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 $\mathsf{E}^{\mathsf{NP}}$ language requiring $Ω(m\sqrt{\log m}/\log\log m)$ 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 $Ω(m\sqrt{\log m})$ 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.
Assuming a Boolean tensor contraction hypothesis (Theorem D.1 of OpenAI's recent preprint refuting the Exponential Time Hypothesis), we study exact feedback vertex set counting in general graphs. For an $n$-vertex, $m$-edge simple graph, we compute the complete size distribution deterministically in $\mathrm{poly}(n+m)\,2^{O(n(\log\log n)^2/\sqrt{\log n})}$ time and polynomial working space. The algorithm combines a whole-edge forest identity, subset-index scalar networks, sparse parity filtering, and disjoint high-degree branching. A binomial inversion recovers the multiplicities introduced by exceptional-set guesses. The counting interface supports fixed vertices, minimum solutions, and unranking. Component markers yield induced-tree and induced-path counts within the same bound. Sparse Hamilton, forest, and Steiner edge-tree counting follow from the scalar encoding.