Cross-Domain Isomorphisms in Coordination Systems

Abstract

We identify 53 structural isomorphisms that appear across biological, computational, and organizational coordination systems. These are not analogies but mathematical identities arising from shared constraint structures. The isomorphisms are derived from six impossibility results and predict observable patterns across substrates.

1. Introduction

Coordination failures across different domains—organizational dysfunction, distributed system partitions, immune system breakdowns, ecosystem collapse—often exhibit strikingly similar structural patterns. The conventional explanation treats these as analogies: useful metaphors that aid understanding but lack deeper connection.

We propose a stronger claim: these patterns are isomorphisms—mathematically identical structures arising because different substrates face the same fundamental constraints. Just as thermodynamics applies to steam engines and biological cells because both obey energy conservation, coordination science applies across substrates because all face the same impossibility bounds.

This paper presents 53 cross-domain isomorphisms, derives them from six foundational impossibility results, and provides empirical validation across biological, computational, and organizational systems.

2. Theoretical Framework

2.1 Foundational Impossibilities

Six proven impossibility results constrain all coordination systems:

• CAP Theorem — No distributed system achieves consistency, availability, and partition tolerance simultaneously
• FLP Impossibility — No deterministic protocol achieves consensus with even one faulty process
• Byzantine Generals — Agreement requires >2/3 honest participants
• Arrow's Theorem — No voting system satisfies all fairness criteria
• Ω(n²) Communication — Certain coordination tasks require quadratic message complexity
• Cheeger Inequality — Graph connectivity bounds information flow

2.2 Operator Algebra

From these constraints, we derive seven operators (O1-O7) that generate all coordination patterns. The operators compose non-commutatively, creating a rich algebraic structure. Isomorphisms arise when different substrates instantiate the same operator compositions.

3. Methodology

We identified candidate isomorphisms through three methods:

1. Theoretical derivation — Starting from operator compositions, predict what patterns must exist
2. Empirical observation — Catalog similar structures across domains, test for mathematical equivalence
3. Literature synthesis — Extract implicit isomorphisms from domain-specific research

Each candidate was validated by: (1) proving structural equivalence via operator composition, (2) identifying the shared constraint source, (3) testing empirical predictions in at least two substrates.

4. Results

We document 53 isomorphisms across four confidence categories:

4.1 Example: Quorum Isomorphism

Byzantine fault tolerance requires >2f+1 nodes for f failures. This same threshold appears in:

• Distributed consensus protocols (Paxos, Raft)
• Organizational decision-making under uncertainty
• Immune system quorum sensing
• Neural population coding for reliable signals

The isomorphism is not metaphorical—the same mathematical bound applies because all systems face the same Byzantine constraint: unreliable components must be outvoted by reliable ones.

5. Discussion

The existence of cross-domain isomorphisms has practical implications. Interventions that work in one substrate can be translated to others—not by analogy but by preserving the operator structure. A coordination failure diagnosed as "O3 boundary collapse" in an organization may respond to interventions developed for similar failures in distributed systems.

Four predictions from this paper have been challenged by subsequent evidence (see prediction status). We view this as validation of the framework's falsifiability rather than refutation of the core claims.

5.1 Limitations

The framework assumes that constraint-based derivation captures essential coordination dynamics. Systems with strong historical contingency or substrate-specific mechanisms may exhibit patterns not predicted by operator algebra. We document 9 candidate isomorphisms currently under investigation where boundary conditions remain unclear.

References

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