A Recursive Prime-Gated Framework for Emergent Coherence in Information Processing

Abstract

This thesis introduces the Self-Emergent Processor (SEP), a recursive computational framework where state transitions are gated by prime numbers, leading to the emergence of coherent patterns from simple rules. SEP formalizes a system that evolves its state S_p = U_p(S_p-1), where U_p incorporates prime p to update patterns, yielding metrics for coherence, stability, and entropy.

We demonstrate through simulations and a C++ implementation (the SEP Engine) that SEP reliably detects structure in data streams, with coherence C = (a·b)/(|a||b|) distinguishing random (C ≈ 0.056) from repetitive data (C = 1.0). Heuristic analogies are drawn to quantum mechanics (interference via pattern reinforcement) and number theory (prime recursion mimicking zeta function products), but these are clearly demarcated as inspirational rather than literal.

Empirical validations include compositionality (coherence difference <0.002 across chunk sizes) and stateful processing (patterns accumulate from 19 to 94 over iterations). Applications in financial time-series show predictive potential (e.g., coherence correlating with market regimes), with benchmarks confirming scalability (280MB processed in ~2 min on GPU). This work provides a verifiable computational tool for studying emergence, inviting collaboration in quantum complexity and number theory to refine its theoretical foundations.

Key Contributions

• Prime-recursive formalism: A novel approach using prime numbers as fundamental time steps
• Empirical PoCs: Demonstrated coherence stability and pattern detection capabilities
• Heuristic bridges: Connections to zeta function and Riemann Hypothesis
• Open-source engine: High-performance C++ implementation for research

Core Framework

State Representation

The SEP state consists of three tiers:

• Short-term memory (STM): New patterns introduced by the latest prime
• Mid-term memory (MTM): Recent patterns with some fading
• Long-term memory (LTM): Persistent patterns that have survived many updates

Update Rule

At the arrival of prime p, we transform state S_p-1 into S_p by:

1. Extract patterns with p
2. Check resonance (match prior)
3. Update state, compute metrics

Key Metrics

Empirical Results

Proof of Concept	Result	Significance
POC1: Data Agnosticism	C=1.0 for repetitive, C≈0.056 for random	Clear distinction between structured and random data
POC2: Stateful Processing	Patterns grow 19→94 over iterations	System accumulates knowledge over time
POC4: Performance	7.8MB/s baseline throughput	Real-time processing capability
POC5: Compositionality	Coherence difference <0.002	Metric stability across different chunk sizes
POC6: Financial Analysis	Coherence 0.4687 on EUR/USD	Correlates with market regimes

Applications

Future Directions

SEP formalizes prime recursion for emergence, validated by PoCs. Heuristics to quantum/number theory inspire, but limits acknowledged. Future work includes:

• Formal mathematical proofs of convergence properties
• Large-scale experiments on diverse data types
• Collaboration with complexity theorists (e.g., Scott Aaronson)
• Exploration of connections to Riemann Hypothesis