Structural Scope, Negative Results, and Epistemic Stop Conditions An Author's Note on the Limits of the ARG Framework

This author’s note documents a negative result obtained during a bounded investigation of a potential connection between the variance-based ARG framework and the Goldbach Conjecture. The analysis demonstrates that no such implication exists, and that the obstruction is structural rather than technical.

Three independent and irreducible barriers are identified:

1. a scope gap between Riemann Hypothesis–level control and Generalized Riemann Hypothesis–level requirements,
2. a structural mismatch between linear variance constraints and bilinear additive correlations, and
3. a fundamental domain gap between multiplicative and additive number theory.

Rather than treating this outcome as a failure, the note argues that it precisely delineates the epistemic boundaries of the ARG framework. The concept of epistemic stop conditions is introduced to formalize when further exploration ceases to yield structural insight, even if additional technical approaches remain possible.

The purpose of this document is methodological. It clarifies framework scope, records principled stopping criteria, and emphasizes the role of negative results in maintaining conceptual discipline and preventing overextension in structural research programs. This work does not propose new conjectures or proofs; it serves as a scope-defining companion to existing ARG publications.