EMA Behind the Canvas BGNP — From Distinction to Obstruction: A Foundational Essay and Scientific Status Record

This foundational essay develops a conceptual and technical path from distinction, number, and mathematical representation to tensor-product structure, locality, localizability, and the local geometry of the BGNP research program. The central question is how a first-order admissible direction can fail to extend to a genuine nearby branch. The essay develops the distinction between tangent compatibility and nonlinear continuation, introduces the second-order obstruction through the cokernel of the linearized constraint map, and connects this local picture to Lyapunov–Schmidt reduction and higher-order continuation. A bounded exact two-dimensional model, ATOM-002, is used to make the obstruction mechanism inspectable: two systems share the same base point, Jacobian, and tangent direction, while differing at second order. The accompanying evidence-binding workflow separates numerical or computational evidence, mathematical validation, scientific identity, and presentation state. The document also records the current scientific status of the broader BGNP investigation, explicitly separating established mathematical facts, computationally supported observations, exact toy-model results, and open questions. This is a research preprint and scientific status record. It is not a peer-reviewed journal publication, and computational evidence reported here is not presented as a substitute for mathematical proof. Version 0.2 · 8 October 2026EMA — Exponential Meta-ArchitectureDOI: 10.5281/zenodo.23235101