Interface Width

w(f, n) = number of primorial stages needed before computation f stabilizes at input size n. W(f) = sup_n w(f, n). Measures computational depth in the primorial tower.

Interface Width

Summary

w(f, n) = number of primorial stages needed before computation f stabilizes at input size n. W(f) = sup_n w(f, n). Measures computational depth in the primorial tower.

Statement

\label{def:interface-width}
Let $f$ be a TTM-computable function and $n \in \mathbb{N}$ an input size.
The \textbf{interface width at size $n$} is
\[
\begin{aligned}
w(f, n) \;=\; \min \bigl\{\, k \in \mathbb{N} \;\big|\;&
  \text{for all inputs $x$ of size $n$,} \\
& \text{the computation of $f(x)$ inspects only primorial levels }
  1, \ldots, k \,\bigr\}.
\end{aligned}
\]
The \textbf{interface width of $f$} is the supremum
\[
W(f) \;=\; \sup_{n \in \mathbb{N}} \, w(f, n) \;\in\; \mathbb{N} \cup \{\infty\}.
\]

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 148
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch56-interface-width-and-tau-admissibility.tex lines 69-85

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Computation.Admissibility
• Name: interface_width_check

Dependencies

• Canonical: III.D51, III.T30, III.T09

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.