1d Ising model: energy inside domains

I am trying to understand some calculations to get the excitation energy $Delta E_text{M} = E_text{M} – E_0$ (M is the number of domain walls) in the 1d Ising model in the absence of a magnetic field:

$$ H = – Jsum_{i=1}^N s_i s_{i+1} $$

As far as I know, you can take two approaches — open boundary conditions and periodic boundary conditions. Open boundary conditions means that the Hamiltonian becomes

$$ H_text{OBC} = – J ( s_1 s_2 + s_2 s_3 + dots + s_{N-1} s_N )$$

while the periodic boundary conditions lead to

$$ H_text{PBC} = – J ( s_1 s_2 + s_2 s_3 + dots + s_{N-1} s_N + s_N s_1 ) .$$

This leads to different ground state energies

$$ E_{0, OBC} = – J ( N – 1 )$$

and

$$ E_{0, PBC} = – J N .$$

If I understood everything correctly, every domain will contribute an energy $- (L_i – 1) J$ (where $L_i$ are just the number of spins in the domain $i$) plus an additional $+J$ per domain wall in the OPC leading to

$$ E_{M, OPC} = – J sum_{i=1}^M (L_i – 1) + J M = – JN + 2JM = E_{0, OPC} + Delta E_{M, OPC} $$

but in the PBC, I would get an energy $- J L_i$ per domain plus $0$ per domain wall since the energy contribution is $propto – JN$. This leads to

$$ E_{M, PBC} = – J sum_{i=1}^M L_i = – JN = E_{0, PBC} $$

I am really confused by this and I feel like I’ve mixed things up quite a lot. I was expecting both results to match (or at least to match in thermodynamic limit).