Explicit Branch Surfaces for Bernardes’ 3D Pythagorean Construction and a Branch-Aware Compatibility Field over Triangle Meshes

Bernardes' three-dimensional extension of the Pythagorean theorem replaces the classical balance of squared side lengths by a family of area-balance relations generated from a base triangle and an auxiliary spatial point. The original construction is implicit. In this paper, each of the three Bernardes branches is reduced explicitly to a quadratic equation in the variable $u=z^2$, yielding a branch-wise sheet representation together with admissibility conditions required to remove spurious algebraic roots. This explicit reformulation reveals that the construction is intrinsically multi-sheeted, branch-dependent, and symmetric with respect to the reflection $z\mapsto -z$. We then define a branch-aware scalar field over triangulated surfaces. The field itself is evaluated from the original Bernardes residuals rather than by solving the quadratic sheets at every query point. That choice is deliberate: the explicit reduction provides analytic structure, clarifies admissibility and degeneracy, and supports visualization of the sheet geometry, while the residual definition supplies a simple branch-symmetric quantity on arbitrary 3D meshes. Numerical experiments on a reference STL-like patch and on isolated triangles show that the field is not a Euclidean distance surrogate. Instead, it behaves as a geometry-aware compatibility descriptor whose strongest alignment with point-to-triangle distance occurs over a finite offset band and whose spatial patterns respond to triangle anisotropy. The contribution of the paper is therefore threefold: an explicit branch-wise formulation of Bernardes' 3D construction, a clarified residual-based field interpretation over triangle meshes, and a reproducible reference implementation with documented numerical settings and figure-generation scripts. The method is positioned as complementary to signed-distance and level-set formulations rather than as a replacement for either.