The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions
arXiv:2609.01399
2026
Dynamics
2 ideas extracted · analyzed Sep 2, 2026
What the math gives to ML
The paper provides a constructive symbolic-dynamics mechanism: a quadratic rotation-symmetric Boolean rule induces a signed binary de Bruijn transfer matrix, and Fourier transformation in an auxiliary parity coordinate block-diagonalizes the finite-type shift into unsigned and signed components. Its dynamical zeta function is exactly the reciprocal characteristic determinant, so periodic-orbit counts are encoded by traces of transfer-matrix powers and the eigenvalues determine recurrence behavior. This can transfer to neural sequence models as a finite-memory parity/automaton module and as a spectral-zeta monitor or regularizer for recurrent dynamics. The transfer is most credible for RNNs, state-space models, and discrete latent sequence architectures rather than generic feedforward networks.
Ideas from this paper
Unverified
2026
Use the determinant and trace-power identities of the rules matrix as a spectral diagnostic for recurrent or state-space training. Penalize unstable or excessively resonant modes through a truncated log-zeta objective, while retaining selected eigenvalues near the unit circle when long memory is desired. This gives a falsifiable transition criterion based on closed-walk growth rather than only gradient norms.
Useful6/10
Difficulty5/10
Novelty6/10
Unverified
2026
Augment an RNN or state-space model with a finite-state binary-context module whose transitions are those of a de Bruijn graph, while a signed transition channel records a quadratic parity function of the recent context. The exact finite-memory branch preserves cancellation-sensitive parity features that a continuous hidden state may forget, and a learned readout can combine it with the ordinary neural state.
Useful6/10
Difficulty5/10
Novelty7/10