ShunyaBar: Differentiable Combinatorial Optimization using Arithmetic Symmetry Breaking ShunyaBar is a dynamical optimization framework grounded in non-commutative geometry and quantum statistical mechanics. The system is formalized as a spectral triple (A, H, D) encoding the arithmetic and geometric structure of the SAT phase space. The associated partition function factorizes over the adèlic ring Aₐ as: Z () = () Tr (e^- L) where () is the Riemann zeta function and L is the constraint graph Laplacian. Core Mechanism We prove that the corresponding Kubo–Martin–Schwinger (KMS) states undergo a phase transition at inverse temperature c = 1, exhibiting full one-step Replica Symmetry Breaking (1-RSB). Applied to combinatorial optimization—such as random 3-SAT near the critical density 4. 26—a quasi-static Renormalization Group (RG) sweep across c = 1 produces dramatic speedups. These are bounded only by the Quantum Adiabatic Theorem, rather than by exponential search. Method Summary ShunyaBar does not perform combinatorial search. Instead, it: Continuously relaxes Boolean constraints into a global dynamical system. Destroys illegal regions of state space by making them energetically unstable. Forces a phase transition via an arithmetic singularity at = 1. Freezes into a discrete Boolean assignment once full satisfaction is achieved. Terminates immediately upon reaching 100% satisfaction (no repair phase). This approach replaces backtracking and clause learning with global consistency enforcement. Performance CDCL search is ineffective as clause learning loses meaning. Result: 100% satisfaction (0/1M violated) in ~9–10 minutes on a single H100 GPU. Case Study 2: Ramsey R (5, 5, 5) at N = 52 Problem: Construct a 3-edge-coloring of K₅₂ with no monochromatic K₅ subgraphs. Search space: 3^1326 10^633. Result: Perfect 3-coloring found in ~17 minutes. This constitutes a constructive lower bound for R (5, 5, 5). Comparison: ShunyaBar vs. NVIDIA TurboSAT Aspect NVIDIA TurboSAT ShunyaBar Core approach Gradient-guided search + CDCL Pure continuous dynamics Uses CDCL Yes (CPU side) No Repair phase Required None Handles High-k Not targeted Native Proof output CDCL certificates Boolean witness + verifier While TurboSAT offloads exploration to GPUs to accelerate classical SAT, ShunyaBar eliminates search entirely, operating in regimes where CDCL ceases to be meaningful. Verification & Reproducibility Instance Type Size Satisfaction Rate Status 129satₙ200 129-SAT N=200, M=10⁶ 100. 00% Verified pythₙ5000 Pythagorean N=5000 100. 00% Verified ramseyₙ52 Ramsey K₅₂, R (5, 5, 5) 100. 00% Verified 3sat₁00k 3-SAT N=23k 94. 90% Partial To verify these results independently: python3 verifyᵣeproducibility. py This script scans the results/ directory, regenerates instances using deterministic generators, and verifies all assignments. ShunyaBar replaces combinatorial search with arithmetic-spectral phase transitions, enforcing global consistency to produce verifiable witnesses in regimes where classical solvers fail.
Sethurathienam Iyer (Wed,) studied this question.