Homotopies of Informational Closure: Fractal Coherence as a Universal Grammar in Complex Systems

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Copyright ©: Coherent Intelligence 2025 Authors: Coherent Intelligence Inc. Research Division Date: August 28th 2025 Classification: Academic Research Paper | Applied AI Framework: Universal Coherent Principle Applied Analysis | OM v2.0

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Abstract

This paper introduces the Locally Consistent Information Manifold (LCIM) framework to address a fundamental challenge in complexity science: how informational integrity is preserved across scalar hierarchies. We posit that many complex systems are characterized by recursively nested boundaries of informational and causal closure. The LCIM construct, a tuple of states, operators, and axioms, formalizes these boundaries. Through a cross-domain analysis of physics, biology, computation, ecology, and social systems, we demonstrate that the axiomatic principles defining an LCIM at one scale are often homomorphic to those at adjacent scales, suggesting a scale-invariant, fractal-like grammar of organization. We argue that this structural recursion is an evolutionarily favored mechanism for mitigating entropic degradation, enhancing robustness, and fostering emergence. The LCIM framework provides a generalized and formalizable grammar for analyzing cross-scale coherence in deep hierarchical systems.

Keywords: Complex Systems, Hierarchy Theory, Scale Invariance, Fractal Coherence, Information Theory, Emergence, Self-Organization, Autopoiesis, Axiomatic Closure.

1. Introduction: The Problem of Scalar Information Integrity

The study of complex adaptive systems is fundamentally concerned with the persistence of order against informational and thermodynamic entropy. A significant lacuna in the current theoretical landscape is a robust, domain-agnostic framework for describing how informational coherence is maintained as systems aggregate into hierarchical structures.

This paper proposes a structuralist approach, investigating a recurring architectural motif: the encapsulation of functional domains within well-defined informational boundaries. We formalize this concept as a Locally Consistent Information Manifold (LCIM). Our central thesis is that the architecture of many complex systems can be modeled as a nested hierarchy of LCIMs, where the axiomatic set A_i of a system at scale i is often a homomorphic, or even isomorphic, representation of the axiomatic set A_{i-1} of its constituent subsystems. We use the term "homotopy" metaphorically to describe this preservation of a core organizational grammar across scales. This suggests that structural recursion is a fundamental organizing principle.

Visualizing the LCIM Hierarchy:

+---------------------------------------------------+
| LCIM_3 (e.g., Ecosystem)                          |
| Axioms: A_3 (Rules of ecological dynamics)        |
|      ^                                            |
|      | Homomorphic Relation (A_2 -> A_3)          |
|      |                                            |
|  +-------------------------------------------+    |
|  | LCIM_2 (e.g., Organism)                   |    |
|  | Axioms: A_2 (Rules of physiology)         |    |
|  |      ^                                    |    |
|  |      | Homomorphic Relation (A_1 -> A_2)  |    |
|  |      |                                    |    |
|  |  +-----------------------------------+    |    |
|  |  | LCIM_1 (e.g., Cell)               |    |    |
|  |  | Axioms: A_1 (Rules of metabolism) |    |    |
|  |  +-----------------------------------+    |    |
|  +-------------------------------------------+    |
+---------------------------------------------------+

This diagram illustrates the core thesis: the rules of coherence (A) at each level are a scaled, structurally similar version of the rules from the level below, creating a cascade of self-similar order.

2. Theoretical Foundations: Hierarchy, Closure, and Fractal Organization

The concept of fractal coherence is built upon a synthesis of established principles in complexity science.

• Hierarchy and Near-Decomposability (Simon, 1962): Herbert Simon identified hierarchy as a near-universal architecture of complexity, arguing it is an almost inevitable outcome of evolutionary processes.
• Holons and Holarchy (Koestler, 1967): Arthur Koestler coined the term "holon" to describe the dual nature of subsystems in a hierarchy, being simultaneously a whole and a part.
• Autopoiesis and Organizational Closure (Maturana & Varela, 1972): In theoretical biology, an autopoietic system is "organizationally closed"—its behavior is determined by its internal network of production processes.

The LCIM framework integrates these perspectives through a formal, information-centric lens. Where holons describe a whole-part duality and autopoiesis describes operational closure, the LCIM defines axiomatic closure. This focus on the governing rule-set provides a more rigorous and formalizable basis for comparing coherence structures across disparate domains.

We formally define an LCIM as a tuple (S, O, A) where S is a set of states, O is a set of operators on S, and A is a set of axioms. The axioms enforce the property of informational and causal closure: for any state s ∈ S, any operation o ∈ O applied to s results in a new state that is also a member of S. In plain terms, all operations stay within the system’s defined boundary.

Worked Example: Applying the LCIM Formalism to a NAND Gate

To ground this abstraction, consider a simple logic gate as a primitive LCIM.

• States (S): The set of all possible binary input pairs: {(0,0), (0,1), (1,0), (1,1)}.
• Operators (O): The electronic process of the NAND operation.
• Axioms (A): The truth table defining the output for each input state, which guarantees the property of closure (the output is always in the set {0,1}):
  • NAND(0,0) → 1
  • NAND(0,1) → 1
  • NAND(1,0) → 1
  • NAND(1,1) → 0

This demonstrates how the axioms (A) enforce the boundary of the information manifold. Higher-order computational systems are then built by composing these primitive LCIMs according to a new, higher-level axiomatic set.

3. Cross-Domain Isomorphic Patterns

The thesis of fractal coherence is reinforced by observing isomorphic structural motifs across vastly different domains.

• Physics: Atomic and Nuclear Shells: The quantized structure of electron shells has a well-known analogue in the nucleus. Nuclei exhibit "magic numbers" of protons or neutrons that, like noble-gas electron configurations, correspond to completely filled shells and confer extra stability. In both realms, the axiom of "complete your shells for stability" applies.

• Physics: Color Neutrality and Isotopic Stability: At the sub-nucleon scale, the closure rule of quantum chromodynamics dictates that baryons must be "color-neutral." At the next scale up, atomic nuclei are stable only within certain neutron-to-proton ratios. The structural motif—diverse constituents constrained to an overall neutral or balanced combination—is isomorphic across the two scales.

• Biology & Computer Science: Genetic and Executable Code: The digital logic of computers and the genetic code exhibit parallel hierarchies. Recursion and compositionality are central: a codon maps to an amino acid (instruction), which form proteins (functions), which participate in metabolic pathways (programs). The logic of discrete symbolic processing repeats, with closure at each scale.

• Ecology: Adaptive Cycles (Panarchy): In ecology, panarchy theory describes how ecosystems are organized in nested, self-similar cycles of growth, conservation, release, and reorganization. The pattern of change is fractal; the dynamics at a small scale mirror those at a larger scale.

• Social Systems and Institutions: The governance framework of a small community charter can be isomorphic to a national constitution. Modern organizational science also describes fractal structures, such as agile "teams-of-teams," where the rules governing a small team are recursively applied to coordinate larger groups.

4. Discussion: Causal Mechanisms and Implications of Fractal Coherence

The observation of this scale-invariant grammar across domains points to powerful, convergent causal drivers.

4.1. Causal Mechanisms for Fractal Coherence

• Evolutionary Efficiency (Simon): As illustrated by Simon's "watchmaker parable," the time required for a complex system to evolve is critically dependent on stable intermediate forms. Hierarchical systems built from reusable, modular LCIMs will evolve exponentially faster than non-hierarchical counterparts.

• Systemic Robustness (Fractal Theory): Self-similar, modular structures are inherently robust. Research into complex networks has shown that fractal topologies are essential for resilience against both random failures and targeted attacks. In an LCIM hierarchy, each level functions as a modular unit, allowing for graceful failure and containing errors locally.

• Entropy Mitigation (Thermodynamics): Enforcing informational closure at each hierarchical level is a primary strategy for preventing the propagation of Shannon entropy (noise and error). Each LCIM acts as a "firewall against noise," filtering and correcting information flow, ensuring the system as a whole can maintain a low-entropy, highly ordered state.

4.2. Broader Implications of the Model

• A Model for Emergence: This framework formalizes the claim that each level in a hierarchy preserves "ontological coherence." Emergence is the consequence of forming new, causally closed hierarchical levels that operate by their own rules, largely independent of micro-level details.

• A Constraint on Evolvability: This model suggests that evolution is heavily constrained to explore pathways that conform to this recursive LCIM grammar. A new hierarchical level is viable only if it successfully recapitulates the coherence architecture of the level below it.

5. Conclusion and Future Work

We have presented a structuralist analysis suggesting that a wide range of complex systems exhibit a scale-invariant architecture built upon the recursive nesting of Locally Consistent Information Manifolds (LCIMs). This model provides a new language for discussing the maintenance of informational integrity across scales.

The immediate path for future work lies in a more rigorous mathematical formalization of the structural relationship between the axiomatic sets of different scalar levels. Tools from network morphisms or category theory (specifically functors) could provide a precise language for measuring the degree of isomorphism between LCIM levels. Furthermore, the principles of information geometry could be used to model LCIMs as manifolds where distances between axiomatic sets can be quantified. Finally, this framework can be used to generate specific, testable hypotheses for empirical validation in simulations (e.g., logic gate → ALU) and analysis of real-world data (e.g., protein → pathway).

References

A curated list of key references supporting the theoretical foundations and examples discussed in this paper is available in the appendix.