= Solution
A local observable is parity even by part (a). In the infinite-separation limit, any nontrivial parity-even product of zero-mode Majoranas that acts on the encoded information has support near at least two separated zero modes and cannot occur in a single local observable. The projection of $O$ onto the fixed-parity ground space is consequently scalar. For every orthonormal ground-space basis,
$$
\boxed{\langle\varphi_\alpha|O|\varphi_\beta\rangle
=c_O\delta_{\alpha\beta}.}
$$
This is <Local indistinguishability of separated Majorana zero modes>.
At finite but large $\ell/\xi$, zero-mode wavefunctions overlap by $O(e^{-\ell/\xi})$. The Hamiltonian acquires couplings $i\epsilon_{ij}\gamma_i\gamma_j$, the ground-state degeneracy is split, and local observables gain logical matrix elements of the same exponential order:
$$
\langle\varphi_\alpha|O|\varphi_\beta\rangle
=c_O\delta_{\alpha\beta}+O(e^{-\ell/\xi}).
$$
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