  Random 3CNF formulas constitute an important distribution for measuring the
average-case behavior of propositional proof systems. Lower bounds for random
3CNF refutations in many propositional proof systems are known. Most notably
are the exponential-size resolution refutation lower bounds for random 3CNF
formulas with $\Omega(n^{1.5-\epsilon}) $ clauses [Chvatal and Szemeredi
(1988), Ben-Sasson and Wigderson (2001)]. On the other hand, the only known
non-trivial upper bound on the size of random 3CNF refutations in a
non-abstract propositional proof system is for resolution with
$\Omega(n^{2}/\log n) $ clauses, shown by Beame et al. (2002). In this paper we
show that already standard propositional proof systems, within the hierarchy of
Frege proofs, admit short refutations for random 3CNF formulas, for
sufficiently large clause-to-variable ratio. Specifically, we demonstrate
polynomial-size propositional refutations whose lines are $TC^0$ formulas
(i.e., $TC^0$-Frege proofs) for random 3CNF formulas with $ n $ variables and $
\Omega(n^{1.4}) $ clauses.
  The idea is based on demonstrating efficient propositional correctness proofs
of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek
(2006). Since the soundness of these witnesses is verified using spectral
techniques, we develop an appropriate way to reason about eigenvectors in
propositional systems. To carry out the full argument we work inside weak
formal systems of arithmetic and use a general translation scheme to
propositional proofs.
