Hardware-Accelerated Hypercomplex Learning via OPMT for high arbitrary dimension

High-dimensional hypercomplex neural networks utilizing Cayley-Dickson algebras have historically been throttled by computational bottlenecks, scaling quadratically $O(N^2)$ due to recursive sign lookups or dense tensor storage overhead. We use Theorem (OPMT) and which collapses the spatial coordinate geometry of arbitrary-dimension hypercomplex multiplication into native, hardware-level bitwise operations. By preserving index order instead of executing component transposition, we achieve exact $\mathcal{O}(1)$ constant-time and $\mathcal{O}(n)$ logarithmic-time sign navigation. We implement this architecture in an optimized GPU kernel using parallelized CUDA streams. Benchmarks on MNIST demonstrate that our Hypercomplex (HC) engine achieves a parameter compression of over 86$\times$ relative to standard dense fully-connected layers ($12,298$ parameters vs. $1,059,850$ parameters) while retaining $97.05\%$ accuracy. Furthermore, under extreme noise conditions, the architectural geometry of the OPMT acts as an intrinsic structural regularizer, outperforming standard Ordinary Least Squares (OLS) by an order of magnitude in mean squared error. Our software framework is open-sourced under the AGPLv3 license.