Boundary Ring and Scalars

Primorial inverse limits; CRT factorization; boundary ring as structured scaffold for scalars. iota_tau = 2/(pi+e) as B/C dominance mediator (earned, not imposed).

Boundary Ring and Scalars

Summary

Primorial inverse limits; CRT factorization; boundary ring as structured scaffold for scalars. iota_tau = 2/(pi+e) as B/C dominance mediator (earned, not imposed).

Statement

%
\label{def:boundary-ring}
The \textbf{profinite boundary ring}
$\hat{\mathbb{Z}}_\tau$ is the inverse limit:
\[
    \boxed{%
    \hat{\mathbb{Z}}_\tau
    := \varprojlim_{k} \mathbb{Z}/M_k
    = \left\{
    (r_1, r_2, r_3, \ldots) \in \prod_{k \geq 1} \mathbb{Z}/M_k
    : \pi_{k,l}(r_l) = r_k \text{ for all } k < l
    \right\}.}
\]
An element of $\hat{\mathbb{Z}}_\tau$ is an infinite sequence
$(r_1, r_2, r_3, \ldots)$ where $r_k \in \mathbb{Z}/M_k$,
and the sequence is \textbf{compatible}:
whenever $k < l$, reducing $r_l$ modulo $M_k$ yields $r_k$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 37
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch39-profinite-boundary-ring.tex lines 94-112

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Boundary.Ring
• Name: Tau.Boundary.BdryRing

Dependencies

• Canonical: I.D07, I.T04, I.D18, I.D28

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.