Chapter 10: τ-Idx — The Alpha-Orbit as Internal Natural Numbers

The passage from generation to denotation begins with a single observation: the alpha-orbit O_α = {α, ρ(α), ρ²(α), …} is the natural numbers. Not by analogy, not by embedding, not by importation — but as an identity. The alpha-orbit, earned from the kernel axioms and populated by the generative act, carries the exact structure of (ℕ, 0, S): a distinguished first element (α), a successor function (ρ), and the properties of injectivity, non-circularity, and induction. We name this identification τ-Idx and construct the three rank transfer maps that link the counting scaffold to the solenoidal orbits.