Necessity of Curvature-Based Reconstruction Obstruction

We study the problem of reconstructing coherent geometric structures from local pre-geometric data. Starting from minimal structural assumptions on comparison along paths—locality, compositional consistency, first-order reparametrisation invariance, and differentiability—we show that any admissible reconstruction system necessarily induces a connection-like infinitesimal comparison operator. We further establish that the failure of infinitesimal path-independence is uniquely captured by a curvature-type tensor, which arises as the universal second-order loop defect. In particular, the vanishing-curvature sector is locally rigid: flat reconstruction systems reduce, at leading nontrivial order, to path-independent transport. As a consequence, every infinitesimally diagnosable reconstruction obstruction necessarily lies in the curvature sector. This provides a structural interpretation of curvature as an unavoidable feature of reconstruction, rather than a postulated geometric ingredient. Finally, we show that the corresponding minimal global scalar obstruction functional is, at leading order, given by the L²-norm of the curvature. This yields a natural derivation of curvature-based functionals from reconstruction principles alone. This manuscript is a preprint version of work currently under submission.