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"General Game Playing (GGP) requires AI agents to reason about arbitrary games described in the Game Description Language (GDL) under strict time constraints, with effectively no opportunity for offline training. Achieving reasonable play across arbitrary games represents a key milestone toward Artificial General Intelligence (AGI). We propose Monte Carlo Vector Search (MCVS), a novel extension of Monte Carlo Tree Search (MCTS) that incorporates a modified PUCT equation with algebraic state analysis. Our key insight is that a broad class of finite games admits a computable algebraic structure captured by our proposed $abc$ model, which embeds game states as vectors in a Hilbert Space. In this vector space, the strategic value of a position is determined by its geometric proximity to known winning configurations, measured via a matrix distance (Manhattan norm) between their weighted matrices. This matrix feature is extracted in polynomial time $O(n^2)$, and a zone-guidance metric biases search toward regions with minimal matrix distance to winning states. Experiments on the benchmark games Breakthrough ($8 \\times 8$), English Draughts, and Chess show that the zone-guided variant of MCVS (without neural network) achieves competitive or superior performance to a strong UCT baseline in Chess and Breakthrough, while results in Draughts are more mixed. These findings validate the practical value of explicit algebraic geometric analysis in resource-constrained general game playing."