Zero-Weight Minimax Equilibrium

1. Mathematical Formulation of $Q^*(s, a)$

In an alternating two-player zero-sum game of perfect information with deterministic transitions $\mathcal{P}(s'|s, a)$, the optimal action-value function $Q^*(s, a)$ satisfies the minimax Bellman equation:

2. Complexity Bounds and Equivalence Classes on Compact Grids

On compact Go manifolds ($7\times7$ and $9\times9$), the full state space collapses under group symmetries and finite game horizons into computationally resolvable equivalence classes:

3. Dihedral Bayesian Bitboards ($D_4$)

The board is canonicalized under the 8 operations of the dihedral group $D_4$ (rotations and reflections), mapping any orientation into its unique numerical minimum:

4. Latent Machine-Unique Concept Projections ($(M - H)$ Subspace)

Following Schut, Hassabis, Kim et al. (PNAS 2025), deep reinforcement learning architectures internalize non-human topological concepts. By projecting dynamic trajectory representations $\Phi(s, a)$ onto the subspace orthogonal to textbook human heuristics ($M - H$), singular value decomposition isolates 5 principal machine concepts:

5. Score-Centered Credit Attribution (KLPO Policy Mirror Descent)

Rather than requiring a learned neural critic or multi-rollout variance reduction (GRPO $N=16$), our engine calculates centered surprise over the local viable action set:

Posterior weights are updated upon terminal resolution via exponential credit modulation:

6. Benson Settlement Invariants and Pass Gating

A diagnostic audit across 8,298 games revealed that 95.7% of $19\times19$ games suffered an artificial 104-point deficit by refusing to pass in settled territories. We enforce Benson's unconditional two-eye theorem to gate valid non-pass candidate actions:

7. Empirical Benchmarks vs. AlphaZero-KataGo

The zero-weight engine was tested across 8,400+ regulation matches against KataGo v1.18.2 (18-block deep ResNet running with Metal acceleration on Apple Silicon):

8. Architectural Comparison: Zero-Weight vs. Neural ResNet

9. Generalization to Frontier Agentic Reasoning

The resolution of $Q^*$ on compact discrete grids provides direct architectural foundations for autonomous LLM reasoning agents:

• Critic-Free Single-Rollout RL ($N=1$): Traditional reinforcement learning requires 16 full rollouts per prompt (GRPO) or a learned critic network (PPO). Score-centered token regression proves that a single trajectory ($N=1$) with auxiliary token draws recovers the exact Policy Mirror Descent gradient, slashing inference compute by $16\times$.
• Decisive Action Isolation: In multi-turn coding and tool-calling agents, boilerplate conversational text has zero centered surprise ($z_t \approx 0$). Score centering automatically concentrates gradient updates exclusively onto decisive tool commands, flags, and regex logic ($z_t \gg 0$).

10. Conclusion

We have shown that $Q^*(s, a)$ on compact Go manifolds ($7\times7$ and $9\times9$) can be resolved without floating-point neural networks. By combining $D_4$ dihedral diamond bitboards, $(M - H)$ subspace concept projections, KLPO score-centered credit attribution, and Benson settlement invariants, our zero-weight engine achieves certified victories over superhuman neural engines at sub-millisecond latency and zero compute cost.

Citation

@techreport{solidSF2026zeroQstar,
  title       = {Zero-Weight Minimax Equilibrium: Resolving Q*(s, a) on Compact Go Manifolds via Dihedral Bayesian Bitboards, Subspace Concept Projections, and Score-Centered Policy Mirror Descent},
  author      = {Norris, Andrew and solidSF Autonomous Systems Lab},
  institution = {solidSF Research},
  year        = {2026},
  month       = sep,
  number      = {SF-2026-QSTAR},
  url         = {https://katago.solidsf.com/paper}
}