Orbit Rays

O_alpha, O_pi, O_gamma, O_eta: the four orbit rays unfolded by rho from each non-omega generator.

Orbit Rays

Summary

O_alpha, O_pi, O_gamma, O_eta: the four orbit rays unfolded by rho from each non-omega generator.

Statement

%
\label{def:orbit-rays}
For each generator $g \in \{\alpha, \pi, \gamma, \eta\}$,
the \textbf{orbit ray} of $g$ is
\[
    O_g := \{\rho^n(g) : n \geq 0\} = \{g, \rho(g), \rho^2(g), \ldots\}.
\]
The \textbf{beacon singleton} is $\{\omega\}$.
By $\KAxiom{6}$, the universe decomposes as
\[
    \Obj(\tau) = \{\omega\} \;\sqcup\; O_\alpha
    \;\sqcup\; O_\pi \;\sqcup\; O_\gamma \;\sqcup\; O_\eta,
\]
where $\sqcup$ denotes disjoint union
(to be proved in Chapter~\ref{ch:ontic-closure}).

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 15
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch06-generative-act.tex lines 149-165

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Orbit.Generation
• Name: Tau.Orbit.OrbitRay

Dependencies

• Canonical: I.X01, I.K3, I.K6

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.