Cylindrical Network Recurrences, Schur Positivity, and Minimality

This paper studies recurrence polynomials for weighted paths in cylindrical networks. An integral potential and periodic subdivision convert arbitrary networks to a local form while preserving vertex-disjoint path families and the cycle polynomial. A transfer-matrix factorization gives explicit starting indices for exterior-power recurrences, including after zero-weight specializations and cancellations. For planar networks, a positive transverse cut produces a finite planar path matrix whose principal minors recover the elementary cycle sums. A repeated-principal-immanant identity yields coefficientwise Schur positivity, a polynomial Polya frequency sequence, and positive real roots under positive edge weights. The results extend to Schur-functor product polynomials, with exact zero-root multiplicities under nonnegative specializations. For strongly connected planar networks with independent edge variables, the paper establishes a full symmetric Galois group and irreducibility of the nonconstant exterior-product polynomials. Endpoint sequences have either the full exterior recurrence or an eventually zero tail, distinguished by a finite window test. Explicit examples identify degeneracies in strict total positivity and unqualified endpoint minimality. The deposit contains the manuscript PDF, LaTeX source, and a computational supplement. Seven exact-arithmetic Python programs check subdivision, signed endpoint path sums, recurrence cutoffs, Schur character and trace identities, degeneracies, and Schur-functor calculations. These finite checks supplement the general proofs.