Scale-Dependent Lower Bounds and a Dense-Family Criterion for a Spectral Gap

A preliminary mathematical research note examining three results relevant to possible scale-dependent approaches to the Yang–Mills mass-gap problem: A counterexample to an inference based on monotonicity. An integrable-loss criterion for preserving positive lower bounds across scales. A dense-family spectral-gap criterion for nonnegative self-adjoint Hamiltonians. The note also outlines a proposed research programme under the YM-Lang Research Project. The mathematical results are presented as conditional statements under their stated assumptions. The proposed Yang–Mills application remains a research programme. This work does not claim to construct a continuum Yang–Mills theory or to prove the physical Yang–Mills mass gap. This is a preliminary research preprint and has not been peer reviewed.