Growth Escape

Tetration escapes the primorial tower: for any tower depth d, there exists c such that 2^^c mod M_d != 2^^c. Quantitative shadow of saturation — the 4th hyperoperation level produces values that outrun any finite primorial approximation.

Growth Escape

Summary

Tetration escapes the primorial tower: for any tower depth d, there exists c such that 2^^c mod M_d != 2^^c. Quantitative shadow of saturation — the 4th hyperoperation level produces values that outrun any finite primorial approximation.

Statement

%
\label{lem:growth-escape}
For any primorial depth~$d$, there exists
a tetration height~$c$ such that
\[
    {}^{\underline{c}}\underline{2} \;\bmod\; M_d
    \;\neq\;
    {}^{\underline{c}}\underline{2}.
\]
That is, the tetration value cannot be faithfully represented
within the modulus~$M_d$.

Proof / Justification

Since $M_d$ is finite for every~$d$,
and ${}^{\underline{c}}\underline{2}$ grows without bound
as $c$ increases
(it is eventually unbounded:
${}^{\underline{c}}\underline{2} > N$ for any $N$,
by induction on~$c$),
there exists~$c$ with ${}^{\underline{c}}\underline{2} > M_d$.
Then ${}^{\underline{c}}\underline{2} \bmod M_d < M_d
\leq {}^{\underline{c}}\underline{2}$, so they differ.
Concretely: ${}^{\underline{4}}\underline{2} = \underline{65536}
> \underline{2310} = M_5$.

Source Context

• Registry source: book-01.jsonl line 76
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex lines 423-435

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Denotation.GrowthEscape
• Name: Tau.Denotation.GrowthEscape.growth_escape

Dependencies

• Canonical: I.D13

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.