Strictness Theorem

The enrichment ladder is strictly increasing: E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Each layer contains genuinely new structure not reducible to the previous: E₁\E₀ = bipolar Hom decomposition, E₂\E₁ = operational closure, E₃\E₂ = self-model consistency.

Strictness Theorem

Summary

The enrichment ladder is strictly increasing: E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Each layer contains genuinely new structure not reducible to the previous: E₁\E₀ = bipolar Hom decomposition, E₂\E₁ = operational closure, E₃\E₂ = self-model consistency.

Statement

%
\label{thm:strictness}
The enrichment tower is strictly nested:
\[
    E_0 \;\subsetneq\; E_1 \;\subsetneq\; E_2 \;\subsetneq\; E_3.
\]
Specifically:
\begin{enumerate}
    \item $E_1 \setminus E_0 \neq \varnothing$.
          Witness: $[A, B]$ with non-trivial bipolar decomposition
          (Proposition~\ref{prop:e1-strictness-witness}).
    \item $E_2 \setminus E_1 \neq \varnothing$.
          Witness: operationally closed proto-code (Part~VI).
    \item $E_3 \setminus E_2 \neq \varnothing$.
          Witness: self-model-consistent code (Book~VII).
\end{enumerate}
Item~(1) is proved in full.
Items~(2) and~(3) carry forward references.

Proof / Justification

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

Source Context

• Registry source: book-03.jsonl line 16
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch06-non-emptiness-and-strictness.tex lines 464-483

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Enrichment.CanonicalLadder
• Name: strictness_8_3

Dependencies

• Canonical: III.D05, III.P01

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.