Geometric Acceleration

GPU Computing on Curved Space

Classical GPU algorithms treat all vertices equally. Checkerboarding assigns the same resources to every grid cell. Branch-and-bound explores speculative paths uniformly. This uniformity assumption is the parallel postulate applied to computation — and like Euclid's fifth postulate, it's false when the space has curvature.

The Davis Manifold framework treats computational problems as geometry. Every CSP, graph algorithm, and iterative solver lives on a manifold with a curvature field. High curvature = high constraint density = where the GPU should focus. The curvature tells you where to invest compute for maximum global payoff.

Γ = m·τ / (K̂max · log|S|)
The Trichotomy Parameter
Γ
Phase selector. Determines which algorithmic phase is optimal. Γ > 1 = propagation suffices. Γ < 0.3 = direct curvature search. Middle = manifold relaxation.
K̂max
Maximum curvature. The tightest constraint density in the instance. This sets the geometric complexity.
τ
Assigned structure. How much of the problem is already determined. High τ = close to solution.

This is the Davis Law C = τ/K reformulated for GPU scheduling. The trichotomy parameter gates every instance into the optimal phase combination. Classical heuristics — MRV, degree ordering, checkerboarding — emerge as degenerate cases when specific curvature weights are zeroed.

Curvature-Guided Wavefront Execution

GPU CONSTRAINT SATISFACTION

260,042 puzzles/second on 15-clue Sudoku

A three-phase GPU pipeline for finite-domain constraint satisfaction. The curvature field Kloc(v) — computed from saturation, scarcity, and coupling — directs GPU threads toward high-curvature cells where they have the greatest marginal impact.

PHASE I
Wavefront Propagation
Bitmask arc consistency + hidden-single detection. Converges in a single ballot pass per iteration. Fully parallelized.
PHASE II
Manifold Relaxation
Softmax probability distributions minimized via curvature-adaptive gradient descent on the Davis energy functional.
PHASE III
Curvature-Directed DFS
Iterative-deepening search with branch selection by V(x), value ordering by ∇K, and holonomy-based dead-branch pruning.
Solver Time (15-clue) vs. Davis GPU
Davis GPU (CUDA) 20.4 ms —
DLX (Dancing Links) 77.8 ms 3.8× slower
CP (Constraint Programming) 3,066 ms 150× slower
DFS (Backtracking) 4,195 ms 206× slower
Davis CPU (Python) 25,012 ms 1,226× slower

Batch throughput: 260,042 puzzles/second (3.85 µs per instance). 65,536-puzzle batches at 100% solve rate. Tested on RTX 5070 (Blackwell, sm_120).

Curvature-Driven PageRank

ITERATIVE SOLVERS ON MANIFOLDS

3–5× speedup on dense graphs, honest failure on sparse

Graph curvature variance determines solver acceleration. For vertex v, the Ollivier-Ricci curvature proxy κ(v) = (degout(v) − d̄) / d̄ measures deviation from graph uniformity. High variance σ²κ enables Gauss-Seidel with SOR to exploit geometric structure.

The honesty principle: sparse graphs (chains, binary trees, road networks) show σ²κ → 0 and zero algorithmic improvement. This is not a limitation — it's a predictable consequence of the field equations.

Graph Type Speedup Iterations
Clustered n=10000 3.59× 51 → 8
Social Network n=10000 2.84× 55 → 11
Clustered n=1000 4.02× 71 → 14
Chain (sparse) 1.00× 71 → 71
Binary Tree (sparse) 1.00× 111 → 111

Curvature-Aware Shortest Paths

SINGLE-SOURCE SHORTEST PATH

80% win rate against optimized Dijkstra

For vertex v, define connectivity curvature κ(v) = (deg(v) − d̄) / d̄. Negative curvature = "flat" region (below-average connectivity). Positive curvature = "curved" hub (above-average connectivity). The algorithm exploits this geometry with four specialized phases.

DAVIS-Δ
Curvature Delta-Stepping
Light/heavy edge classification informed by local curvature.
DAVIS-CORRIDOR
Geodesic Scouting
Multi-hop propagation in flat regions (deg ≤ 4).
DAVIS-DEEP
Adaptive Propagation
2–3 hop lookahead with curvature-adaptive thresholds.
DAVIS-HUB
Batch Relaxation
High-curvature vertices (deg ≥ 2d̄) processed in batches.
Graph Type Davis Wins Avg Speedup
Sparse 5/5 1.75×
Clustered 5/5 1.23×
Grid 3/5 1.24×
Road-like 2/5 1.19×
Dense 5/5 1.03×

The Non-Decoupling Theorem

WHY FLAT MODELS FAIL

Quantifying the cost of ignoring geometry

Theorem: A system with principal fiber bundle structure and connection ω can be modeled by decoupled (connection-free) components if and only if the curvature 2-form Ω vanishes identically.

When curvature is nonzero: (1) any flat model incurs irreducible error bounded below by the Yang–Mills functional ‖Ω‖²; (2) error concentrates where K is maximal; (3) when the bundle has non-trivial characteristic classes, no flat connection exists — decoupled models are categorically impossible.

The accumulated error is the holonomy debt. Berry's geometric phase is its canonical physical instance. The decoupling assumption — that components can be processed independently and combined later — is the parallel postulate. On curved spaces, parallel lines don't exist.

C = τ/K is a special case: the decoupling assumption (K → 0) forces capacity to diverge and tolerance to collapse simultaneously.

The Equations

Kloc(v) = ws·σ(v) + wr·ρ(v) + wc·κ(v)
Local Curvature
σ
Saturation. Boundary curvature — how constrained is this point?
ρ
Scarcity. Fiber dimension — how rare is this resource?
κ
Coupling. Connection rigidity — how linked to neighbors?
V(v) = (Kloc(v) + Σu∈N(v) Kloc(u)) / |D(v)|
Information Value
V(v)
Variable ordering. Which vertex to process next — the one that propagates the most information globally.
|D(v)|
Domain size. How many choices remain at this vertex.
E[γ] = λ₁∫ds + λ₂∫K(s)ds + λ₃∫‖Hol(s) − I‖ds
Davis Energy Functional
∫ds
Path length. Total computational distance.
∫K·ds
Curvature load. Total constraint density encountered.
‖Hol − I‖
Holonomy deficit. Accumulated inconsistency along the path.

Domain Agnostic

The framework applies to any problem expressible as variables with finite domains and pairwise constraints:

Patent Pending: Curvature-guided wavefront execution, three-phase GPU pipeline with trichotomy gating, information value functional for variable ordering, and holonomy-based branch pruning are subject to U.S. Provisional Patent Applications filed 2025–2026 by Bee Rosa Davis. Academic use with citation permitted. Commercial use may require license.
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