A Scale-Invariant Geometric Threshold for Photonic Transmission: Deriving the Exact Optical Attenuation Limit via Spatial Propagation Dynamics

The global expansion of high-capacity optical networks is fundamentally bottlenecked by signal attenuation and chromatic dispersion, forcing reliance on empirically spaced Erbium-Doped Fiber Amplifiers (EDFAs). Current network architectures utilize probabilistic Bit Error Rate (BER) margins and Shannon-Hartley capacity estimates to predict signal degradation, often resulting in hardware over-expenditure and unpredictable packet loss. This paper introduces a deterministic topo-dynamical framework for photonic transmission. By modeling the coherent propagation of the optical wave packet as a continuous spatial expansion operator competing against the localized structural decay of Rayleigh scattering, we derive a scale-invariant geometric threshold (Λ∗). We mathematically demonstrate that this invariant defines the absolute physical maximum unamplified transmission distance. By integrating this continuous limit into existing Software-Defined Networking (SDN) controllers as a firmware-level active routing algorithm, we outline a generalized, parameter-free methodology for achieving maximum-efficiency, zero-packet-loss transmission in global optical networks.