A Recursive Construction Improving the Lower Bound on the Shannon Capacity of $C_7$

We give a recursive reformulation and extension of the independent set of size $134753$ in $C_7^{10}$ constructed by N. Itty, C. D. Rosin, C. Carstensen, and D. Reichman (arXiv:2607.21517v1). We prove a product lemma that combines gadgets of different dimensions while preserving the required independence conditions. Starting from the size-$367$ independent set in $C_7^5$ of S. C. Polak and A. Schrijver (Information Processing Letters 143 (2019), 37-40), the construction gives an explicitly specified independent set in $C_7^{200}$. Consequently, $$\Theta(C_7)\geq 3.2587891539086910161967650155\ldots .$$ An accompanying program verifies the finite assertions about the five-dimensional base gadget and performs the exact integer computations used in the recursion.