Letting C be the fraction of primes with the property that altering any bit destroys primality, it occurs to me I can prove limsup C <= 3/8 by arguing that every prime p=1 mod 60 has p+2=composite, p=7(60) ==> p-2 composite, p=11(60) ==> p-2 compo, p=13(60) ==> p+2 compo, p=23(60) ==> p-2 compo, and similarly for -1,-7,-11,-13,-23. One could of course get much stronger upper bounds by considering congruence classes to appropriate higher moduli by computer; the best moduli for this purpose perhaps are of the form 2^j * 3*5*7*11*13*... I still lack a proof that lim C exists, merely have limsupC<=3/8 and liminfC>10^(-29). But I suspect an appropriate subsequence of the upper bounds alluded to above, converges to limC from above.