A Reproducible Certificate Framework for Conditional Lower-Bound Quantification of Spectral Mass Gaps

This preprint presents a reproducible certificate framework for conditional lower-bound quantification of spectral mass gaps. The framework separates abstract spectral assumptions, physical-sector identification, monotone scale binding, integer MeV interval arithmetic, and claim-boundary control. The central result is a machine-checkable conditional theorem: if an abstract certified lower witness satisfies mcertₐbs ≤ mₛtarₐbs, and a monotone scale-binding map S satisfies S (mcertₐbs) = 1500 MeV, then S (mₛtarₐbs) ≥ 1500 MeV = 1. 500 GeV. As an illustrative application, the framework is applied to a calibrated four-dimensional SU (3) pure Yang–Mills reference sector. Using a reference center of 1653 MeV and conservative margins of 26 MeV, 100 MeV, and 27 MeV, the integer interval calculation yields a conditional lower-bound surface of 1500 MeV = 1. 500 GeV. This work does not claim an unconditional solution of the Clay Millennium Yang–Mills mass gap problem, nor an exact physical QCD glueball mass. It provides a conditional, reproducible, and formally portable certificate framework designed for auditability and translation into proof assistants such as Lean and Coq.