ShunyaBar: Spectral–Arithmetic Phase Transitions for Combinatorial Optimization

Record type: Preprint (software + paper bundle) — published v2, December 31, 2025 DOI: 10.5281/zenodo.18214172 Record URL: https://zenodo.org/records/18214172 All-versions DOI: 10.5281/zenodo.18096757 License: CC-BY-4.0 Authors / Creators: Sethurathienam Iyer (ORCID: 0009-0008-5446-2856)

Description#

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 encoding the arithmetic and geometric structure of the SAT phase space.

The associated partition function factorizes over the adèlic ring as:

Z(\beta) \;=\; \zeta(\beta) \cdot \mathrm{Tr}\!\left(e^{-\beta L}\right)

where $\zeta(\beta)$ 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 $\beta = 1$ , exhibiting full one-step Replica Symmetry Breaking (1-RSB). Applied to combinatorial optimization — such as random 3-SAT near the critical density — a quasi-static Renormalization Group (RG) sweep across $\beta = 1$ 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 $\beta = 1$ .
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.

Blog Post: theory.shunyabar.foo
Live API: navokoj.shunyabar.foo

Performance & Industrial Benchmarks#

SAT 2024 Industrial Track#

Navokoj (the implementation of ShunyaBar) achieved a 92.57% perfect solution rate on the SAT 2024 industrial benchmarks (4,199 problems), tested across three engines:

Engine	Perfect Rate	Speed	Quality	Use Case
PRO	92.57%	7.9/sec	99.92%	Mission-critical
MINI	31.37%	10.6/sec	99.55%	Balanced
NANO	3.24%	12/sec	96.41%	Real-time

Case Study 1: 129-SAT (Ultra-High-k Regime)#

Problem: $k = 129$ SAT, $\alpha = 1000\times$ over-constrained.
Challenge: Locality is destroyed; 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 $K_{52}$ with no monochromatic subgraphs.
Search space: $\approx 3^{\binom{52}{2}} \approx 3^{1326}$ .
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
`129sat_n200`	129-SAT	1,000,000 clauses	100.00%	Verified
`pyth_n5000`	Pythagorean	5,000	100.00%	Verified
`ramsey_n52`	Ramsey	$K_{52}$	100.00%	Verified
`3sat_100k`	3-SAT	100,000	94.90%	Partial

To verify these results independently:

python3 verify_reproducibility.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.

Abstract#

This record contains the paper, datasets, solver outputs, and verification artifacts accompanying ShunyaBar, a spectral–arithmetic dynamical system for combinatorial optimization.

We introduce a non-commutative spectral triple whose partition function factorizes as $\zeta(\beta) \cdot \mathrm{Tr}(e^{-\beta L})$ , exhibiting a phase transition at $\beta = 1$ . This phase transition enables global consistency enforcement without combinatorial search.

Included are fully verifiable witnesses for large-scale SAT instances (including 129-SAT with 1,000,000 clauses), Ramsey R(5,5,5) constructions, reversible pebbling benchmarks, and independent verification scripts. All claims are reproducible from the attached artifacts.

Files in the Archive#

File	Size	MD5
`arithmetic_symmentry_breaking.zip`	193.3 MB	`647faf9a05a9bd14591e6317f0862fdd`

Total: 193.3 MB across 1 file.

Citation#

BibTeX#

@article{ShunyaBarSpectralArithmetic2025,
  author    = {Iyer, Sethu},
  title     = {{ShunyaBar: Spectral--Arithmetic Phase Transitions for Combinatorial Optimization}},
  year      = {2025},
  publisher = {Zenodo},
  version   = {v2},
  doi       = {10.5281/zenodo.18214172},
  url       = {https://doi.org/10.5281/zenodo.18214172},
  license   = {CC-BY-4.0}
}

APA#

Sethurathienam Iyer. (2025). ShunyaBar: Spectral–Arithmetic Phase Transitions for Combinatorial Optimization (Version v2). Zenodo. https://doi.org/10.5281/zenodo.18214172

Versions#

Version	DOI	Date
v2	10.5281/zenodo.18214172	Dec 31, 2025
v1	10.5281/zenodo.18096758	Dec 30, 2025

Cite-all DOI (always resolves to latest): 10.5281/zenodo.18096757

Indexed In#

OpenAIRE
Zenodo (publisher)
CERN Data Centre (infrastructure)

Keywords: Boolean Satisfiability

Metadata#

Field	Value
Resource type	Preprint
Publisher	Zenodo
Language	English
Created	January 11, 2026
Modified	January 13, 2026
License	CC-BY-4.0