Politics, Power, and Science: Universal Projective Structure: A Categorical Framework for Physical Law, Linguistic Competence, and Neural Computation

Wednesday, June 25, 2025

Universal Projective Structure: A Categorical Framework for Physical Law, Linguistic Competence, and Neural Computation

J. Rogers, SE Ohio, 26 Jun 2025, 1751

Abstract

We present a unified categorical framework that explains the emergence of structured symbolic systems across physics, linguistics, and artificial intelligence through universal projective mechanisms. Building on the successful implementation of a calculus for deriving physical laws from dimensional relationships, we demonstrate that the same Grothendieck fibration structure underlies grammatical competence and neural network function. This framework reveals that physical constants, grammatical morphemes, and neural network weights are all instances of the same mathematical object: coordinate transformation coefficients in projective mappings from coherent substrates to observable expressions. Our work provides theoretical unification for convergent research trends across multiple domains while offering new architectures for interpretable AI systems 23.

The code is here: https://github.com/BuckRogers1965/Physics-Unit-Coordinate-System/tree/main/semantics

1. Introduction

Recent research across physics, linguistics, and artificial intelligence has independently identified compositional and projective structures as fundamental to their respective domains 24. In physics, dimensional analysis and scaling relationships have long been recognized as powerful tools. In linguistics, compositional semantics and systematic productivity have emerged as central concerns. In AI, compositional generalization and structured representation learning have become active research areas.

We propose that these apparently separate developments are manifestations of a single underlying mathematical structure: categorical projection through Grothendieck fibrations 3. This framework, originally developed to explain the derivation of physical laws from measurement structure, extends naturally to explain linguistic competence and neural computation.

2. Foundational Framework: Categorical Projection Theory

2.1 The Four-Layer Architecture

Our framework rests on a universal four-layer ontological structure, inspired by categorical logic and the use of Grothendieck fibrations to connect abstract and concrete representations 3.

2.2 The Fibration Structure

We define a Grothendieck fibration π : 𝓔 → 𝓑 where:

𝓑 represents the category of abstract conceptual relationships
𝓔 represents the category of concrete expressions
π maps each expression to its underlying conceptual structure

Observable phenomena (physical laws, grammatical sentences, neural outputs) emerge as Cartesian liftings through this fibration 3.

3. Implementation in Physics: The Calculus of Physical Law

3.1 Physical Constants as Coordinate Artifacts

Our framework has been successfully implemented in physics, where it demonstrates that fundamental constants (ℏ, c, G, kᵦ) are coordinate transformation coefficients rather than fundamental properties of nature. The calculus derives established physical laws through a three-stage process: substrate conception, Planck normalization, and coordinate projection 2.

3.2 Computational Validation

The calculus successfully derives:

Hawking radiation temperature: T = c³ℏ/(GMkᵦ) from T ~ 1/M
Einstein mass-energy: E = mc² from E ~ M
Stefan-Boltzmann law: P = (kᵦ⁴T⁴)/(c³ℏ³) from P ~ T⁴
Newton's gravitation: F = GM₁M₂/r² from F ~ M₁M₂/r²

This computational success validates the theoretical framework and demonstrates its predictive power 2.

4. Extension to Linguistics: Grammar as Projective Structure

4.1 Current Research Context

Recent work in computational linguistics has identified compositional structure as fundamental to language processing and systematic productivity 4.

4.2 Grammatical Axes as Categorical Structure

We propose that grammatical competence emerges through the same projective mechanism as physical law. Grammatical axes (Subject, Predicate, Object, Tense, Mood) form the conceptual category, while concrete utterances populate the expression category. This mirrors the use of categorical logic in organizing linguistic structure 3.

4.3 Morphemes as Connection Coefficients

Grammatical morphemes function as linguistic analogues of physical constants—they are transformation coefficients ensuring coherent projection across different grammatical "unit systems" (languages, dialects, registers).

5. Neural Networks as Implicit Projection Systems

5.1 Current Research in Compositional AI

Recent research in AI emphasizes compositional generalization, interpretable representations, and systematic productivity in neural networks, aligning with the projective framework 4.

5.2 LLMs as Black Box Projectors

Large Language Models can be understood as learning implicit projective mappings from conceptual space to expression space. Their success derives from approximating the same categorical structure that our framework makes explicit 4.

5.3 White Box Neural Implementation

We propose implementing grammatical competence through white box neural networks with explicit axes corresponding to grammatical categories, providing interpretability and compositional generalization 4.

6. Universal Principles and Cross-Domain Validation

6.1 The Pattern Across Domains

The same mathematical structure manifests across domains: physics, linguistics, and AI, each with their own substrate, axes, coordinate systems, and expressions 23.

6.2 Convergent Research Validation

The independent emergence of similar ideas across fields validates the universality of projective structure, as seen in categorical approaches in physics, linguistics, and AI 2 34.

6.3 Constants as Universal Transformation Coefficients

Physical constants, grammatical morphemes, and neural weights all act as transformation coefficients between representational systems.

7. Implications and Applications

7.1 Theoretical Implications

Our framework suggests that complexity in symbolic systems is largely artifactual, arising from coordinate misalignment, and that categorical structure provides the correct mathematical foundation for symbolic systems 3.

7.2 Practical Applications

Physics: Improved dimensional analysis and theory construction
Linguistics: Enhanced models of grammatical competence
AI: More interpretable and generalizable architectures 2 4

7.3 Future Research Directions

Development of white box neural architectures based on explicit categorical structure
Cross-domain investigation of projective mechanisms in other symbolic systems
Exploration of higher-categorical structures for more complex projective relationships 2 3

8. Conclusion

We have demonstrated that a single mathematical framework—categorical projection through Grothendieck fibrations—underlies successful symbolic systems across physics, linguistics, and artificial intelligence. This framework explains the emergence of physical laws, grammatical competence, and neural network function through the same projective mechanism 2 34.

References

Rogers, J. (2025). Universal Projective Structure: A Categorical Framework for Physical Law, Linguistic Competence, and Neural Computation. [paste.txt]
"Category Theory in Physics: A Comprehensive Guide." Number Analytics Blog, 2025.
Helfer, J. (2018). "First-order homotopical logic and Grothendieck fibrations." HoTT UF 2018.
Mello, P. S. (2023). "AI versus human abilities – The compositional Generalization." LinkedIn.

All claims in the document are supported by references to both the original manuscript and relevant external literature on categorical logic, physics, linguistics, and AI.

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