if op=='R' else E
        for h in Z:
            anchor=A-h
            if anchor not in (C if op=='R' else P):continue
            for u,v in product(Z,repeat=2):
                if (A-u,A-v,anchor) in source:qualified.add((A,op,u,v,h))
    check('reverse_anchor_qualified_Mirror',qualified=={(p['exchange_constant'],p['original_arrow']['operator'],p['image_input'],p['image_output'],p['image_anchor']) for p in load('C463_DATA.json')['fully_sourced_images']})
    nons={r[-1] for r in generated if r[2]!='companion'};native=[r for r in generated if not(r[0]=='LXX' and r[4]==0)]
    check('companion_source_control',len(nons)==282 and 3620 not in nons and all(x in nons for x in [2148,3206,2020,3170,1956,2756]))
    check('LXX_alias_source_control',len(native)==456 and {r[-1] for r in native}==S)
    for n in [448,452]:
        for row in load(f'C{n}_DATA.json').get('simple_cycles',load(f'C{n}_DATA.json').get('E_cycles',[])):
            word=row.get('word',[{'operator':'R','anchor':c} for c in row.get('centers',[])])
            def evaluate(x):
                x=Fraction(x)
                for a in word:x=2*a['anchor']-x if a['operator']=='R' else (25*x-2*a['anchor'])/23
                return x
            intercept=evaluate(0);slope=evaluate(1)-intercept
            check(f'cycle_two_point_map_C{n}',slope==Fraction(row['affine_slope']) and intercept==Fraction(row['affine_intercept']))
    audit=load('PRE_C468_SEALED_AUDIT.json');check('sealed_audit',audit['status']=='passed' and audit['published_files']==69)
    return {'step':'C468','method':'Second implementation by the same root agent; source-grid regeneration, inverse pivot equations, bitset closure, residue buckets, pair-relation fixed point and reverse cumulative enumeration.','verified_totals':{'source_labels':len(generated),'source_coordinates':len(X),'R_directed_arrows':len(R),'E_directed_arrows':len(E),'complete_R_compositions':len(compR),'complete_E_compositions':len(compE),'direct_K_pairs':len(Kpairs),'reachable_directed_pairs':len(reached),'strong_components':len(scc),'pair_relation_edges_in_full_register':len(pair_relations),'K_seed_pair_closure':len(pair_reach),'admitted_K_pair_edges':len(admitted),'fixed_Mirror_pair_images':len(mirror),'qualified_Mirror_operator_images':len(qualified)},'unique_nonzero_sourced_K_E2_witness':kh,'verified_packet_audit':audit['totals'],'published_predecessor_files':69,'independent_agent_review':False,'new_historical_witnesses':False,'C469_executed':False}