Apophatic Optimization: Subtractive Gradient Filtering and Baseline Relaxation in Non-Equilibrium Systems

We introduce a subtractive optimization paradigm grounded in the Nicht-Theorie framework[1]. Traditional gradient descent variants minimize loss by accumulating additiveparameters, frequently inducing systemic friction (W) and speculative high-entropy noise.The apophatic-engine reformulates updates via direct noise subtraction (¬X), assertionpressure reduction (PA), and continuous relaxation toward an invariant baseline (B0 = 0).We formally detail the algorithmic execution and validate convergence across canonicalsynthetic and real-world benchmarks. Empirical results demonstrate > 91% parametersparsity while maintaining classification accuracy on Fashion-MNIST, complete isolationunder Heavy-Tail Cauchy noise (∥w∥ = 0.2067 vs SGD ∥w∥ = 146.27), and machine-precisionsubstrate recovery (5.51 × 10−14) at the Kolmogorov limit. The reference implementation isavailable via PyPI [2] and maintained as an open-source benchmark suite [3].