Growth Base

The unique positive real number b satisfying the self-reproduction condition with log b = 1. Characterizes e as the growth rate eigenvalue of the nu-iterator.

Growth Base

Summary

The unique positive real number b satisfying the self-reproduction condition with log b = 1. Characterizes e as the growth rate eigenvalue of the nu-iterator.

Statement

%
\label{def:growth-base}
A real number $b > 0$ is a \textbf{growth base}
for the primorial tower
if the stage-to-stage refinement factor
$f_k(b) := b^{1/p_{k+1}}$
satisfies the \textbf{self-reproduction condition}:
\[
    \boxed{%
    \lim_{k \to \infty}\;
    p_{k+1} \cdot \bigl(f_k(b) - 1\bigr)
    \;=\;
    \log b.}
\]
The constant~$e$ is the unique growth base
for which $\log b = 1$,
equivalently the unique $b > 0$ satisfying
\[
    \lim_{n \to \infty}\;
    n \cdot \bigl(b^{1/n} - 1\bigr)
    \;=\; 1.
\]

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 70
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch26-e-earned.tex lines 339-362

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookII.Transcendentals.EEarned
• Name: e_compound_check

Dependencies

• Canonical: II.D30, II.T23

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.