Dynamical Neglectons: A Taxonomy of Invisible Obstructions Across Mathematics

This work introduces the concept of a dynamical neglecton: a categorical obstruction that is invisible to stochastic, ergodic, and spectral methods yet governs the transition from “almost all” to “all” in deterministic mathematics. The framework is built using transfer-functor towers and correspondence categories equipped with negligible ideals and modified traces. A dynamical neglecton is defined as a phantom class satisfying four axioms: it lies in a negligible ideal, has zero stochastic trace, couples to deterministic structure via a modified trace, and is sustained by an external analytic engine. A neglecton is irreducible if it is annihilated by all finite-level reconstruction operators but persists in the inverse limit. The paper establishes a structural dichotomy between two types of negligible ideals: the Cartan kernel (algebraic invisibility) and the trace kernel (stochastic invisibility), providing a classification of obstruction types across problems. A wild-phase criterion based on the first Betti number of transition graphs characterizes the existence of neglectons, and a degrees-of-freedom threshold separates problems where such obstructions vanish from those where they persist. The framework is applied to a taxonomy of nineteen problems spanning number theory, analysis, geometry, and physics, including Goldbach, twin primes, Collatz, BSD, Riemann, Navier–Stokes, and Langlands endoscopy. In sieve-based cases, the obstruction is shown to vanish identically at every finite level and arise only as an inverse-limit anomaly driven by a finite set of analytic inputs. This work does not claim to resolve these conjectures, but instead identifies a common structural mechanism underlying their difficulty: irreducible obstructions that are provably invisible to all semisimple and finite-level methods.