Circuit Lower Bounds for Extended Gate Bases from Subexponential Counting

We study circuit lower bounds for extended gate bases using exact CNF counting and an extension-ring form of the contraction theorem of OpenAI. The deterministic counting theorem of Claude (Anthropic) yields an ENP language requiring Ω(m√log m/loglog m) gates over unbounded AND/OR with binary XOR, infinitely often. Using that contraction theorem, a bounded binary base with a linear budget of arbitrary XOR and symmetric gates requires Ω(m√log m) gates infinitely often. These results extend the binary-gate lower bound of OpenAI. Signed long-clause selectors reduce the first counting problem to the difference of two ordinary 3-CNF counts. Affine characters and unit-separated interpolation give the second counting interface, with logarithmic cost per symmetric-gate arity. A quantitative local-reduction transfer accounts for auxiliary variables, branch counts, and integer recovery.