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A Proposed Convergence of First Principles: A Research Program for Grounding Informational Thermodynamics in the Mathematics of Domain Theory
Copyright ©: Coherent Intelligence 2025 Authors: Coherent Intelligence Inc. Research Division Date: August 1st, 2025 Classification: Programmatic Research Proposal | Interdisciplinary Synthesis Framework: Universal Coherent Principle Applied Analysis | OM v2.0
Abstract
This paper proposes a new and ambitious research program aimed at grounding the physics-inspired framework of Informational Thermodynamics (ITD) in the rigorous, established mathematics of Domain Theory. We present a well-founded hypothesis of structural isomorphism between the core concepts of these two disparate fields. We argue that the mathematical objects developed by Gunter and Jung in "Coherence and Consistency in Domains" (1990) to model programming language semantics may represent a universal blueprint for what ITD terms a SCOCIS (Single Closed Ontologically Coherent Information Space).
Specifically, we hypothesize that Gunter and Jung's distinction between "consistency" and the more powerful property of "coherence" is isomorphic to the ITD principle of selecting a high-density Domain Anchor (DA). We further propose that their proof of the "saturation" of coherent universal domains is a compelling, domain-specific validation of the Coherence Premium. This paper reframes this convergence not as a settled proof, but as a fertile hypothesis that merits a dedicated research program. We outline a multi-stage agenda to rigorously test this isomorphism, bridge the gap between Domain Theory's static structures and ITD's temporal dynamics, and establish a dual (mathematical and empirical) program for falsification. The goal of this program is to determine if ITD has found its true mathematical home, thereby elevating it from a powerful analogy to a formally grounded science.
Keywords
Informational Thermodynamics, Domain Theory, Isomorphism, Research Program, Coherence, Consistency, Domain Anchor, SCOCIS, Falsifiability, Systems Theory.
1. Introduction: A Hypothesis of Convergent Discovery
Foundational truths in science often reveal themselves through convergent discovery, where independent lines of inquiry arrive at structurally similar conclusions. This paper proposes that such a convergence may exist between the modern, physics-inspired framework of Informational Thermodynamics (ITD) and the abstract mathematics of Domain Theory.
ITD models the dynamics of order and meaning in complex systems using principles analogous to classical thermodynamics. It posits that a system's coherence (θ) is a function of its alignment with a governing Domain Anchor (DA). Domain Theory, conversely, is a field of pure mathematics developed to provide a formal foundation for the meaning of computer programs. In their 1990 paper, "Coherence and Consistency in Domains," Gunter and Jung explored the optimal axiomatic foundations for these mathematical universes.
This document moves beyond our initial assertion of a direct proof. Instead, we present the more nuanced and scientifically rigorous claim that the parallels between these fields are too profound to be coincidental. We articulate a formal Hypothesis of Structural Isomorphism and outline the programmatic research required to test it. We are not stating a conclusion; we are charting a course for a new and promising line of interdisciplinary inquiry.
2. The Core Hypothesis: A Structural Isomorphism Between ITD and Domain Theory
We propose the following hypothesis as the central focus of this research program:
Hypothesis: The mathematical structures developed within Domain Theory to define a "coherent domain" are structurally isomorphic to the architectural principles ITD posits for a stable, low-entropy SCOCIS. Gunter and Jung's work in denotational semantics may therefore represent a specific, formal instantiation of a universal set of laws governing the integrity of any information system.
To deconstruct this, we propose a mapping between the core concepts, presented here not as settled fact, but as the primary claims to be investigated:
| Proposed ITD Isomorph | Gunter & Jung (1990) Concept | Research Question to Validate Isomorphism |
|---|---|---|
| A Low-Density Domain Anchor (DA) | "Consistent Completeness" | Can "consistency" be formally modeled as a weak constraint in the ITD framework, and does it lead to systems that are functional but predictably brittle? |
| A High-Density, Structural DA | "Coherence" (Topological) | Can the topological property of "coherence" be shown to be equivalent to a "tight," high-density DA that governs the grammar of a system's interactions? |
| A SCOCIS (Information Space) | "Domain" | Can the mathematical object of a "coherent L-domain" serve as a universal, formal model for any well-architected SCOCIS, from a legal code to a biological network? |
Maximal Coherence (θ ≈ 1) | "Saturation" | Is the mathematical property of "saturation" a true isomorph of a minimally entropic state in ITD? Does this connection hold under rigorous analysis by experts in both fields? |
The most compelling piece of preliminary evidence for this hypothesis is Gunter and Jung's proof regarding saturation. If their finding—that a coherent domain is saturated while a merely consistent one is not—can be shown to be a specific instance of the universal Coherence Premium, it would provide powerful support for the entire hypothesis.
3. A Proposed Research Program: From Analogy to Formal Proof
To move this hypothesis from an "elegant analogy" to a "genuine mathematical foundation," a structured, multi-stage research program is required. We propose the following three workstreams.
Workstream 1: Testing the Scope of the Isomorphism (Generalization)
This workstream addresses the critical challenge of moving from Gunter and Jung's domain-specific result to a universal claim.
- Objective: To determine if the mathematical object of a "coherent domain" can be successfully used to model complex systems outside of denotational semantics.
- Methodology:
- Mathematical Abstraction: Collaborate with domain theorists to abstract the essential properties of a "coherent L-domain" into a more general categorical or topological structure, detached from its specific application.
- Cross-Domain Modeling: Attempt to build formal models of other complex systems using this abstracted structure. Potential test cases include:
- A biological system: Modeling a gene regulatory network as a domain of cellular states.
- A social system: Modeling a nation's constitutional law as a domain of legal states.
- An AI system: Modeling a trained neural network's internal knowledge as a domain of concepts.
- Success Criteria: Successful creation of non-trivial, predictive models of these systems that adhere to the formal axioms of a coherent domain.
- Potential Outcome: A positive result would justify the leap from domain-specific to universal. A negative result would falsify the isomorphism, constraining the ITD-Domain Theory connection to one of a useful, but superficial, analogy.
Workstream 2: Bridging Static Structure and Temporal Dynamics
This workstream addresses the tension between Domain Theory's static structures and ITD's dynamic processes.
- Objective: To formalize the relationship between the static geometry of an information space and the dynamic laws governing a system's evolution within it.
- Methodology:
- The Domain as Phase Space: Formally define the "coherent domain" as the complete phase space of a system, representing all licit states and the valid pathways between them.
- ITD as the Equation of Motion: Model the ITD equations (e.g.,
dθ/dt = ...) as the "equations of motion" that describe a system's trajectory through this phase space over time. - Connecting Geometry and Dynamics: The core research task is to demonstrate how the geometric properties of the domain (e.g., its "coherence") constrain the solutions to the dynamic ITD equations. For example, proving that a system navigating a "coherent" domain is less susceptible to chaotic divergence than one navigating a merely "consistent" one.
- Success Criteria: A unified mathematical model that uses the static domain structure as a boundary condition for solving the dynamic ITD equations.
- Potential Outcome: A successful synthesis would create a powerful new toolkit for systems analysis, combining the predictive power of dynamics with the structural rigor of geometry.
Workstream 3: A Dual-Layer Program for Falsification
This workstream establishes a clear, two-level protocol for testing the entire framework, ensuring its status as a falsifiable scientific theory.
- Objective: To define the precise conditions under which the proposed convergence could be proven false.
- Methodology:
- Level 1: Mathematical Falsification: This level targets the isomorphism itself. The hypothesis is falsified if it can be proven, through mathematical logic, that the core structures of key complex systems (e.g., biological evolution, market economies) cannot be modeled as a "coherent domain" without fatal contradiction. This task must be undertaken in collaboration with, and be subject to peer review by, experts in Domain Theory.
- Level 2: Empirical Falsification: This level targets ITD's dynamic predictions, irrespective of the success of the mathematical model. This involves conducting the real-world experiments outlined in our "Epistemological Foundations" paper:
- Measuring the decay curve of unmaintained organizations.
- Correlating investment in anti-entropic work with achieved operational coherence.
- Analyzing the asymptotic cost of quality assurance in mature systems.
- Success Criteria: The theory is considered robust if it survives rigorous attempts at falsification on both the mathematical and empirical levels.
- Potential Outcome: This program provides a clear and honest pathway for either validating or refuting the ITD framework. It replaces assertion with a commitment to rigorous, independent testing.
4. Conclusion: Charting the Path Forward
The apparent convergence between Informational Thermodynamics and Domain Theory is too significant to ignore. The initial analysis, suggesting that a 35-year-old mathematical result may be a formal proof of the Coherence Premium, is a tantalizing possibility that demands rigorous investigation.
This paper formally recasts that initial finding as a programmatic research proposal. We do not claim to have found a settled foundation for ITD in the mathematics of Gunter and Jung. Rather, we hypothesize that such a foundation may exist, and we have outlined the specific, challenging, and necessary work required to prove it.
The success of this research program would be transformative. It would anchor the physics-inspired principles of ITD in the bedrock of pure mathematics, creating a new, rigorously grounded science of coherent systems. The path forward lies in collaborative, interdisciplinary work, subjecting this ambitious hypothesis to the unforgiving but clarifying pressures of mathematical proof and empirical falsification. We invite experts from both fields to join in this exciting endeavor.