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We give a general, direct and explicit construction of lower-boundedgeneralized twisted modules satisfying a universal property for agrading-restricted vertex (super)algebra $V$ associated to an automorphism $g$of $V$. In particular, when $g$ is the identity, we obtain lower-boundedgeneralized $V$-modules satisfying a universal property. Let $W$ be alower-bounded graded vector space equipped with a set of "generating twistedfields" and a set of "generator twist fields" satisfying a weak commutativityfor generating twisted fields, a generalized weak commutativity for onegenerating twisted field and one generator twist field and some otherproperties that are relatively easy to verify. We first prove the convergenceand commutativity of products of an arbitrary number of generating twistedfields, one twist generator field and an arbitrary number of generating fieldsfor $V$. Then using the convergence and commutativity, we define a twistedvertex operator map for $W$ and prove that $W$ equipped with this twistedvertex operator map is a lower-bounded generalized $g$-twisted $V$-module.Using this result, we give an explicit construction of lower-boundedgeneralized $g$-twisted $V$-modules satisfying a universal property startingfrom vector spaces graded by weights, $\mathbb{Z}_{2}$-fermion numbers and$g$-weights (eigenvalues of $g$) and real numbers corresponding to the lowerbounds of the weights of the modules to be constructed. In particular, everylower-bounded generalized $g$-twisted $V$-module (every lower-boundedgeneralized $V$-module when $g$ is the identity) is a quotient of such auniversal lower-bounded generalized $g$-twisted $V$-module (a universallower-bounded generalized $V$-module).

A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra