2 added two more points

All the 12 or more approaches to geometry over the field with one element are tentatives to create such intermediate languages. But you seemed to ask more about a pre-existing area of it's own which may serve as a bridge - in this direction there are

$\bullet$ Alexandru Buium's theory of Arithmetic Differential Equations which brings the theory of differential equations into play as an intermediate language.

$\bullet$ I remember Alexandru Buium saying that there is also a theory of difference equations, different from his, but don't know more about this.

$\bullet$ Shai Haran uses probability theory as an intermediate language in his book "Mysteries of the Real Prime". It connects to the quantum theory column, but I don't know whether it's in the same way that Fraenkel Frenkel suggested.

$\bullet$ Homotopy theory might be a future candidate for a column of its own, on the one hand via motivic homotopy theory as it is getting available over more and more general base schemes with more and more general coefficients and thus moving towards arithmetic. And on the other hand possibly via ring spectra which may serve as the base deeper than the integers which is hoped to trigger the translation process one day...

$\bullet$ You write that the possession of Zeta functions is too weak to make the theory of dynamical systems an intermediate language. But there is certainly much more connecting dynamical systems with Arithmetic, as shown in Deninger's work.

$\bullet$ The works of Bost, Connes, Marcolli, Meyer, Laca and others connect Arithmetic to the Theory of Operator Algebras, and via those again to dynamical systems, quantum physics and Thermodynamics

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All the 12 or more approaches to geometry over the field with one element are tentatives to create such intermediate languages. But you seemed to ask more about a pre-existing area of it's own which may serve as a bridge - in this direction there are

$\bullet$ Alexandru Buium's theory of Arithmetic Differential Equations which brings the theory of differential equations into play as an intermediate language.

$\bullet$ I remember Alexandru Buium saying that there is also a theory of difference equations, different from his, but don't know more about this.

$\bullet$ Shai Haran uses probability theory as an intermediate language in his book "Mysteries of the Real Prime". It connects to the quantum theory column, but I don't know whether it's in the same way that Fraenkel suggested.

$\bullet$ Homotopy theory might be a future candidate for a column of its own, on the one hand via motivic homotopy theory as it is getting available over more and more general base schemes with more and more general coefficients and thus moving towards arithmetic. And on the other hand possibly via ring spectra which may serve as the base deeper than the integers which is hoped to trigger the translation process one day...