Inverse Fields and Signature Geometry

This monograph develops a unified theory of inverse fields and signature geometry, advancing the claim that the central task of modern modeling is not the estimation of functions but the reconstruction of generative fields from their observable projections. Across domains ranging from high-energy physics to cognition, empirical data arise as outputs of composite operators acting on latent structures. We formalize this situation within Cognitive Field Theory (CFT), wherelocal kernels constitute an atlas of admissible microstructures and global potentials  encode nonlocal closure. Observations are expressed as operator projections, rendering inference an intrinsically inverse and ill-posed problem.