Meaning
A representable physical section is a section module that has passed the additional requirements for a physical Hilbert-space realization.
GAT I requires faithful normal coefficient representations and coherent correspondence realizations; a chosen countable cofinal directed system with strictified composition; isometric comparison maps on the declared tail; compatible reference vectors; compatible local actions and Hamiltonian data; and cyclicity of the induced reference for the descended thin local action. These data justify the stated Hilbert inductive limit.
Example (illustrative)
A nested sequence C²→C³→C⁴ with isometric inclusions illustrates how an ordinary Hilbert limit can be formed. In the physical construction, one must additionally certify the coefficient actions, references, coherence, dynamics, and cyclicity.
Scope and limits
A module pseudosection alone does not supply this limit. Different choices of strictification are not canonical without a coherent comparison theorem.
Source
GAT I — Grassmannian Gluing, Spectral Trichotomy, and Conditional Sectorial Type II. GAT I, Definition 3.6.