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Recurring Architectures in Modern Physics: Propagation, Interaction, Holonomy, and Recursive Closure Toward a Structural Theory of Recurrent Physical Processes

Borros Arneth

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.2139/ssrn.7301540

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Many physical systems generate an arbitrarily large or infinite family of histories from a finite set of elementary processes. Multiple scattering, repeated reflections, Green-function insertions, network traversals, resonant tunneling, and repeated loop windings are familiar examples. This work develops a structural description of such systems based on propagation, interaction, holonomy or closed-path monodromy, and recursive closure, collectively denoted PIHR. The central step is to replace a particular factorization of the physical equations by the complete weighted behaviour of their recurrent histories. Primitive return processes generate words in a history alphabet, and the corresponding physical amplitudes define a noncommutative weighted behaviour series. Its Hankel matrix encodes the response of every admissible past to every admissible continuation. Established realization theory then implies that finite Hankel rank is equivalent to the existence of a finite-dimensional linear realization, that the minimal realization dimension equals this rank, and that minimal realizations are unique up to similarity. These classical results are formulated here as a PIHR Master Theorem for recurrent physical architectures. A further consequence follows from the Cayley–Hamilton theorem: recursion-depth amplitudes generated by a finite minimal return kernel necessarily satisfy a finite linear recurrence. Thus an exponentially proliferating family of physical histories can possess finite architectural memory. A three-arm quantum network provides an explicit example in which the number of histories grows by a factor of nine at each return while the coherent amplitude sequence is governed by a recurrence of order at most three. Structural equivalence of minimal PIHR realizations yields a corresponding transfer principle for realization dimension, kernel spectra, recurrence laws, and boundary generating functions. PIHR is therefore proposed not as a replacement for established physical theories, but as a framework for identifying when infinite recurrent physical behaviour is generated by a finite, canonical internal architecture.

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Modern theoretical physics is formulated using several highly successful but conceptually distinct mathematical structures. Quantum mechanics is based on Hilbert spaces and linear operators, quantum field theory is formulated through Green functions and propagators, gauge theories rely on fiber bundles and holonomy, while general relativity describes gravitation through differential geometry. Although these mathematical frameworks are deeply interconnected, they are traditionally introduced as independent axiomatic constructions. In this work, we propose a unified mathematical reconstruction framework in which Green functions, Hilbert spaces, gauge transport, and effective geometry emerge from a single primitive transport architecture. The fundamental object is neither a wavefunction nor a spacetime manifold, but an operator-valued reconstruction architecture consisting of directed transport graphs, interaction vertices, closure relations, and recursive consistency maps. Green operators arise naturally as inverse transport operators, Hilbert spaces appear as completions of admissible architectural amplitudes, gauge fields emerge from holonomy of closed transport loops, and effective geometry is reconstructed as the continuum limit of transport consistency. A central Reconstruction Theorem is formulated, establishing that every admissible reconstruction architecture induces Green operators, Hilbert reconstruction spaces, holonomy observables, effective gauge connections, and continuum geometry. As an application, a discrete set of architectural quantum numbers (a,b,c,d) is introduced to characterize propagation, interaction, topology, and recursion. A phenomenological mass reconstruction formula is subsequently derived as an application rather than a fundamental assumption. The proposed framework does not replace established physical theories. Instead, it offers a common mathematical origin from which several of their fundamental structures arise naturally. The work therefore represents a structural reformulation of theoretical physics based on operator-valued transport architectures.

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Phenomenological Approaches to Physics

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Many physical systems generate an arbitrarily large or infinite family of histories from a finite set of elementary processes. Multiple scattering, repeated reflections, Green-function insertions, network traversals, resonant tunneling, and repeated loop windings are familiar examples. This work develops a structural description of such systems based on propagation, interaction, holonomy or closed-path monodromy, and recursive closure, collectively denoted PIHR. The central step is to replace a particular factorization of the physical equations by the complete weighted behaviour of their recurrent histories. Primitive return processes generate words in a history alphabet, and the corresponding physical amplitudes define a noncommutative weighted behaviour series. Its Hankel matrix encodes the response of every admissible past to every admissible continuation. Established realization theory then implies that finite Hankel rank is equivalent to the existence of a finite-dimensional linear realization, that the minimal realization dimension equals this rank, and that minimal realizations are unique up to similarity. These classical results are formulated here as a PIHR Master Theorem for recurrent physical architectures. A further consequence follows from the Cayley–Hamilton theorem: recursion-depth amplitudes generated by a finite minimal return kernel necessarily satisfy a finite linear recurrence. Thus an exponentially proliferating family of physical histories can possess finite architectural memory. A three-arm quantum network provides an explicit example in which the number of histories grows by a factor of nine at each return while the coherent amplitude sequence is governed by a recurrence of order at most three. Structural equivalence of minimal PIHR realizations yields a corresponding transfer principle for realization dimension, kernel spectra, recurrence laws, and boundary generating functions. PIHR is therefore proposed not as a replacement for established physical theories, but as a framework for identifying when infinite recurrent physical behaviour is generated by a finite, canonical internal architecture.
Modern theoretical physics is formulated using several highly successful but conceptually distinct mathematical structures. Quantum mechanics is based on Hilbert spaces and linear operators, quantum field theory is formulated through Green functions and propagators, gauge theories rely on fiber bundles and holonomy, while general relativity describes gravitation through differential geometry. Although these mathematical frameworks are deeply interconnected, they are traditionally introduced as independent axiomatic constructions. In this work, we propose a unified mathematical reconstruction framework in which Green functions, Hilbert spaces, gauge transport, and effective geometry emerge from a single primitive transport architecture. The fundamental object is neither a wavefunction nor a spacetime manifold, but an operator-valued reconstruction architecture consisting of directed transport graphs, interaction vertices, closure relations, and recursive consistency maps. Green operators arise naturally as inverse transport operators, Hilbert spaces appear as completions of admissible architectural amplitudes, gauge fields emerge from holonomy of closed transport loops, and effective geometry is reconstructed as the continuum limit of transport consistency. A central Reconstruction Theorem is formulated, establishing that every admissible reconstruction architecture induces Green operators, Hilbert reconstruction spaces, holonomy observables, effective gauge connections, and continuum geometry. As an application, a discrete set of architectural quantum numbers (a,b,c,d) is introduced to characterize propagation, interaction, topology, and recursion. A phenomenological mass reconstruction formula is subsequently derived as an application rather than a fundamental assumption. The proposed framework does not replace established physical theories. Instead, it offers a common mathematical origin from which several of their fundamental structures arise naturally. The work therefore represents a structural reformulation of theoretical physics based on operator-valued transport architectures.