BAHA: Branch-Aware Holonomy Annealing

What It Solves#

Simulated annealing gets stuck in local minima because it treats the solution landscape as one smooth surface. BAHA detects when the landscape fractures (thermodynamic fracture) and uses complex-plane branch enumeration to jump to entirely different solution basins.

Key Innovation#

Thermodynamic Fracture Detection: BAHA monitors

\rho(\beta) = \left| \frac{d}{d\beta} \log Z(\beta) \right|

to detect when the landscape shatters into multiple disconnected basins.

Lambert W Branch Enumeration: When fracture is detected, BAHA uses both $W_0$ (principal) and $W_{-1}$ (secondary) branches of the Lambert W function to enumerate new basins and jump selectively.

Why it works: The ln K / ln ln K scaling of the phase transition is a fingerprint of prime-weighted systems. No other weighting function produces this exact asymptotic.

Architecture#

FractureDetector
├── Monitors: β-history, log Z history
├── Computes: ρ(β) = |d/dβ log Z|
└── Flags when: ρ > threshold

LambertWOptimizer
├── Computes: K* = exp(-C·W(-1/C))
├── Enumerates: W₀ and W₋₁ branches
└── Jumps to: New basin estimated via Monte Carlo

BranchAwareOptimizer
├── Estimates: log Z via Monte Carlo
├── Scores: Each branch by fitness
└── Selects: Jump to highest-scoring basin

Results#

Benchmark	BAHA	Simulated Annealing	Improvement
Spin Glass (64 spins)	100%	6%	4169%
Graph Isomorphism (N=50)	100%	40%	60% absolute
Overall pass rate	84%	—	across 26 problem domains

Website#

Live: sethuiyer.github.io/baha

Key Files#

baha/README.md — Overview and quick start
baha/docs/README.md — Detailed documentation
shunyabar.lua/src/shunyabar.lua — Lua implementation (lines 196-500)

Connection to Core Vision#

BAHA is the phase transition engine of the Arithmetic Manifold. It detects when the landscape fractures (via $\rho(\beta)$ ) and provides the navigation mechanism (Lambert W branches) to escape. All other algorithms use BAHA or a BAHA-like detection when operating near criticality.