Spectral Factorization and the Emergence of qdRIS Computing: From Integer Decomposition to Reconfigurable Impedance Processors

We revisit the classical problem of integer factorization through the qdRIS framework, a geometric-spectral formalism based on the log-metric time coordinate and the Quaternionic Laplace Transform (QLT) built on the fundamental time quantum h0. In this representation, factorization becomes a spectral decomposition problem in phase space, where prime factors appear as resonant modes. We present the qdRIS method, illus- trateitsoperationonseveralexamples,andshowthatalthoughconceptuallypowerful,itsim- plementation on classical digital hardware remains inefficient due to the inherently sequential nature of current architectures. This observation motivates the introduction of a new hard- ware paradigm : the Reconfigurable Impedance Processor (RIP), an analog high–resolution computing device based on wave propagation and interference. We describe its architecture, advantages, engineering challenges, and the broader implications for a new generation of post–Turing and post–quantum computing systems.