Metric, Grounding, and Polysemy
Fisher Information on Context Space
Summary
This sprint completes the Prim Hom-set enumeration for the 5 primitive algebras, proving
the Yoneda-Grounding theorem: the Unique Grounding Theorem is a concrete instantiation of the Yoneda
Lemma restricted to the subcategory Prim. The Fisher information metric on context space X is shown
to be well-defined and positive semi-definite, with strict positive-definiteness proven for a toy model.
Status Summary
Proven(4)
- •Prim Hom-set enumeration (5×5 matrix)
- •Yoneda-Grounding theorem
- •Fisher metric well-defined, positive semi-definite
- •Toy model: strictly positive-definite
Framework(1)
- •Polysemy refinement (two sub-definitions)
Key Ideas
Prim Hom-Set Enumeration
ProvenThe 5×5 matrix of Hom-sets between primitive algebras (Pt, Ln, An, Cv, En) has pairwise distinct row profiles.
Yoneda-Grounding Theorem
ProvenThe Unique Grounding Theorem is a concrete instantiation of the Yoneda Lemma restricted to the subcategory Prim.
Fisher Information Metric
ProvenThe Fisher information metric on context space X is well-defined and positive semi-definite.
Toy Model: Strictly Positive
ProvenFor the toy model (X = [0,1], 3 words, rotation connection), the Fisher metric is strictly positive-definite.
Polysemy Refinement
FrameworkDefinition split into polysemous (genuine) vs. frame-resistant expressions.
Not Yet Addressed
- Fisher metric for higher-dimensional X
- Full Riemannian geometry (Christoffel symbols, geodesics)
- Metric completion: does Fisher make X complete?
Prerequisites
- foundations/paradigm.md
- foundations/formal-foundations.md
- category-of-languages
- foundation-securing
- probability-information
Related Open Problems
Source Document
frontier/expedition-two/metric-and-grounding.md