No Barrier Theorem

At E₂, no encoding gap exists between external and internal computation. TTM τ-Nativity means programs ARE τ-addresses: code = data, no translation layer. The 1st Edition’s Representation Barrier was a category error (asking E₂ question with E₀ tools).

No Barrier Theorem

Summary

At E₂, no encoding gap exists between external and internal computation. TTM τ-Nativity means programs ARE τ-addresses: code = data, no translation layer. The 1st Edition’s Representation Barrier was a category error (asking E₂ question with E₀ tools).

Statement

\label{thm:no-barrier}
Let $\Pi$ be a $\tau$-admissible NP problem with TTM verifier $V$
of interface width $k_0$
(Definition~\ref{def:tau-admissibility}).
At $\Elayer{2}$, the self-referential structure of the $\tau$-Tower
Machine (Theorem~\ref{thm:ttm-tau-nativity}) eliminates the
encoding gap:
\begin{enumerate}
\item[(i)] Programs, data, and decoders are all $\tau$-addresses
      in $\hat{\mathbb{Z}}_{\T}$. No encoding map
      $E \colon \{0,1\}^* \to \mathrm{Addr}(\tau)$ is needed.
\item[(ii)] Faithfulness, reduction-preservation, and
      witness-completeness are satisfied by the identity on
      $\hat{\mathbb{Z}}_{\T}$.
\item[(iii)] The $\tau$-Admissibility Collapse
      (Theorem~\ref{thm:tau-admissibility-collapse}) holds without
      encoding overhead:
      $\tau\text{-}P_{\mathrm{adm}} = \tau\text{-}NP_{\mathrm{adm}}$
      is a theorem about $\tau$-addresses acting on $\tau$-addresses.
\end{enumerate}

Proof / Justification

No immediate manuscript proof block was extracted in this pilot run.

Source Context

• Registry source: book-03.jsonl line 158
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch59-why-there-is-no-barrier.tex lines 140-161

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Computation.CompBiSquare
• Name: no_barrier_check

Dependencies

• Canonical: III.T33, III.D56, III.D49, III.D50, III.T30

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.