TSO: Non-Integrable Exponential Structures in Time–Space Oscillation Theory

This work explores a novel frontier in the Time–Space Oscillation (TSO) framework, revealing how non-integrable exponential functions—long considered mathematically pathological—can acquire physical significance. By treating particles as intrinsic oscillators embedded in a multi-slice, curved time–space manifold, we show that certain internal phases fail to integrate globally yet remain locally meaningful. Remarkably, ratios of these non-integrable exponentials form discrete patterns reminiscent of prime number distributions and other algorithmic structures. Through a systematic study spanning local phase amplification, spectral decomposition, causal hierarchies, and emergent logical sequences, the paper bridges the gap between manifold-induced instabilities and emergent discrete order. Beyond theoretical physics, these findings hint at connections to number theory, computational structures, and pseudo-random sequences, suggesting that TSO may provide a unified framework where geometry, oscillatory dynamics, and algorithmic patterns coexist. This research opens new avenues for understanding how fundamental oscillatory processes can generate complex discrete structures without invoking traditional integrability assumptions. For readers new to the Time–Space Oscillation framework, a structured overview of the four foundational TSO conceptual papers is available in “Time Space Oscillations. Introduction papers” (Zenodo DOI: 10.5281/zenodo.1767154). This companion document summarises the core principles of TSO —TSO concept, Time dilation and Gravity equivalence, TSO & EM, TSO & QM—and provides direct links to the full manuscripts. It serves as the recommended starting point for understanding how the present photon model fits into the broader TSO structure. For an overview of related work and publications, visit:https://ndl1971.github.io/time-space-oscillations/