Constructive Existence of Quantum Yang--Mills Theory on $\mathbb{R}^4$ with a Strictly  Positive Mass Gap for Each Compact Simple Gauge Group $G$

Santos  Borom

The Clay Mathematics Institute problem on Yang--Mills theory and the mass gap (Jaffe--Witten)~\cite{JaffeWitten2000} asks for a mathematically rigorous, nonperturbative construction of four--dimensional quantum Yang--Mills theory with a strictly positive mass gap for each compact simple gauge group $G$. In earlier work~\cite{borom}, Borom proved a \emph{Yang--Mills Mass Gap Theorem} which shows that any \emph{axiomatic K\"all\'en--Lehmann--good Yang--Mills datum} (KL-good, for short) ---a continuum OS/BRST Yang--Mills sector on $\mathbb R^4$ satisfying a set of four K\"all\'en--Lehmann, IR/UV Guard, and Symmetric Law-State (Quandrant~I)/BRST--health conditions automatically yields a four--dimensional OS/BRST Yang--Mills theory with a strictly positive spectral gap in the BRST--physical colour sector. The present article constructs, for each compact simple gauge group $G$, at least one such axiomatic KL--good datum in a fully continuum, Wilson--free framework. Phase~A starts from a \emph{benchmark minimal master potential} $\mathcal T_G^{\mathrm{bench}}$ and builds a chronogeometric and matter weave, called the $\tau$-kairos weave, whose infinite--volume limit defines an emergent Fourfold Algebra/Gel'fand--Naimark--Segal (GNS) \emph{rig} and a continuum an Osterwalder--Schrader (OS) / Becchi--Rouet--Stora--Tyutin (BRST) Yang--Mills sector on $\mathbb R^4$ with a TT--typed, Einstein--safe \emph{bench} geometry and a Quadrant~I Yang--Mills law/state datum on \emph{certified} plateaux \cite{borom}. Phase~B extends this OS/BRST datum to a fully typed Fourfold Algebra/GNS rig, constructs TT--Einstein--safe plateaux and a benchmark Yang--Mills plateau $(\mathcal P_{\mathrm{YM}},\mathcal D_{\mathrm{YM}})$, and defines a benchmark Yang--Mills deliverable bundle and canonical glueball typed meters $W_O^{(\tau)}$ which are BRST--closed, Nielsen--stable, and insensitive to pure TT excitations. Phase~C imports the KL Guards architecture, builds an IR window and metric Guards from the TT sector, establishes triadic KL representations with positive spectral measures for all canonical YM meters, verifies IR Guard bounds and decoupling of non--YM light modes, and confirms that the colour quadrant functional remains Symmmetric-Asymmetric Law-State Quadrant~I/BRST--healthy. These results prove that the benchmark Fourfold Algebra/GNS Yang--Mills rig satisfies all four axiomatic K\"all\'en--Lehmann--good. Applying the Yang--Mills Mass Gap Theorem of \cite{borom} to this benchmark datum, we obtain a \emph{benchmark Yang--Mills mass gap theorem}: for each compact simple $G$ there exists a certified Tau--Kairos weave and associated Fourfold Algebra/GNS rig whose emergent continuum OS/BRST Yang--Mills sector on $\mathbb R^4$ satisfies the Osterwalder--Schrader axioms and has a strictly positive mass gap $m_{\mathrm{gap}}(G)>0$ in the BRST--physical colour channel. In particular, within the $\tau$--kairos weave framework, we realise the spectral content of the Clay Yang--Mills existence and mass gap problem by constructing, in the continuum and without reference to lattice regularisations, a four--dimensional Yang--Mills theory with a rigorously controlled mass gap.

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