Zeta renormalization and pressure at infinity for an infinitely cusped tree lattice

What the math gives to ML

The paper develops a constructive way to assign finite determinant-like quantities to an infinite transition system whose ordinary Euler product diverges because infinitely many short cycles recur. Its transferable asset is the combination of height damping, trace-class operators, and explicit dilogarithmic subtraction of the divergent integrated-pressure contribution. A neural analogue is a stability and expressivity diagnostic for infinitely deep or weight-tied networks: damp Jacobian transitions by layer height, compute a renormalized log-determinant, and use it as a regularizer or monitoring signal. This is most promising for recurrent, state-space, or deep-equilibrium models rather than ordinary finite transformers.