  It has been argued that small corrections to evolution arising from
non-geometric effects can resolve the information paradox. We can get such
effects, for example, from subleading saddle points in the Euclidean path
integral. But an inequality derived in 2009 using strong sub-additivity showed
that such corrections {\it cannot} solve the problem. As a result we sharpen
the original Hawking puzzle: we must either have (A) new (nonlocal) physics or
(B) construct hair at the horizon. We get correspondingly different approaches
to resolving the AMPS puzzle. Traditional complementarity assumes (A); here we
require that the AMPS experiment measures the correct vacuum entanglement of
Hawking modes, and invoke nonlocal $A=R_B$ type effects to obtain unitarity of
radiation. Fuzzball complementarity is in category (B); here the AMPS
measurement is outside the validity of the approximation required to obtain the
complementary description, and a effective regular horizon arises only for
freely infalling observers with energies $E\gg T$.
