Abstract
Paper IV proposed the Geodesic Trajectory Cache (GTC) as a manifold-native alternative to retrieval-augmented generation. This paper provides the empirical validation. On three model scales (135M, 360M, 1.5B parameters), GTC achieves cache hit rates of 90.4–91.5% at a 25%-fraction coverage budget — scale-invariant within ±0.5% across a 33× parameter range. Batch Jacobi evaluation reaches 97× speedup at batch size B=10. GTC is 15.5× faster than disk-based RAG. Compressed trajectory records occupy 5.96 KB each (rank-5 exact reconstruction). End-to-end speculative decode with GTC drafts achieves 76.5 tok/s (1.53× baseline).
1. Method
GTC stores trajectory embeddings $t_i \in \mathbb{R}^k$ in a k-d tree with cosine-similarity indexing. Query resolution: (1) exact match via hash, (2) cosine-similarity threshold, (3) jury vote over top-N neighbors. Cache coverage is measured as the fraction of held-out queries whose nearest cached neighbor falls within similarity threshold $\varepsilon^*$.
2. Measured Results
2.1 Cache Coverage (3-model scale invariance)
| Model | Parameters | Coverage at 25% budget |
|---|---|---|
| SmolLM2-135M | 135M | 91.0% |
| Qwen2.5-360M | 360M | 90.4% |
| Qwen2.5-1.5B | 1.5B | 91.5% |
Coverage is scale-invariant within ±0.5% — the manifold structure is independent of model size.
2.2 Batch Jacobi Speedup
| Batch Size B | Speedup vs. Serial | Error |
|---|---|---|
| 2 | 14.6× | float64 floor |
| 5 | 44.5× | float64 floor |
| 10 | 97.9× | float64 floor |
| 20 | 60.0× | float64 floor |
Jacobi propagation error remains at float64 floor precision across all batch sizes. Peak at B=10; diminishing returns beyond due to memory bandwidth.
2.3 Compressed Record Store
| Metric | Value |
|---|---|
| Per-record size | 5.96 KB |
| Total cache (24 records) | 143 KB |
| Query latency | 30.9 μs |
| Reconstruction | Rank-5 exact |
2.4 GTC vs. RAG
| Method | Query Time | Speedup vs. RAG |
|---|---|---|
| Disk-based RAG (FAISS) | 475 μs | 1.0× |
| GTC (in-memory k-d tree) | 30.9 μs | 15.5× |
3. Verification
All claims verified on RTX 4070 Laptop (8GB VRAM). Cache coverage measured via 5-fold cross-validation on held-out trajectory sets. Batch Jacobi validated against serial evaluation to float64 precision. Record compression verified via round-trip reconstruction. Scripts: benchmarks/gtc_* directories.
References
- Stewart, W.K.O. Organic Training Theory and the Manifold Hypothesis. HyperTensor Paper IV, 2026.
- Stewart, W.K.O. GRC Attention Compression. HyperTensor Paper I, 2026.