No feasible interpolation for TC⁰-Frege proofs

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC⁰-Frege systems. More specifically, we show that unless factoring is feasible, neither Frege nor TC⁰-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-size TC⁰-Frege. In particular, we show how to carry out the proof for the Chinese Remainder Theorem, which may be of independent interest. As a corollary, we obtain that TC⁰-Frege as well as any proof system that polynomially simulates it, is not automatizable (under a hardness assumption).