The World is Not Smooth: Sign-Chain Architecture, Euler Surplus, and the Failure of the Affine Category Across Mathematics and Physics

We introduce a discrete, refinement-stable framework for analyzing systems whose behaviouris not captured by smooth or locally affine descriptions. The construction is based on an orderedstate graph GT , an ordering functional T , and a finite-support admissibility condition defininglocal temporal transitions. The only invariant retained is the discrete second variation∆2c T (p) = T (p2) − 2T (p) + T (p0)evaluated on admissible two-step chains.From this we define a refinement-limit sign structure S∞(p) and the associated interfaceset L∞, consisting of points at which incompatible second-variation signs persist at arbitrarilyfine scales. The central result is a strict dichotomy: either the system admits affine temporalcollapse (global single-sign evolution), or it exhibits persistent non-collapse through a nonemptyinterface set. This dichotomy is invariant under admissible refinement and cannot be removedby local approximation.We show that a range of problems across analysis, geometry, number theory, logic, andcomplexity share the same structural form when expressed in this framework: their decisiveobstruction corresponds to the forced presence of L∞. In each case, the classical formulationoperates on an affine projection that suppresses this invariant, while the admissible formulationretains it.The contribution of this work is therefore not a collection of solutions within existing smoothformulations, but a category-level reformulation. It identifies refinement-persistent sign struc-ture as the governing invariant and reduces diverse problems to a common question: whethertheir defining relations admit affine collapse or force persistent non-collapsing evolution.