A countable norm construction for free Banach lattices under arbitrary convexity implications

We give an explicit countable construction of the norm of the free Banach lattice of an arbitrary prescribed convexity type over a pre-ordered Banach space. Starting with the ordinary free Banach lattice, each step takes the gauge of a closed solid absolutely convex hull determined by the original convexity implications. If $p_k$ is the seminorm obtained after $k$ steps, then the free norm is $p_\omega=\inf_{k<\omega}p_k$. No continuity is required of the increasing numerical functions in the implications, and every antecedent is retained in full. The proof uses disjoint cutoffs that repair the antecedents while preserving the value of one positive functional. The same estimate gives quotient closure and an exact restriction of each successor gauge to finite-dimensional vector sublattices. Applying classical compactness to the resulting class yields a local norm formula over finite-dimensional linear input spaces and finite sets of complete implications. Consequently every strict decrease of the free norm is witnessed at a finite input dimension, a finite set of implications, and a finite stage. For each fixed separable input, a countable subfamily of complete conditions already determines the same free norm. For one presentation of the AM-space type, no finite stage computes the norm on all lattice expressions. The formula also gives an intrinsic isometry criterion, without providing the convenient geometric test asked for by de Jeu and Jiang.

GitHub repository: https://github.com/eulogica/unrestricted-free-lattice-characterization