From Infinite Path Families to Finite Resolvents: A PIHR Resummation Calculus for Recursive Physical Systems Green Functions, Multiple Scattering, Holonomy, and Minimal Realization in Quantum Networks
Borros Arneth
Recursive formulations in theoretical physics often generate very large or formally infinite families of alternative histories, even when the internal state space controlling their generation is finite. This work develops a sector-resolved resummation framework based on propagation, interaction, closed-path monodromy or holonomy, and recursive closure (PIHR). Propagation, interaction, and holonomy are treated as primitive dynamical data, whereas recursion is treated as a closure operation acting on a finite return kernel. For systems admitting a unique first-return factorization, an exact resummation theorem is established: all repeated histories are generated by powers of one return operator and summed by its resolvent whenever the history series converges. The framework is mapped onto Dyson resummation, Lippmann–Schwinger scattering, the Bethe–Salpeter equation, Foldy–Lax multiple scattering, Fabry–Pérot interference, double-barrier tunneling, and Aharonov–Bohm transport. A three-arm flux-threaded quantum network then provides a quantitative benchmark. With complete junction mixing, the number of arm-resolved histories grows exponentially with recursive depth, while the exact coherent amplitude remains encoded in a fixed three-dimensional return operator. Through eight internal returns the cumulative path count already exceeds one hundred forty-five million, although the all-orders solution still requires only a three-dimensional resolvent. Using standard linear-systems realization theory, we additionally give a controllability-and-observability criterion under which this three-dimensional representation is minimal among linear realizations of the recurrence-generating function. The resolvent identity and the individual physical models are established results; the proposed contribution is the PIHR factorization, its cross-domain dictionary, and the quantitative separation of path-growth complexity from finite realization dimension.
•Aug 16, 2026•0 citations