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\label{prop:e1-strictness-witness}
Let $A, B \in E_0$ lie in distinct spectral sectors of the ABCD chart.
Then the $H_\tau$-enriched Hom object $[A, B]$ satisfies:
\begin{enumerate}
    \item $[A, B] \in E_1$
          (it is $H_\tau$-enriched).
    \item $[A, B]_+ \neq 0$ and $[A, B]_- \neq 0$
          (the bipolar decomposition is non-trivial).
    \item $[A, B] \notin E_0$
          (it cannot be reduced to an $E_0$-object).
\end{enumerate}
In particular, $E_1 \neq E_0$.

(1) Self-enrichment (Book~II, Part~VIII)
equips every Hom space with $H_\tau$-module structure.

(2) By the spectral decomposition theorem (I.T12),
objects in distinct ABCD sectors produce non-zero projections
onto both $\chi_+$ and $\chi_-$ characters
of the algebraic lemniscate (I.D18).

(3) Suppose $[A, B]$ were an $E_0$-object.
Then it would have an NF address
$\NF([A, B]) = (r_1, \ldots, r_d)$
recording \emph{which} object it is,
not its \emph{internal Hom structure}.
The NF address carries no idempotent splitting,
no $j^2 = +1$ unit, no bipolar projection.
The type mismatch---scalar address versus module structure---is
irreducible.