Bernardes Branch-Aware Pythagorean Residual Fields for Exploratory Mesh-Envelope Diagnostics

Abstract This paper studies a branch-aware Pythagorean residual field as a conservative exploratory primitive for mesh-envelope visualization and seam-network initialization. The field is not a signed-distance field, not an offset method, and not a fabrication-ready panelizer. Its purpose is narrower: to expose area-balance compatibility structures induced by local mesh triangles and to test whether these structures can provide candidate contour cues for later, constraint-based panel-layout tools. The formulation reduces each single-triangle branch relation to a quadratic equation in u = z 2 , applies an admissibility test to remove roots introduced by squaring, and evaluates a locality-gated residual field over procedural benchmark meshes. The implementation reports point-triangle, point-plane, and curvature-proxy comparisons, raw benchmark metrics, threshold sensitivity, grid/scale/rotation/noise checks, and failure cases. The results show that the field can generate branch-compatible contour networks that differ from ordinary proximity fields, but they also expose significant limitations: non-nearest-triangle responses, coordinate-frame dependence of the projected query layer, mesh-resolution sensitivity, and lack of manufacturability guarantees. The contribution should therefore be read as a reproducible geometry-processing diagnostic and seam-initialization study, not as a validated replacement for distance fields, robust offsets, or production architectural panelization. In addition to point-plane and point-triangle distance baselines, we compare $\PhiK$ against a curvature-saliency descriptor on procedural benchmarks, documenting systematic differences in branch-compatible contour structure. We provide the complete implementation, benchmark-generation scripts, tabulated outputs, and reproduction instructions in an open GitHub repository to facilitate further exploratory use and critical follow-up work