Shifted S-templates and improved lower bounds for Schur numbers

We present an extension of the template-based approach for Schur numbers developed by Rowley. This new form of template construction, which we call shifted S-templates, was discovered during a conversation with ChatGPT 5.5 Pro, then refined, verified, and extended to multiple shifted S-templates. These new templates generalize the first ones by giving more flexibility in the coloring. Using this added flexibility with the new way to color the special label cells of the template, we exhibit a template which yields the recurrence $S(k+2)\ge 10S(k)+2$, improving on the classical Abbott-Hanson recurrence $S(k+2)\ge 9S(k)+4$ for the same step. Combined with the known bounds $S(6)\ge 536$ and $S(11)\ge 203828$, this implies $S(8)\ge 5362$ and $S(13)\ge 2038282$, improving the previously listed lower bounds $5286$ and $2011290$.