Expedition Two • Front EMarch 12, 2026
Metaphor and Projection
Formalizing Non-Literal Meaning
Summary
This document formalizes metaphor as a mathematical operation within the gauge bundle framework.
Metaphor is defined as a projection from full meaning in G to a constrained subspace, combined with
context transport. The Polysemy-Holonomy theorem is completed with a full proof, establishing the
equivalence between polysemy and non-trivial holonomy.
Status Summary
Proven(2)
- •Meaning assignment μ: T → Γ(E) defined
- •Polysemy-Holonomy theorem (full proof)
Framework(1)
- •Metaphor as projection + transport
Key Ideas
Meaning Assignment as Section
ProvenMeaning assignment μ: T → Γ(E) maps terms to sections of the meaning bundle.
Metaphor as Projection
FrameworkMetaphor is formalized as projection from G to a constrained subspace, composed with context transport.
Polysemy-Holonomy Full Proof
ProvenComplete proof of the equivalence: polysemy ⟺ non-trivial holonomy under non-degeneracy.
Not Yet Addressed
- Quantitative measure of metaphorical distance
- Compositional metaphor (nested projections)
- Learning metaphorical mappings from data
Prerequisites
- foundations/paradigm.md
- gauge-bundles
Related Open Problems
Source Document
frontier/expedition-two/metaphor-and-projection.md