3.7. Reach🔗

What the certificate tier can do depends on how much of n - 1 the untrusted search can factor:

• The supported elaboration ceiling is 512 bits. The release probes include table-smooth certificates from 31 through 511 bits and a 512-bit certificate whose search discovers an above-table factor with bounded rho work.

• The bounded search reports exhaustion rather than claiming compositeness. A separate 512-bit probable-prime probe exercises this path, while a 513-bit input is rejected before search begins.

• Negative answers are conclusive only when a size check, table lookup, exact trial decision, or one of the thirteen fixed Miller-Rabin bases supplies a witness. Passing all fixed bases is not itself a primality result.