On 2D Inverse Problems/Papirus

We develop a dictionary between the finite circular planar graph theory (inverse boundary problems on discrete electrical network) and infinite analytic number theory. The relationship is build by interpolation, extrapolation, discretization and sampling of the boundary Dirichlet-to-Neumann map (response given by non-commutative continued fraction or a Schur complement of Laplacian) and , where s is a complex number or a vertex (intersection of two geodesics) in the medial graph. The natural numbers are realized as eigenvalues and the nodal zeros of eigenfunctions of the uniform disc. For the self-dual network the Dirichlet-to-Neumann map is the square root of boundary Laplacian operator or second derivative, which is equivalent to simple shift monodromy operator for the anti-symmetric conductivity. We conjecture the relationship between the collinear eigenvalues of the monodromy matrix (because of total positivity in alternating medial graph basis) and non-trivial zeros of zeta-function, viewed as a half trace of the DtN map or energy integral for its analytic continuation or Mellin transform of fractional part function, repeated in Euclidean algorithm encoded in continued fractions of a layered structure.