A Treatise on Physical Reasoning: Solving Combinatorial Constraints via Continuum Hamiltonian Relaxation

A Treatise on Physical Reasoning: Solving Combinatorial Constraints via Continuum Hamiltonian Relaxation

This treatise presents the mathematical, physical, and hardware architectural framework of Physical Reasoning—the autonomous resolution of continuous combinatorial constraints via Hamiltonian field relaxation instead of sequential symbolic search.

Conventional computing architectures govern physical processes through discrete symbolic abstractions, which inherently impose an irreducible latency floor, hard-switching dissipation in power electronics, and quadratic deadlocks (O(N²)) in multi-body kinetics. This work establishes a continuous operational paradigm by mapping computational constraints directly into a continuous potential landscape (V_bias) governed by a unified non-linear field relaxation dynamics:

• Continuum Hamiltonian Foundation: Variational derivation of spatial restoring forces via complex Wirtinger functional calculus, analytical 1D topological soliton solutions, and irrotational superfluid transport.
• Information-Theoretic Chaos Gating: Dynamic mobility modulation 𝒲(H) regulated by differential Shannon entropy, suppressing numerical instabilities and gain chattering while guaranteeing global asymptotic stability via the Lyapunov-LaSalle invariance principle.
• Macro-Scale Dynamics: Collective transport of 10⁴ discrete agents through geometric chokepoints via hydrodynamic nozzle acceleration, ensuring collision-free transit (d_min > r_core) and asymptotic condensation onto compact orbital attractors.
• Micro-Scale Silicon Realization: Sub-cycle state evaluation under single-cycle hardware timing closure (τ_core = 25.0 ns at 40.00 MHz) with balanced low-skew spatial parallelism (Δt_skew ≡ 0.0 ns), verified through second-order unitary Taylor discretization and Zero-Voltage Switching (ZVS) boundary state locking.

Complete functional analytic proofs in complex Sobolev spaces (H¹ ⊆ L²) and truncation error bounds for hardware discretization are included, providing a theoretical and empirical foundation for continuous physical reasoning at native hardware timescales.

Bibliographic Citation:
Hyun, J. (2026). A Treatise on Physical Reasoning: Solving Combinatorial Constraints via Continuum Hamiltonian Relaxation (1st ed.). QuantNature Research Group. DOI: 10.5281/zenodo.23276002