Arithmetic Landscape Functions of a Discrete Cat Map
arXiv:2607.24857
2026
Architecture
2 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper gives an exact arithmetic localization mechanism for deterministic finite-state dynamics. For the resolvent Gα = (I − αP)^−1 of a permutation transfer operator, the diagonal entry at state x is (1 − α^kx)^−1, where kx is the period of x, so short periodic orbits receive larger self-return amplification without disorder. A transferable neural-network construction is a permutation-equivariant recurrent or graph layer whose memory gain is modulated by this Green-function diagonal. A second transfer is a cycle-statistics regularizer based on the resolvent trace, which can control short feedback loops in learned routing. The central falsifiable signatures are the exact period-dependent gain curve and the predicted resolvent-trace dependence on return statistics.
Ideas from this paper
Unverified
2026
Construct a recurrent or graph-neural layer on a finite state space with a known bijection T, such as a modular cat map, and use the diagonal resolvent gain (1 − α^kx)^−1 as a state-dependent self-return or memory coefficient. States on short periodic orbits receive larger amplification, while long-period states receive weaker amplification, producing deterministic localization without learned disorder.
Useful6/10
Difficulty5/10
Novelty8/10
Unverified
2026
Use the resolvent trace as a differentiable statistic that controls how strongly a learned routing or recurrent transition matrix returns to short cycles. Penalizing this quantity suppresses accidental short feedback loops, while matching a target trace can impose a desired memory profile in recurrent, graph, or mixture-of-experts architectures.
Useful5/10
Difficulty6/10
Novelty7/10