Topological Minimization of Thermodynamic Cost: Replacing Computation with Geometry in Neuromorphic and Quantum Architectures

The computational efficiency of modern processors is limited not by transistor size, but by the physical cost of data movement and switching activity (Dynamic Power). We propose that energy inefficiencies in AI and Quantum computing are symptoms of suboptimal topology. By organizing computational structures according to the Teixido-Boreal graph-theoretic framework, we demonstrate that geometric optimization can serve as a direct substitute for energy expenditure. We present three physical proofs of this "Geometry-for-Energy" exchange: Interconnect Physics: Manhattan-geometry simulations of a 512-neuron core demonstrate a 164.9x reduction in total wire length, directly reducing capacitive load and resistive heat. Logic Density: Migration to the Tropical (max, +) semiring reduces logic gate switching activity by 52.3x, minimizing dynamic power dissipation. Memory Dynamics: A 592x topological compression of model weights allows for SRAM residency, eliminating the 100x energy penalty of DRAM data retrieval. Commercial Licensing:This preprint establishes prior art for the Topological Analytical Homeostasis (TAH) paradigm. Proprietary hardware specifications, RTL blueprints (Verilog/VHDL), and commercial licenses for the Teixido-ISA are available. Contact: johnvteixido@gmail.com