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Frontier Research

Deep mathematical investigations from the Universal Language expeditions — category theory, differential geometry, computability, and practical applications.

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Research Topics
32
Key Ideas
21
Proven Results
11
In Development

Expedition One

First systematic exploration — category theory, gauge geometry, and computational foundations.

The Category of Languages

Front B

Functorial Semantics for the Universal Language

This document establishes Lang(Σ_UL) as a well-defined category whose objects are Σ_UL-algebras

5 proven1 framework6 key ideas

The Gauge Bundle of Meaning

Front A

Differential Geometry of Context and Interpretation

This research applies differential geometry to the problem of meaning in context. The context

3 proven2 framework5 key ideas

Numbers and Computability

Front C

Arithmetic and Algorithmic Foundations

This work establishes the computational foundations of the Universal Language. Natural numbers

4 proven1 framework5 key ideas

Applications for LLMs and AI

Front D

Cognitive Infrastructure for AI Systems

This document catalogs practical applications of UL-structured artifacts for LLM systems and

5 framework5 key ideas

Expedition Two

Second systematic exploration — metrics, grounding, metaphor, and foundation securing.

Metric, Grounding, and Polysemy

Front D

Fisher Information on Context Space

This sprint completes the Prim Hom-set enumeration for the 5 primitive algebras, proving

4 proven1 framework5 key ideas

Metaphor and Projection

Front E

Formalizing Non-Literal Meaning

This document formalizes metaphor as a mathematical operation within the gauge bundle framework.

2 proven1 framework3 key ideas

Foundation Securing

Front C

Adjoint Functors and Left Adjoints

This document constructs the left adjoints to the forgetful functors in the Erlangen chain.

3 proven3 key ideas

Rigor Standards

All frontier research follows a four-label rigor system documented in the methodology:

Proven
Framework
Conjectured
Analogy
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