  The consequences of the gauge-coupling dependent normalization-factor of
$1/g^{\alpha}$ in the transfer-matrix of 2d U(1) lattice gauge theory are
explored. It is seen by the $\alpha=1$ choice that the lowest energy develops a
minimum at coupling $g_*=1.125$, leading to a \textit{multi-valued} Gibbs
energy similar to the systems with the first-order phase transition. It is
argued how the $1/g$ normalization may be regarded as a lost normalization in
the commonly used change of variable to the dimensionless angle-variables.
Based on the continuum limit at the next-leading order and the Ostrogradsky
formulation of higher-order time-derivatives theories, it is argued that the
spectrum at continuum is compatible only with the $\alpha=1$ choice.
