Multiplicative Calculus for Hardness Detection and Branch-Aware Optimization: A Computational Framework for Detecting Phase Transitions via Non-Integrable Log-Derivatives

Multiplicative Calculus for Hardness Detection and Branch-Aware Optimization: A Computational Framework for Detecting Phase Transitions via Non-Integrable Log-Derivatives

We present a novel computational framework that characterizes algorithmic hardness through the lens of multiplicative calculus.

Unlike classical additive calculus, where derivatives measure smooth local change, multiplicative calculus monitors the log-derivative of partition functions and detects fractures.

Fractures are the points where the integral of |Z'/Z| diverges. At these fractures, standard optimization methods fail because the control parameter lives on a multi-sheeted analytic surface. We introduce a branch-aware optimizer that uses the Lambert-W function to navigate between sheets, reducing complexity from exponential to linear in the number of fractures. We demonstrate this framework on spin glass models, synthetic game trees, neural network-guided game search, and the Traveling Salesman
 

Problem, showing that fracture detection reliably identifies phase transitions and enables principled pruning of search spaces.

Github: https://github.com/sethuiyer/baha

  Keywords: multiplicative calculus, algorithmic hardness, phase transitions, optimization, Lambert-W function, fractals, computational complexity

  Version: 1.0.0

  Programming languages: Python, Lua