Preprint  ·  February 2026

Curvature-Guided
Wavefront Execution
on the Davis Manifold

GPU-accelerated constraint satisfaction guided by intrinsic Riemannian curvature. A geometry-aware solver that crushes the world's hardest Sudoku at 1,226× speedup over Python CPU — and generalizes to any finite-domain CSP.

40,128× faster than Kona 1.0 Yann LeCun’s EBM solves one puzzle in 313ms. We solve 270,000 per second.
Bee Rosa Davis
Brown University (MS) · 27 Years Aerospace
DOI: 10.5281/zenodo.18511755
Patent Pending · U.S. PPA Feb 2026
Performance Summary
GPU throughput~270K puzzles/sec
Per-instance latency3.7 µs
Speedup vs. Python CPU1,226×
Hardest-known solved11 / 11
Avg solve time (v4)7.8 ms
Scroll
1,226×
vs. Python CPU · 15-Clue Extreme Sudoku · Consumer Laptop

Computational resources should flow toward regions of highest curvature — where they have the greatest marginal impact on constraint resolution. Classical GPU approaches treat every vertex equally. The Davis manifold tells you where the hard work is, before the work begins.

01 — The Approach

Three-Phase GPU Pipeline

The solver implements a three-phase CUDA pipeline derived from the Davis Field Equations. A trichotomy parameter Γ automatically classifies each instance by its geometric complexity and routes it to the optimal phase combination — no manual algorithm selection required.

I
Wavefront Propagation
Bitmask-based arc consistency and hidden-single detection, parallelized across warps. Converges in a single ballot pass per iteration.
II
Manifold Relaxation
Softmax-parameterized probability distributions minimized via curvature-adaptive gradient descent on the Davis energy functional. Fully data-parallel.
III
Curvature-Directed DFS
Iterative-deepening depth-first search with branch selection by information value and holonomy-based dead-branch pruning.

Classical heuristics — MRV, degree heuristic, checkerboarding — all emerge as degenerate cases of the curvature-guided ordering when specific curvature weight components are set to zero. The Davis manifold framework subsumes and unifies them.

02 — The Mathematics

Intrinsic Curvature as Scheduling Signal

The constraint graph induces a discrete Riemannian manifold in the sense of Regge calculus — not a metaphor, but a genuine geometric structure with a metric tensor, connection, and holonomy group. The local curvature field measures three independent scalar invariants of the constraint fiber bundle.

Local Curvature
$$K_{\text{loc}}(v) = w_s \cdot \sigma(v) + w_r \cdot \rho(v) + w_c \cdot \kappa(v)$$
$\sigma$ = saturation (boundary curvature) · $\rho$ = scarcity (fiber dimension) · $\kappa$ = coupling norm (connection rigidity)
Information Value
$$V(v) = \frac{K_{\text{loc}}(v) + \displaystyle\sum_{u \in N(v)} K_{\text{loc}}(u)}{\left|D(v)\right|}$$
Curvature-weighted variable ordering — selects the vertex whose resolution propagates the most information globally.
Davis Energy Functional
$$E[\gamma] = \lambda_1 \int_\gamma ds + \lambda_2 \int_\gamma K_{\text{loc}}(s)\, ds + \lambda_3 \int_\gamma \left\| \mathrm{Hol}_\gamma(s) - I \right\| ds$$
Path length + curvature-weighted complexity + holonomy deficit. A geodesic of this functional resolves constraints with minimal total effort.

The trichotomy parameter $\Gamma = \dfrac{m \cdot \tau}{\hat{K}_{\max} \cdot \log |S|}$ classifies instances by the ratio of assigned structure to geometric complexity, automatically gating them into the optimal phase combination.

Solver in Action

Curvature-Guided Solve Process

Watch the solver trace a geodesic through the constraint manifold. At each step, the cell with the highest information value $V(c)$ is resolved first, and the curvature field collapses monotonically toward zero.

Curvature-guided solve animation
Curvature-Guided Cell Selection
Background color encodes $K_{\text{loc}}$. Warm colors = high curvature = solved first. The solver works boundary-first, targeting constraint bottlenecks before the unconstrained interior.
Energy functional descent
Energy Functional Descent
$E[\gamma]$ drops from 62.2 → 0. Total curvature $\Sigma K$ and entropy $\Sigma H$ collapse simultaneously. No non-monotonic excursions — the $V(c)$ ordering traces a near-geodesic path.
03 — Results

Benchmark Performance

Evaluated on 15-clue extreme Sudoku (66 empty cells, Γ ≈ 0.19) — the hardest class of well-posed puzzles. All 11 hardest-known instances solved in under 9ms on a consumer laptop GPU (NVIDIA RTX 5070, Blackwell architecture). Solver v4 averages 7.8ms per puzzle.

GPU Performance · Hardest Known (v4)
AI Escargot8.5 ms
Inkala 20108.1 ms
Golden Nugget8.5 ms
Easter Monster8.5 ms
17-clue Coloin6.4 ms
CUDA kernel time, excluding host transfer · Avg 7.8 ms across 11 puzzles
Speedup vs. Baseline Solvers
vs. Kona 1.0 EBM (LeCun)40,128×
vs. DLX (Dancing Links)3.8×
vs. Python CPU1,226×
Batch throughput (10K)278K/s
Batch throughput (65K)268K/s
Kona: 313ms avg, 96.2% accuracy (source) · Davis: 7.8ms, 100%
Head-to-Head · Single Instance
Davis GPU (v4)
7.8 ms
DLX
29.6 ms
Kona 1.0 EBM
313 ms
Python CPU
9,565 ms
Hardware
Test Configuration
GPURTX 5070 Laptop
ArchitectureBlackwell (SM 12.0)
CUDA Cores4,608
Memory8 GB GDDR7
04 — Applicability

Beyond Sudoku: Domain-Agnostic

The framework is not a Sudoku solver. It is a general-purpose GPU execution strategy for any finite-domain constraint satisfaction problem. Any CSP expressible as variables with finite domains and pairwise constraints admits a Davis manifold whose curvature field guides GPU scheduling.

SAT Solving
Clause-variable interaction graphs exhibit locality that gives the curvature field meaningful spatial structure for wavefront ordering.
Graph Coloring
Curvature identifies chromatic pressure. Register allocation in compilers reduces to graph coloring of the interference graph.
Scheduling
The curvature field identifies operations at the intersection of tight precedence chains and oversubscribed machines.
The Researcher
Bee Rosa Davis

Bee Rosa Davis

MS Digital Forensics, Brown University
27 years aerospace & security engineering
ORCID: 0009-0009-8034-4308
Independent researcher developing geometric frameworks for computation, security, and AI. Creator of the Davis manifold theory and its applications across cybersecurity, medicine, and constraint optimization. Overall Mother of the House of Diwa.
Read more →
Selected Papers
The Field Equations of Semantic Coherence — Zenodo, 2025
The Davis Manifold — Zenodo, 2025
Geometric Intelligence Series
Hidden Variable — From a childhood IQ diagnosis to NASA and algorithmic justice, a manifesto on finding what systems hide.
The Geometry of Sameness — Proves production ML and Riemannian manifolds tell the same story, yielding error guarantees that survive coordinate changes.
The Geometry of Medicine — Reimagines health as geometric coherence on the Davis Manifold — stability as a path, not the absence of disease.
The Geometry of Fuel — From the Davis Law to topological vacuum rectification, a unified framework connecting inference, physics, and energy itself.
Acknowledgment · Black History Month 2026

David Harold Blackwell

The GPU architecture targeted in this work bears the name of David H. Blackwell (1919–2010), the first Black scholar inducted into the National Academy of Sciences and a pioneer of dynamic programming, Bayesian statistics, and game theory. The sequential decision-making under uncertainty at the heart of constraint satisfaction is precisely the domain Blackwell formalized.

Open Source

Reproduce the Results

$ git clone https://github.com/nurdymuny/sudoky-energy.git
$ cd sudoky-energy
$ pip install -e .
$ python scripts/benchmark_hardest.py
View Repository → Read Paper (PDF) → Zenodo Record →
Patent Notice: The curvature-guided wavefront execution method, three-phase GPU pipeline with trichotomy gating, information value functional, and holonomy-based branch pruning are the subject of U.S. Provisional Patent Application filed February 2026 by Bee Rosa Davis.