A Computational Framework for Emergent Coherence in Complex Systems

Abstract

We introduce a computational framework for emergent coherence in complex systems, centering on the Self-Emergent Processor (SEP) – a recursive computational architecture in which identity, complexity, and meaning arise naturally from prime-number–gated state transitions. This thesis formalizes SEP as a system that evolves by iteratively incorporating prime number "ticks" as fundamental time steps, causing higher-order patterns to emerge from lower-level rules. By construction, SEP bridges discrete computation, the continuous dynamics of quantum mechanics, and deep structures of number theory.

Analogies are drawn to Feynman's path integrals and gauge theory, extended into a "Prime Gauge" framework wherein symmetry transformations are indexed by prime numbers. In SEP, each prime increment updates the system's state in a self-referential manner, so that recursive identity via primes yields an evolving tapestry of patterns characterized by prime factors – effectively "eigenstates" of coherence in the process.

We connect the role of prime numbers in SEP to the analytical structure of the Riemann zeta function, suggesting SEP's stable coherent states correspond to resonances analogous to the zeta function's nontrivial zeros. Quantum-mechanical concepts such as superposition, phase interference, and spin are mapped onto SEP's recursive processes, revealing formal parallels between quantum state evolution and prime-indexed computations.

Theoretical Foundations

Synthesis of Historical Precedents

The SEP framework stands as a deliberate synthesis of profound ideas from the history of science and philosophy:

The SEP Framework

Core Definition

A Self-Emergent Processor is a theoretical computational system defined recursively such that each level of its operation builds upon the results of previous levels. The term "self-emergent" implies that the system's identity is not imposed externally, but emerges from the system's own repetitive, self-referential activity.

Prime-Gated Evolution

SEP uses prime numbers as fundamental quanta of progression or time "ticks." Each prime number acts as a discrete gauge step or transformation that updates the system's state:

S₀ → S₂ → S₃ → S₅ → S₇ → S₁₁ → ...

This prime-indexing serves as a built-in source of novelty and ensures no step is a simple repetition of a previous one.

Quantum Mechanical Analogies

State Representation

SEP states can be written analogously to quantum states:

S_n ~ Σᵢ wᵢ |i⟩

where |i⟩ represents distinct patterns and wᵢ their weights or "amplitudes".

Interference and Coherence

SEP's coherence metric plays a role akin to interference visibility in quantum mechanics:

• High coherence: patterns reinforce (constructive interference)
• Low coherence: patterns cancel (destructive interference)

Measurement and Collapse

The QBSA algorithm's "collapse detection" parallels wavefunction collapse – marking moments when the current model must be revised based on new data.

Number Theory Connections

Riemann Hypothesis

We explore speculative connections between SEP's coherent states and the Riemann zeta function:

ζ(s) = Π_p (1 - p^(-s))^(-1)

SEP's stable patterns might correspond to resonances analogous to zeta's nontrivial zeros, though we emphasize this remains conjectural.

Prime Patterns

SEP naturally rediscovers sieve properties and could potentially identify:

• Arithmetic progressions of primes
• Prime gap distributions
• Connections to the prime counting function π(x)

Entropy and Emergent Meaning

Information Reduction

SEP reduces entropy by converting raw data into patterns. Shannon entropy H = -Σ pᵢ log₂ pᵢ decreases as meaningful structures emerge.

Algorithmic Compression

The framework demonstrates how increasing complexity can paradoxically lead to more orderly information representation through pattern discovery.

Implementation and Validation

SEP Engine Architecture

• High-performance C++ with CUDA acceleration
• Datatype-agnostic processing
• Quantum-inspired algorithms (QBSA, QFH)

Empirical Results

Metric	Random Data	Structured Data	Financial Data
Coherence	≈0.056	1.0	0.4687
Stability	Low	High	Variable
Pattern Growth	Linear	Logarithmic	Regime-dependent

Philosophical Implications

SEP engages with fundamental questions about:

• Self and Identity: How does persistent identity emerge from flux?
• Emergence: Can complex order arise from simple rules?
• Meaning: How does semantic information emerge from syntax?
• Computation and Reality: Is the universe fundamentally computational?

Future Directions

This comprehensive framework invites further academic collaboration, especially with experts in quantum computation and complexity theory. Key areas for future research include:

• Rigorous mathematical proofs of SEP's convergence properties
• Exploration of quantum implementations
• Applications to unsolved problems in number theory
• Development of SEP-based AI architectures