Run deterministic neural-guided MCTS on a supplied two-player tree, starting at root_state. Inputs are Python dictionaries keyed by state names:

• transitions maps each nonterminal state to its integer-action-to-child dictionary.
• legal_actions gives increasing action lists; policy_logits gives matching neural logits.
• players gives every state's current player, 1 or -1.
• value_predictions gives nonterminal neural values; terminal_values gives terminal outcomes. Both use that state's current-player perspective.

The root starts expanded; all edge visit counts and total values start at zero. Priors are softmax probabilities of the supplied logits. Run exactly simulations traversals. Selection maximizes

Here $s$ is a state, $a$ and $b$ are legal actions, $S$ is the score, $c_{\mathrm{puct}}$ is cpuct, $P$ is the prior, $N$ is the visit count, and $Q$ is the mean value from the parent player's perspective, initially zero. Exact ties choose the lowest action.

Each traversal ends at the first terminal or unexpanded state. Use its supplied value, expanding it only if nonterminal. Each traversed edge gains one visit and the leaf value signed for its parent player; its mean is accumulated value divided by visits. At the root and positive temperature,

Here $N_a$ is a root visit count, $\tau$ is temperature, and $\pi_a$ is the final policy. At zero temperature, assign probability one to the most-visited action, breaking ties by lowest action.

Return a three-item tuple of NumPy arrays in the root's legal-action order: int64 visit counts, float64 mean values, and float64 policy. Each has shape (A,), where A is the root's legal-action count.