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A Four-sector Resummation Theorem for Recursive Green-function Problems Propagation, Interaction, Holonomy, and Recursive Closure from Dyson Equations to Aharonov-Bohm Networks

Borros Arneth

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.2139/ssrn.7284361

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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.

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Common Structures Across Physical Theories: A PIHR Meta-Theory of Recursive Organization From Green Functions and Multiple Scattering to Interference, Holonomy, and Quantum Networks

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Different areas of physics often employ very different microscopic concepts, variables, equations, and experimental interpretations while nevertheless exhibiting similar forms of mathematical organization. Green-function methods, quantum scattering, relativistic bound-state equations, multiple-wave scattering, optical interference, resonant tunneling, gauge-dependent transport, and quantum networks all contain examples in which propagation between states is modified by local interactions, closed paths retain phase information, and repeated processes generate large or infinite families of histories. This work proposes PIHR as a structural meta-theory for describing such common organization. PIHR separates four roles: propagation, interaction, holonomy or closed-path monodromy, and recursive closure. The framework does not replace the underlying physical theories and does not identify systems with different microscopic meanings. Instead, it introduces an abstraction layer in which domain-specific quantities are mapped to common structural roles and compared at that level. A first-return factorization theorem is established for linear recursive systems, showing when arbitrarily many repeated histories can be represented exactly by a finite internal resolvent. Dyson resummation, Lippmann–Schwinger scattering, Bethe–Salpeter resummation, Foldy–Lax multiple scattering, Fabry–Pérot interference, quantum double-barrier tunneling, Aharonov–Bohm transport, and quantum graphs are then interpreted as distinct object-level realizations of a common meta-level organization. A three-arm multi-cycle quantum network provides a nontrivial quantitative test: the number of explicit histories grows exponentially with recursive depth while the exact representation remains three-dimensional. Classical controllability and observability criteria further determine when this finite representation is minimal. The resulting distinction between microscopic history complexity and structural realization complexity suggests that common architectures can exist across physical domains without implying equivalence of the underlying physics. PIHR is therefore proposed not as a new fundamental interaction or microscopic law, but as a testable meta-theoretical language for recurrent mathematical organization across physical theories and formalisms.

•Aug 16, 2026•0 citations
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.
Different areas of physics often employ very different microscopic concepts, variables, equations, and experimental interpretations while nevertheless exhibiting similar forms of mathematical organization. Green-function methods, quantum scattering, relativistic bound-state equations, multiple-wave scattering, optical interference, resonant tunneling, gauge-dependent transport, and quantum networks all contain examples in which propagation between states is modified by local interactions, closed paths retain phase information, and repeated processes generate large or infinite families of histories. This work proposes PIHR as a structural meta-theory for describing such common organization. PIHR separates four roles: propagation, interaction, holonomy or closed-path monodromy, and recursive closure. The framework does not replace the underlying physical theories and does not identify systems with different microscopic meanings. Instead, it introduces an abstraction layer in which domain-specific quantities are mapped to common structural roles and compared at that level. A first-return factorization theorem is established for linear recursive systems, showing when arbitrarily many repeated histories can be represented exactly by a finite internal resolvent. Dyson resummation, Lippmann–Schwinger scattering, Bethe–Salpeter resummation, Foldy–Lax multiple scattering, Fabry–Pérot interference, quantum double-barrier tunneling, Aharonov–Bohm transport, and quantum graphs are then interpreted as distinct object-level realizations of a common meta-level organization. A three-arm multi-cycle quantum network provides a nontrivial quantitative test: the number of explicit histories grows exponentially with recursive depth while the exact representation remains three-dimensional. Classical controllability and observability criteria further determine when this finite representation is minimal. The resulting distinction between microscopic history complexity and structural realization complexity suggests that common architectures can exist across physical domains without implying equivalence of the underlying physics. PIHR is therefore proposed not as a new fundamental interaction or microscopic law, but as a testable meta-theoretical language for recurrent mathematical organization across physical theories and formalisms.