  We build rigorously the attractive five-dimensional model where bulk fermions
propagate along the $ \mathcal{S}^1 / \mathbb{Z}_2$ orbifold and interact with
a Higgs boson localised at a fixed point of the extra dimension. The analytical
calculation of the fermion mass spectrum and effective Yukawa couplings is
shown to require the introduction of either Essential Boundary Conditions (EBC)
imposed by the model definition or certain Bilinear Brane Terms (BBT) in the
action, instead of the usual brane-Higgs regularisations. The obtained fermion
profiles along the extra dimension turn out to undergo some discontinuities, in
particular at the Higgs brane, which can be mathematically consistent if the
action is well written with improper integrals. We also show that the
$\mathbb{Z}_2$ parity transformations in the bulk do not affect the fermion
chiralities, masses and couplings, in contrast with the EBC and BBT, but when
extended to the fixed points, they can generate the chiral nature of the theory
and even select the Standard Model chirality set-up while fixing as well the
fermion masses and couplings. Thanks to the strict analysis developed, the
duality with the interval model is scrutinised.
