From Boltzmann Stochasticity to Hamiltonian Integrability: Emergence of Topological Crystals and Synthetic Planck Constants

February 1, 2026Open Access

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The aim is to explore the emergence of topological crystals in neural networks using Hamiltonian dynamics under extreme regularization.
Trained a spectral neural network using Hamiltonian Processing Units to learn Hamiltonian evolution on a 2D torus.
Applied extreme regularization pressure ($\lambda \approx 7.5 \times 10^{34}$) to the model.
Monitored stability and topological invariants during training.
Achieved perfect validation accuracy (1.0000) while 99.996% of parameters collapsed toward zero.
Identified significant topological invariants with Berry phases of $\pm 10.26$ rad and winding numbers $\pm 2$.
Showed marginal stability indicated by a Lyapunov exponent of $\lambda_{max} = +0.00175$.

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This research demonstrates topological protection in Hamiltonian neural networks, suggesting new ways to encode physical laws in continuous systems.

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Gris Iscomeback (2026) studied this question.

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