Alpha: A Port-Hamiltonian Neural Architecture for Scalable Reasoning with Adaptive Complexity

We introduce Alpha, a neural architecture for scalable reasoning that replaces pairwise token attention with a geometrically structured dynamical system over a phase space. The architecture is founded on three principles: (1) reasoning as state evolution governed by a Port-Hamiltonian system with provable passivity and contraction guarantees, (2) long-context processing via a four-level hierarchical memory system with formal load-balancing through Sinkhorn routing, and (3) adaptive computation depth as an independent scaling axis controlled by an iterative cognition loop with a proposer-critic-verifier system. We prove that the Port-Hamiltonian recurrence with a quadratic Hamiltonian admits exact parallel computation via the Hillis-Steele associative prefix scan in O(n log n) operations (Theorem 5), achieving GPU efficiency competitive with linear recurrent models while preserving energy-based stability guarantees. We establish Lyapunov global asymptotic stability (Theorem 7), Lipschitz bounds on memory retrieval ensuring contraction (Theorem 9), Sinkhorn doubly-stochastic routing with load-balance guarantees (Theorem 12), and termination of the adaptive halting mechanism. A central contribution is the Adaptive Complexity Class Allocation mechanism: Alpha does not operate at a fixed computational complexity class. Instead, it estimates per-instance problem difficulty from the verifier confidence trajectory and allocates reasoning depth T = O(D(t) · log n), where D(t) is a learned difficulty score. Easy problems receive depth O(log n) (NC²), moderate problems receive O(log² n) (NC³), and hard problems receive O(log³ n) (NC⁴) or beyond. The effective complexity class is an emergent property of the halting distribution, not a hyperparameter. This contrasts fundamentally with fixed-depth transformers, which are limited to TC⁰ per forward pass. We present component-level empirical validation confirming seven core theoretical predictions on dual NVIDIA RTX 6000 GPUs: pH passivity (d = 1,536, 500 steps), parallel scan with 22–32× speedup at L ≤ 16,384, Sinkhorn load balancing within 0.04% of target, RFF retrieval with 2.2% relative error and monotonic convergence, OMD convergence vs. SGD divergence across 31 orders of magnitude, adaptive halting with ET = 11.1 vs. degenerate ET = 1.0, Hopfield working memory with 100% energy monotone decrease and exponentially precise retrieval, and adaptive depth allocation correctly mapping problem difficulty to NC complexity classes 1 through 4. The architecture is specified at a 250M-parameter reference configuration.