A finite toric gap does not imply arbitrary perturbation stability

Since [F1] identifies the code as the ground eigenspace of the self-adjoint finite operator [F5], its orthogonal projector ${\mathrm{\Pi }}_L$ commutes with $H_L$: for $\psi$ in the code and $\eta \perp \mathcal{C}$, $⟨\psi ,H_L\eta ⟩=⟨H_L\psi ,\eta ⟩=E_g⟨\psi ,\eta ⟩=0$. The complement is nonzero by [F4], so the restriction of $H_L$ to it has a least eigenvalue. By [F1] that eigenvalue is $E_g+{\mathrm{\Delta }}_L$. The other finite-volume properties [F2] and [F3] concern this same unperturbed Hamiltonian and code.

Define $V_L=-{\mathrm{\Delta }}_L(I-{\mathrm{\Pi }}_L)$. The operator $I-{\mathrm{\Pi }}_L$ is a nonzero orthogonal projection by [F4], so $V_L$ is self-adjoint and $\mathrm{\parallel }{V}_L\mathrm{\parallel }={\mathrm{\Delta }}_L$. It vanishes on the code and subtracts ${\mathrm{\Delta }}_L$ from every eigenvalue on the orthogonal complement. The construction uses the global code projector and is not asserted to belong to a local perturbation class.

The code eigenvalue remains $E_g$ after adding $V_L$. Every excited eigenvalue $E\ge E_g+{\mathrm{\Delta }}_L$ becomes $E-{\mathrm{\Delta }}_L\ge E_g$, and a nonzero eigenspace with $E=E_g+{\mathrm{\Delta }}_L$ now also has eigenvalue $E_g$. Thus the original code is no longer the full ground space: the energy gap above that four-dimensional code space is zero and the perturbed ground space is strictly larger. The finite unperturbed gap, local indistinguishability, and barrier statements therefore do not by themselves prove stability against arbitrary perturbations. No thermal evolution or material model was specified, so no memory-lifetime or physical-phase conclusion follows from these finite operator facts.