Hodge Coercivity and Global Dynamics in Two-Field Edge-Cochain Systems with MHD-Type Cancellation
arXiv:2608.19360
2026
Dynamics
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper provides a transferable stability mechanism for graph- or complex-valued neural dynamics: quadratic two-field interactions can exchange energy exactly while contributing zero to the total-energy derivative, provided they are built from a skew-symmetric anticommutator and projected onto the divergence-free cochain space. Dissipation is globally coercive exactly when the relevant Hodge-Laplacian kernel, namely the harmonic one-cochains, is trivial; this yields an absorbing ball and a compact global attractor. A practical neural implementation is a Hodge-structured latent ODE or recurrent layer with fixed Laplacian dissipation, trainable skew interaction operators, and an explicit harmonic-mode branch rather than silently assuming coercivity.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Replace an unconstrained graph-neural latent ODE or recurrent transition with two edge-cochain states whose linear drift is Hodge-Laplacian dissipation and whose quadratic coupling is generated by a skew-symmetric anticommutator. The coupling remains expressive while cancelling from the total energy, so the long-time envelope is determined by the Hodge spectral gap rather than uncontrolled nonlinear growth.
Useful8/10
Difficulty6/10
Novelty6/10
△ Mechanism confirmed, baseline not beaten
2026
Do not force Hodge dissipation onto harmonic edge modes, because these modes are precisely the obstruction to global coercivity. Split the latent state into dissipative coexact modes and a finite-dimensional harmonic branch, and use harmonic-decoupled interactions so each harmonic coordinate defines an invariant affine fibre with its own attractor.
Useful7/10
Difficulty6/10
Novelty7/10