Recursive-algebraic solution of the closed string tachyon vacuum equation

We develop a recursive algebraic framework for solving the closed string tachyon vacuum equation, derived from the hyperbolic recursion relations of F{\i}rat and Valdes-Meller. We restrict to the sector of zero-momentum Lorentz-scalar states. Lorentz symmetry ensures that this sector is closed under the equations of motion. In this sector, we introduce a seam-graded expansion and show that the equation is entirely algebraic at every order: the unknown at each grade enters only through point evaluations at the systolic length, so each grade reduces to a matrix inversion with no Fredholm equations. The expansion is formal; convergence in the multi-level system is the subject of ongoing work. This work was conducted with a publicly available version of Claude Code (Anthropic, Claude Opus 4.6). The complete research repository, including all computations, adversarial review logs, and the full human-AI collaboration history, is publicly available at https://github.com/mk2427/csft-tachyon-vacuum.