%
\label{thm:structural-incompatibility}
%   I.D76, II.D69, II.D68
Properties (U) and (A) of the Infinity Trade-Off
(Definition~\ref{def:infinity-trade-off})
are structurally incompatible.
No mathematical framework can simultaneously satisfy both.

[Proof sketch]
Assume both (U) and (A) hold simultaneously.
\begin{enumerate}
    \item[\textup{(i)}]
          From~(U): there exists a unique $\omega$ with
          $\rho(\omega) = \omega$ and the diagonal discipline K5
          blocks unrestricted cardinality arguments
          (I.T35, I.T36).
    \item[\textup{(ii)}]
          From~(A): Archimedean density implies
          that between $0$ and $\omega$
          lie uncountably many objects.
          Cantor's diagonal argument constructs
          uncountable subsets.
    \item[\textup{(iii)}]
          But K5 blocks the diagonal argument
          within $\tau$ (I.T35):
          the self-application required
          for diagonalization exceeds
          the allowed iteration depth.
    \item[\textup{(iv)}]
          Contradiction: (A) requires the diagonal argument
          to construct uncountable sets;
          (U) requires K5 to block it.
\end{enumerate}
The incompatibility is not a limitation
of current proof techniques.
It is a structural impossibility:
the axioms required for (U)
contradict the axioms required for (A).