%
\label{thm:trade-off-atomicity}
%   II.T43, II.D68, II.D69, II.D91, II.D82, II.D93
The gain--cost structure of Category~$\tau$
is \textbf{atomic}:
accepting the foundational commitments
(K0--K6, $\jj^2 = +1$)
\emph{simultaneously forces}
all 16 Mode~C costs
and \emph{simultaneously enables}
all 23 Mode~D gains
and all 13 Mode~E earned results.
No partial selection is possible:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Costs are inseparable.}
          Each Mode~C refusal traces to K5, K6, or $\jj^2 = +1$
          (Chapter~\ref{ch:refuses}).
          You cannot remove any refusal
          without removing the axiom that forces it,
          which would also remove
          the Mode~D gains that the axiom enables.

    \item[\textup{(ii)}]
          \textbf{Gains are inseparable.}
          Each Mode~D gain traces to categoricity (I.T08),
          which depends on K6,
          or to the forced sign (I.T10),
          which depends on prime polarity (I.T05).
          You cannot import a single gain into ZFC
          without importing the axioms that produce it,
          which would also import the Mode~C costs.

    \item[\textup{(iii)}]
          \textbf{The ledger is a package deal.}
          The 49~gains and 16~costs
          are not a menu to choose from.
          They are the complete, indivisible output
          of a single axiomatic commitment.
\end{enumerate}

[Proof sketch]
The key observation is that
all Mode~D gains pass through
one of two bottlenecks:
categoricity (I.T08, depending on K6)
or the forced sign (I.T10, depending on I.T05).
All Mode~C costs pass through
the same two bottlenecks plus K5.
Since K5, K6, and I.T05 are axioms (or immediate consequences),
they cannot be partially adopted.
Accepting the axiom system is all-or-nothing.