A TEBAC Program for Yang--Mills Existence and Mass Gap: Gauge-Invariant Construction, Geometric Regularization, and Spectral Coercivity

This preprint is the first roadmap module of a TEBAC program for the Yang--Mills existence and mass gap problem. It does not claim a completed proof of the Millennium problem. Its purpose is to place the TEBAC strategy in a form legible to mathematical physicists, gauge theorists, and constructive quantum field theorists. The target is the construction, for every compact simple Lie group \(G\), of a non-trivial quantum Yang--Mills theory on Euclidean \(\mathbb R^4\), satisfying axiomatic quantum field theory requirements at least at the Osterwalder--Schrader or Wightman level, together with a positive mass gap. The program is organized as a five-module chain: \[\mathrm{YM\text{-}I}\Longrightarrow\mathrm{YM\text{-}II}\Longrightarrow\mathrm{YM\text{-}III}\Longrightarrow\mathrm{YM\text{-}IV}\Longrightarrow\mathrm{YM\text{-}V}.\] YM-I fixes the classical gauge-theoretic input and the gauge-invariant observable language. YM-II targets finite-cutoff Euclidean Yang--Mills measures with an auxiliary TEBAC geometric regularization layer. YM-III formulates the Osterwalder--Schrader reconstruction target. YM-IV isolates the mass-gap theorem as a uniform infrared spectral coercivity problem. YM-V records the continuum-limit, non-triviality, and regulator-independence criteria required for a final theorem. The auxiliary nine-dimensional TEBAC language is treated strictly as a geometric regulator and bookkeeping device. Every final statement is required to descend to the four-dimensional gauge-invariant observable sector. Universal TEBAC regularity/coercivity diagram: Navier--Stokes:high-frequency cascade ==> spectral / energy coercivity ==> global regularity BSD:rank / height inflation ==> height / regulator coercivity ==> rank and leading-term closure Yang--Mills:massless infrared packetization ==> spectral mass-gap coercivity ==> existence + mass gap +-------------------------------+-------------------------------+----------------------------------+| Navier--Stokes | BSD | Yang--Mills |+-------------------------------+-------------------------------+----------------------------------+| high-frequency cascade | rank / height inflation | massless infrared packetization || | | || spectral / energy coercivity | height / regulator coercivity | spectral mass-gap coercivity || | | || global regularity | rank and leading-term closure | existence + mass gap |+-------------------------------+-------------------------------+----------------------------------+ The diagram is methodological and organizational only. It illustrates the recurring TEBAC pattern: a potential instability or non-closure channel is converted into a coercive obstruction and then into a theorem-level closure target. In the present Yang--Mills module, this pattern is subordinated to the standard constructive quantum field theory requirements of gauge invariance, reflection positivity, Osterwalder--Schrader reconstruction, continuum-limit control, and a positive spectral gap.