ex po ne nt ial ly ma ny clu st er s in th e th er mo dy na mi c lim it , as th e cl au se de ns it y is inc rea sed tow ard s the SAT-u nSAT thr esh old for lar ge eno ugh

k

. The Ham min g distance between a solution that lies in one cluster and that in another is

O

(

n

)

. Next, we encode

k

-SAT formulae as structures on which

FO

(

LFP

)

captures polynomial time. By asking

FO

(

LFP

)

to extend partial assignments on ensem- bles of random

k

-SAT, we build distributions of solutions. We then construct a dynami c graph ical model on a pro duct space that captur es all the informatio n ﬂows through the various stages of a

LFP

computation on ensembles of

k

-SAT structures. Distributions computed by

LFP

must satisfy this model. This model is directed, which allows us to compute factorizations locally and parameterize using Gibbs potentials on cliques. We then use results from ensembles of factor graphs of random

k

-SAT to bound the various information ﬂows in this di- rected graphical model. We parametrize the resulting distributions in a manner that demonstrates that irreducible interactions between covariates — namely, those that may not be factored any further through conditional independencies — cannot grow faster than

poly(log

n

)

in the

LFP

computed distributions. This characterization allows us to analyze the behavior of the entire class of polyno- mial time algorithms on ensembles simultaneously. Using the aforementioned limitations of

LFP

, we demonstrate that a pur- ported polynomial time solution to

k

-SAT would result in solution space that is a mixture of distributions each having an exponentially smaller parametriza- tion than is consistent with the highly constrained d1RSB phases of

k

-SAT. We show that this would contradict the behavior exhibited by the solution space in the d1RSB phase. This corresponds to the intuitive picture provided by physics about the emergence of extensive (meaning

O

(

n

)

) long-range correlations be- tween variables in this phase and also explains the empirical observation that all known polynomial time algorithms break down in this phase. Our work shows that every polynomial time algorithm must fail to produce solutions to large enough problem instances of

k

-S AT in the d1RS B phase. This shows that polynomial time algorithms are not capable of solving

NP

-complete problems in their hard phases, and demonstrates the separation of

P

from

NP