  The current state-of-the-art methods for showing inapproximability in PPAD
arise from the $\varepsilon$-Generalized-Circuit ($\varepsilon$-GCircuit)
problem. Rubinstein (2018) showed that there exists a small unknown constant
$\varepsilon$ for which $\varepsilon$-GCircuit is PPAD-hard, and subsequent
work has shown hardness results for other problems in PPAD by using
$\varepsilon$-GCircuit as an intermediate problem.
  We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be
thought of as $\varepsilon$-GCircuit pushed to the limit as $\varepsilon
\rightarrow 1$, and we show that the problem is PPAD-complete. We then prove
that $\varepsilon$-GCircuit is PPAD-hard for all $\varepsilon < 0.1$ by a
reduction from Pure-Circuit, and thus strengthen all prior work that has used
GCircuit as an intermediate problem from the existential-constant regime to the
large-constant regime.
  We show that stronger inapproximability results can be derived by reducing
directly from Pure-Circuit. In particular, we prove tight inapproximability
results for computing $\varepsilon$-well-supported Nash equilibria in
two-action polymatrix games, as well as for finding approximate equilibria in
threshold games.
