This paper establishes a sharp dichotomy governing the continuum scaling limit of lattice SU(2) Yang–Mills theory at weak coupling. For any subsequence β\u2096 → ∞, the smoothed lattice susceptibility either collapses to zero (Regime I), producing a trivial factorised limit, or remains bounded below (Regime II), producing a non-degenerate continuum quantum field theory.
\nIn Regime II we construct a canonical susceptibility-normalised observable by fixing the smoothed susceptibility at unit physical scale. We then show that the associated renormalised Schwinger functions admit subsequential limits with uniform Osterwalder–Schrader (OS) bounds. The limiting theory is O(4)-invariant, has a positive mass gap, and possesses a non-vanishing connected two-point function, establishing nontriviality of the continuum limit. OS reconstruction therefore yields a non-trivial Wightman quantum field theory on Minkowski space with a positive spectral gap.
\nThe argument uses no renormalisation-group analysis. Instead it relies only on three structural inputs established in companion papers:
(i) exponential clustering of lattice correlations,
(ii) existence of subsequential continuum limits satisfying the OS axioms,
(iii) restoration of O(4) rotational invariance in the scaling limit.
The dichotomy is accompanied by an equivalence between susceptibility behaviour and physical mass behaviour: Regime I corresponds to m_phys → ∞ and triviality, while Regime II corresponds to bounded physical mass and nontriviality. Consequently, for SU(2) the Clay Millennium Yang–Mills mass-gap problem is reduced to a single remaining lattice input: an upper bound on the lattice mass gap ensuring that Regime II holds (i.e. that the physical mass does not diverge along the scaling trajectory).
\nThis work is part of the Emergence as Regime Formation programme and is intended as the final reduction step in a sequence establishing the continuum construction of SU(2) Yang–Mills theory from lattice spectral/analyticity mechanisms.
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