A Metric-Free Algorithm for Extracting the Exact Part of Differential 1-Forms via Directional Primitives

We introduce a constructive and metric-free method for extracting the exact component of differential 1-forms defined on star-shaped domains of R^n.

The method is based on directional primitives combined with a non-redundant summation principle, yielding an explicit algorithmic operator that directly produces a primitive for the exact part of a given 1-form. Unlike classical approaches relying on Hodge theory, this construction does not require any Riemannian metric, elliptic PDEs, or global functional analysis.

The algorithm applies uniformly to both exact and non-exact 1-forms. In the exact case, it reconstructs a primitive explicitly. In the non-exact case, it extracts the exact component while separating the cohomological obstruction. The method is purely local, relies only on one-dimensional integrals, and is well suited for symbolic and numerical implementation.

Several explicit examples are provided, including nontrivial polynomial and rational 1-forms, illustrating the non-redundancy mechanism and the full step-by-step extraction procedure. This work provides a constructive alternative to classical existence results such as the Poincaré lemma and complements Hodge-type decompositions in computational contexts.