Multiplicative Calculus for Hardness Detection and Branch-Aware Optimization

Record type: Working paper (with code) — published v1, January 26, 2026 DOI: 10.5281/zenodo.18373732 Record URL: https://zenodo.org/records/18373732 All-versions DOI: 10.5281/zenodo.18373730 License: CC-BY-4.0 Authors / Creators: Sethurathienam Iyer (ORCID: 0009-0008-5446-2856)

Description#

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
Traveling Salesman Problem (TSP)

Showing that fracture detection reliably identifies phase transitions and enables principled pruning of search spaces.

GitHub: github.com/sethuiyer/baha
Live site: sethuiyer.github.io/baha

Core Contributions#

1. Multiplicative Calculus as a Diagnostic#

The log-derivative $Z'/Z$ acts as a complexity seismograph. Where additive derivatives vanish (extrema) the system is calm; where $|Z'/Z|$ diverges, the optimization landscape fractures. The integral

\mathcal{F} \;=\; \int_{\beta_a}^{\beta_b} \left| \frac{Z'(\beta)}{Z(\beta)} \right|\, d\beta

diverges precisely at phase transitions.

When the control parameter $\beta$ crosses a fracture, the system jumps to a new analytic sheet of the multi-valued Lambert W function. Standard optimizers get stuck on the original sheet. BAHA detects the sheet change and continues optimization on the new branch — recovering a linear-in-fractures complexity profile.

3. Cross-Domain Validation#

Domain	What we measure	What we show
Spin glass models	Energy fracture density	Linear scaling in number of fractures
Synthetic game trees	Branch-point count	Principle pruning reduces search tree depth
Neural-guided game search	Log-derivative divergence	Pruning enabled by fracture detection
Traveling Salesman Problem	Lambert W sheet crossings	Phase-transition-aware search beats classical heuristics

Repository#

The accompanying code is available at github.com/sethuiyer/baha.

Version: 1.0.0
Languages: Python, Lua
Type: Research paper + reference implementation

Keywords#

multiplicative calculus · algorithmic hardness · phase transitions · optimization · Lambert-W function · fractals · computational complexity

Abstract#

This working paper introduces BAHA — Branch-Aware Holonomy Annealing — a computational framework for hardness detection and branch-aware optimization grounded in multiplicative calculus. The framework treats phase transitions as analytic fractures of the partition function $Z(\beta)$ and exploits the Lambert W function to navigate multi-sheeted control surfaces. Across spin-glass benchmarks, synthetic game trees, NN-guided game search, and TSP, the framework reliably identifies phase transitions and enables principled, fracture-aware pruning of the search space. The reference implementation in Python and Lua is attached.

Files in the Archive#

File	Size	MD5
`research_paper.zip`	2.2 MB	`0e8e7c2520ffb8ad222a5b1eb9fda6ff`

Total: 2.2 MB across 1 file.

Citation#

BibTeX#

@article{MultiplicativeCalculusBAHA2026,
  author    = {Iyer, Sethu},
  title     = {{Multiplicative Calculus for Hardness Detection and Branch-Aware Optimization: A Computational Framework for Detecting Phase Transitions via Non-Integrable Log-Derivatives}},
  year      = {2026},
  publisher = {Zenodo},
  version   = {v1},
  doi       = {10.5281/zenodo.18373732},
  url       = {https://doi.org/10.5281/zenodo.18373732},
  license   = {CC-BY-4.0}
}

APA#

Sethurathienam Iyer. (2026). Multiplicative Calculus for Hardness Detection and Branch-Aware Optimization: A Computational Framework for Detecting Phase Transitions via Non-Integrable Log-Derivatives (Version v1). Zenodo. https://doi.org/10.5281/zenodo.18373732

Versions#

Version	DOI	Date
v1	10.5281/zenodo.18373732	Jan 26, 2026

Cite-all DOI (always resolves to latest): 10.5281/zenodo.18373730

Indexed In#

OpenAIRE
Zenodo (publisher)
CERN Data Centre (infrastructure)

Metadata#

Field	Value
Resource type	Working paper
Publisher	Zenodo
Language	English
Created	January 26, 2026
Modified	January 27, 2026
License	CC-BY-4.0