Primorial Ladder

Primorial numbers Prim(k) = p₁·p₂·…·pₖ form an inverse system ℤ/Prim(1)ℤ ← ℤ/Prim(2)ℤ ← …. Inverse limit = profinite completion Ẑ_τ. Connection to Book I’s primorial presheaf (I.D83). Canonical cofinal filtration.

Primorial Ladder

Summary

Primorial numbers Prim(k) = p₁·p₂·…·pₖ form an inverse system ℤ/Prim(1)ℤ ← ℤ/Prim(2)ℤ ← …. Inverse limit = profinite completion Ẑ_τ. Connection to Book I’s primorial presheaf (I.D83). Canonical cofinal filtration.

Statement

%
\label{def:primorial-ladder}
Let $(p_n)_{n \geq 1}$ enumerate the primes
of $\tau$-Idx in order.
The $k$-th \textbf{primorial} is
\[
    M_k \;:=\; \prod_{n=1}^{k} p_n.
\]
For $k \leq \ell$, the \textbf{reduction map}
$\pi_{\ell \to k} :
\mathbb{Z}/M_\ell\mathbb{Z} \to \mathbb{Z}/M_k\mathbb{Z}$
is reduction modulo $M_k$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 38
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch28-omega-germs.tex lines 81-93

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Spectral.PrimorialLadder
• Name: PrimorialStage

Dependencies

• Canonical: I.D83

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.