ROL: Recursive Observer/Language Architecture
for Cognitive Subsumption

1. Introduction

Modern AI systems increasingly depend on large language models (LLMs) accessed via API calls. This creates brittleness: API outages, cost accumulation, and latency constraints limit deployment. We propose cognitive subsumption—the progressive absorption of external AI capabilities into a local, self-contained dynamical system.

The ROL architecture achieves this through three mechanisms:

1. Oscillatory pattern storage: Concepts are represented as synchronized oscillator groups
2. Amplitude-gated dynamics: Only active patterns participate in computation
3. Embedding-based coupling: Semantic relationships from neural networks determine oscillator interactions

Unlike approaches that attempt to distill neural network weights, ROL imports only the semantic structure encoded in embedding spaces. This is both computationally tractable and semantically meaningful.

2. Background: Kuramoto Oscillators

The Kuramoto model [Kuramoto, 1984] describes coupled oscillators that synchronize through phase interactions:

where $\theta_i$ is the phase of oscillator $i$, $\omega_i$ is its natural frequency, and $K$ is the coupling strength. When $K$ exceeds a critical value, oscillators spontaneously synchronize—a phase transition analogous to pattern formation in neural systems.

The order parameter measures synchronization:

where $r = 0$ indicates incoherence and $r = 1$ indicates perfect synchronization.

3. ROL Architecture

3.1 Token Representation

Each token (concept) is represented by a group of $M$ oscillators. For token $k$:

Oscillators within a token are strongly coupled, promoting internal coherence. The token's activation state is characterized by:

3.2 Coupling Structure

The coupling matrix $K_{ij}$ encodes relationships:

• Intra-token coupling $K_{intra}$: Strong, promotes synchronization within tokens
• Inter-token coupling $K_{inter}$: Determined by learned transitions or embedding similarity

4. Amplitude Gating

The original Kuramoto model lacks amplitude dynamics. All oscillators couple equally regardless of activation state. This prevents pattern separation and winner-take-all behavior essential for language processing.

4.1 The Gating Function

We introduce a sigmoidal gating function that modulates coupling based on amplitude:

where:

• $a$ = oscillator amplitude
• $\beta$ = gating sharpness (we use $\beta = 15$)
• $a_{th}$ = activation threshold (we use $a_{th} = 0.2$)

4.2 Effective Coupling

The effective coupling between oscillators becomes:

This product ensures:

1. Bilateral gating: Both oscillators must be active ($g > 0.5$) for coupling
2. Amplitude scaling: Coupling strength proportional to activation product
3. Pattern isolation: Inactive patterns don't interfere with active computation

4.3 Gated Phase Dynamics

The complete phase dynamics with gating:

where $\eta_i(t)$ is a small noise term ($\sigma = 0.015$) preventing metastable trapping.

5. Amplitude Propagation

Phase dynamics alone cannot produce sequential activation. Activating token A cannot cause token B to become active without explicit amplitude propagation.

5.1 Amplitude Dynamics

We introduce coupled amplitude dynamics:

where:

5.2 Coherence Gating

The propagation term includes source coherence $c_j$, ensuring amplitude propagates only from well-formed (synchronized) patterns:

This prevents noise from propagating through the transition graph.

5.3 Competition

Winner-take-all dynamics through soft competition:

where $a_{max} = \max_k a_k$ and $\alpha = 0.3$. The strongest token suppresses others, but not so aggressively as to prevent transition propagation.

6. Cognitive Subsumption via Embeddings

6.1 The Weight Conversion Problem

Previous approaches to cognitive subsumption proposed converting neural network weights to oscillator parameters (e.g., weights to frequencies). This is:

• Computationally intractable: Billions of weights, unclear mapping
• Semantically unclear: What does a weight mean for an oscillator?
• Architecturally dependent: Different networks have incompatible weight structures

6.2 The Embedding Solution

Neural networks encode semantic relationships in their embedding spaces. Two concepts with similar embeddings are semantically related. We use this directly:

For concepts with embedding vectors $\mathbf{e}_i, \mathbf{e}_j$, compute cosine similarity:

6.3 Similarity-Based Dynamics

Embedding similarity directly determines excitation vs. inhibition:

where:

• $\alpha_{exc} = 0.5$ (excitation strength)
• $\alpha_{inh} = 0.6$ (inhibition strength)
• $s_{th} = 0.5$ (similarity threshold)

6.4 Subsumption Process

The complete cognitive subsumption pipeline:

┌─────────────────┐      ┌─────────────────┐      ┌─────────────────┐
│  External API   │      │    Embedding    │      │   ROL System    │
│  (LLM/Mistral)  │─────▶│   Extraction    │─────▶│   Similarity    │
│                 │      │                 │      │    Dynamics     │
└─────────────────┘      └─────────────────┘      └─────────────────┘
         │                                                 │
         │              ┌─────────────────┐               │
         └─────────────▶│   Behavioral    │◀──────────────┘
                        │    Learning     │
                        │  (transitions)  │
                        └─────────────────┘
    

1. Extract embeddings from external model (one-time cost)
2. Compute similarity matrix $S = [s_{ij}]$
3. Run similarity dynamics for activation spread
4. Learn transitions by observing API input/output pairs
5. Predict internally when confident; fall back to API when uncertain
6. Progressive reduction: API calls decrease as internal model strengthens

7. Experimental Results

7.1 Test Suite Summary

7.2 Sequence Generation

Learning chain $0 \rightarrow 1 \rightarrow 2 \rightarrow 3$ and generating from token 0:

Result: Generated sequence [0, 1, 2, 3] matches expected.

7.3 Embedding Cluster Separation

Four semantic clusters (animals, vehicles, emotions, colors) with 5 items each:

7.4 Real Embedding Validation

Using Mistral 7B via llama-server for real embeddings:

Semantic similarity tests with real embeddings:

8. Scaling Analysis

8.1 Computational Complexity

The similarity-based dynamics have per-timestep complexity:

arising from all-pairs interaction in the activation update.

8.2 Empirical Scaling

Tested on synthetic clustered embeddings (256-dimensional):

8.3 Scaling Verification

Comparing empirical vs. theoretical $O(n^2)$:

Sub-quadratic empirical scaling at small $n$ due to Python/NumPy overhead. Approaches theoretical $O(n^2)$ at larger scales.

8.4 Memory Analysis

Practical limit: ~10,000 concepts on 16GB system with full similarity matrix.

8.5 Optimization Paths

For production scale (>10K concepts):

9. Discussion

9.1 The "Always Thinking" Property

Unlike feedforward neural networks that are static between inputs, ROL exhibits continuous dynamics:

Step 0:   alpha: 0.68, beta: 0.05
Step 50:  alpha: 0.66, beta: 0.59
Step 100: beta: 1.04, alpha: 0.51, gamma: 0.39
Step 150: beta: 1.31, gamma: 0.98, alpha: 0.32
Step 200: gamma: 1.53, beta: 1.00
Step 250: gamma: 1.81, beta: 0.93

Without any input after initial activation, the system spontaneously evolves through the learned transition chain: $\alpha \rightarrow \beta \rightarrow \gamma$. This is the always thinking behavior characteristic of biological neural systems.

9.2 Comparison to Neural Networks

9.3 Limitations

1. Scale: $O(n^2)$ limits concept count without optimization
2. Sequence length: Long sequences require careful decay tuning
3. Complex reasoning: Not yet demonstrated for multi-step inference
4. Embedding quality: System inherits biases from source embeddings

9.4 The Observer Term

The recursive observer term from the original formulation:

had minimal measurable effect in our experiments. The system synchronizes effectively with standard Kuramoto coupling. The observer term may provide benefits in regimes we have not yet explored (e.g., near-critical dynamics, very large systems).

10. Conclusion

We have presented the ROL architecture with three critical additions to the original specification:

1. Amplitude gating: $K_{ij}^{eff} = K_{ij} \cdot g(a_i) \cdot g(a_j) \cdot a_i \cdot a_j$
2. Amplitude propagation: $\frac{da_i}{dt} = -\lambda a_i + I_i + \sum_{j \rightarrow i} \gamma w_{ji} a_j c_j - C_i$
3. Embedding-based subsumption: Use similarity, not weight conversion

With these additions, ROL demonstrates:

• Pattern storage and recall via synchronized oscillator groups
• Language-like sequence generation through amplitude propagation
• Cognitive subsumption of external AI models via embedding import
• Progressive API dependency reduction (100% elimination achieved)
• Scalability to 2,000+ concepts with maintained accuracy

The path from API dependency to autonomous operation is demonstrated at toy scale. Production deployment requires optimization for scale (sparse methods, GPU acceleration) and validation on larger vocabularies.

"Cognitive subsumption is achieved not by copying weights, but by importing semantic structure. The oscillatory system absorbs the network's relationships, not its parameters."

References

1. Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. Springer.
2. Strogatz, S. H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D, 143(1-4), 1-20.
3. Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. PNAS, 79(8), 2554-2558.
4. Mikolov, T., et al. (2013). Efficient estimation of word representations in vector space. arXiv:1301.3781.
5. Vaswani, A., et al. (2017). Attention is all you need. NeurIPS.

Appendix: Implementation Parameters

Validated parameter set for reproducibility:

Implementation Files

C:\AthenaSystem\observer\
├── tuned_system.py        # Core oscillatory dynamics (4/4 tests)
├── subsumption.py         # Behavioral learning (6/6 tests)
├── embedding_v3.py        # Embedding subsumption (4/4 tests)
├── athena_integration.py  # Real embedding client
├── real_embedding_test.py # Mistral 7B validation
├── scale_test.py          # O(n²) benchmarks
├── ROL_Paper_Supplement.md  # Formal specifications
└── ROL_Implementation_Summary.md  # Development notes