SlimeTree (v4.4): A Structural Framework for Approximate-Commutative Collapse in Inference Graphs

SlimeTree is a structural framework for reducing computational redundancy in inference graphs through approximate-commutative collapse.

█ CORE INSIGHT

Inference graphs in deep learning exhibit significant structural redundancy: repeated attention/MLP motifs, recurring Jacobian patterns, and blocks whose execution order has limited semantic impact. SlimeTree exposes this redundancy via approximate commutativity derived from local linear signatures.

SlimeTree does not accelerate kernels—it eliminates computations that need not be performed. Its gains arise from reduced computational volume, not faster operations.

█ KEY CONTRIBUTIONS

1. **Local Linear Signature**: Operator similarity measure via averaged Jacobians
   - S(a_i) = (1/K) Σ J_a(x_k), x_k ~ N(0, I_d)
   - Variance stabilizes at Var < 0.008 for K ≥ 100

2. **Similarity Graph**: Avoids transitivity failures of exact commutation
   - D(a_i, a_j) = ‖S(a_i) − S(a_j)‖_F
   - Edge iff D < τ (threshold)
   - Classes by connectivity, not equivalence

3. **Hierarchical Collapse**: Ward clustering within connected components
   - Practical O(n log n) complexity
   - h-hop restriction for scalability

4. **Quotient Graph**: Collapsed representation preserving inter-cluster dependencies

█ EXPERIMENTAL RESULTS

| Setting | Collapse | RMSE | FLOPs Reduction |
|---------|----------|------|-----------------|
| DAG (A) | 0.62 | 0.007 | 58% |
| Transformer (B) | 0.58 | 0.009 | 52% |
| RNN (C) | 0.58 | 0.011 | 54% |

Output fidelity: ε < 0.01 (L2 over 1000 inputs)

█ COMPLEXITY

- Signature computation: O(nKd²)
- Similarity graph (h-hop): O(nm̄ʰd²), nearly linear for h=1
- Clustering: O(n log n)

█ CONNECTION TO SS THEORY

This framework implements the core principle of Slime Structure (SS) Theory:

"When roles are marked, order is redundant."

Local linear signatures mark structural roles; approximate commutativity identifies where order is redundant; collapse eliminates the redundancy.

Cross-referenced with SS Theory Theorem 1 for deeper commutativity integration.

█ SPECULATIVE NOTES

- Dual-time abstraction resembles predictive coding's fast-error / slow-prediction separation
- Order-sensitive dependency failures may underlie hallucination in generative models
- Removing redundant paths may expose or eliminate such failures

█ VERSION NOTE (v4.4)

Refined τ thresholds in sensitivity analysis for better cluster stability. Framework-focused: toy code available separately at slimetree.ai.

█ RELATED WORK

Patent Pending: Japan 2025-183827 (SlimeTree data structure)
Toy Code: https://www.slimetree.ai/pre-print/17945058-slimetree-v4-4-zenodo/code/