Template Invariance Under Reflection

The layer template (Carrier, Predicate, Decoder, Invariant) is preserved under the Langlands₁ reflection. The four components change substance but preserve structure. Proved via commutativity with the enrichment functor.

Template Invariance Under Reflection

Summary

The layer template (Carrier, Predicate, Decoder, Invariant) is preserved under the Langlands₁ reflection. The four components change substance but preserve structure. Proved via commutativity with the enrichment functor.

Statement

%
\label{thm:template-invariance-reflection}
The Langlands$_1$ reflection bridge
$\Lambda_1 : \mathrm{Char}(\Lemniscate) \Rightarrow \mathrm{Char}_1(\Lemniscate)$
preserves the four-component layer template:
\begin{enumerate}
    \item[\textup{(i)}]
          If $\chi \in \Sector{g}$ for
          $g \in \{\alpha, \pi, \gamma, \eta\}$,
          then $\Lambda_1(\chi) \in \Sector{g}^{(1)}$.
    \item[\textup{(ii)}]
          If $\chi \in \MixedSector$,
          then $\Lambda_1(\chi) \in \MixedSector^{(1)}$.
    \item[\textup{(iii)}]
          Each restriction
          $\Lambda_1|_{\Sector{g}} : \Sector{g} \to \Sector{g}^{(1)}$
          is injective on isomorphism classes.
    \item[\textup{(iv)}]
          The template
          $(\mathrm{Carrier}, \mathrm{Predicate}, \mathrm{Decoder}, \mathrm{Invariant})$
          at $E_0$ maps bijectively to the template at $E_1$.
\end{enumerate}

Proof / Justification

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

Source Context

• Registry source: book-03.jsonl line 30
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch11-the-yoneda-langlands-reflection.tex lines 249-272

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Sectors.LanglandsReflection
• Name: template_invariance_8_3

Dependencies

• Canonical: III.D15, III.D05

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.