Physical P=NP

The τ-Admissibility Collapse (III.T33) applies to all physically realizable Turing machines: τ-P_adm = τ-NP_adm for physical computation. P=NP is the natural state of physically instantiated computation.

Physical P=NP

Summary

The τ-Admissibility Collapse (III.T33) applies to all physically realizable Turing machines: τ-P_adm = τ-NP_adm for physical computation. P=NP is the natural state of physically instantiated computation.

Statement

\label{thm:physical-p-equals-np}
For all physically realizable computation,
$\mathrm{P} = \mathrm{NP}$.
More precisely: if $\Pi$ is an NP problem whose every physically
realizable verifier is a physical $\Elayer{2}$ agent, then
$\Pi \in \tau\text{-}P_{\mathrm{adm}}$.

Proof / Justification

Physical verifier $V$ is $\tau$-admissible
(Theorem~\ref{thm:physical-admissibility}).
The $\tau$-Admissibility Collapse
(Theorem~\ref{thm:tau-admissibility-collapse}, Ch.~58) gives
$\tau\text{-}P_{\mathrm{adm}} = \tau\text{-}NP_{\mathrm{adm}}$.
Since $\Pi \in \tau\text{-}NP_{\mathrm{adm}}$,
we have $\Pi \in \tau\text{-}P_{\mathrm{adm}}$.

Source Context

• Registry source: book-03.jsonl line 201
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch78-physical-turing-machines.tex lines 223-231

Lean / Formalization Notes

• Formalization: not_applicable
• Module: None
• Name: None

Dependencies

• Canonical: III.T51, III.T33

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.