Pure SU(2) Lattice Gauge Theory on a Desktop, Paper 4: Smeared Variational Correlators, the Scalar Glueball, and the Series' Direct Statement About the Mass Gap

This fourth paper of the series measures the lightest excitation of pure SU(2) gauge theory — the scalar glueball, lattice irrep A₁⁺⁺ — from smeared, variationally optimized Euclidean correlators on the asymmetric engine of Paper 3, under an analysis whose operator basis, GEVP reference time, conditioning rule, plateau criterion, and chain lengths were frozen and registered before any production correlator existed. Here "mass gap" means, throughout, the leading exponential decay rate of gauge-invariant correlators in the discretized, finite-volume Euclidean theory. The dataset comprises nine production streams at 16³×32 (β ∈ {2.40, 2.45, 2.50}, three independent streams each, 49,000 measurements per stream) plus a 20³×40 control, with every measurement archived incrementally. The machinery carries four analytic calibrations, including a strong-coupling anchor: at β = 0.90 and 1.10 the measured scalar decay lands on the four-plaquette-tube prediction am = −4 ln u at 0.1σ and 0.2σ. The pre-registered plateau rule, applied mechanically, quotes am₀₊₊ = 0.8145(67) at β = 2.45 (window t ∈ [2,3]) and 0.6801(110) at β = 2.50 (single-point window, stated as such) — the independent fit route agreeing at 0.7σ and 1.1σ — and returns INDETERMINATE at β = 2.40, where the signal-to-noise cliff — this paper's named, budgeted constraint — killed the eigenvalue signal at t = 1, and at the control volume. The discretized theory's leading decay rate is therefore nonzero at these parameters — am/σ_am ≈ 120 and 60 — and what that does and does not mean is fixed by a binding boundary paragraph (Section 1). Two ratios are formed. m₀₊₊/√σ = 3.310(173) and 2.868(103), using Paper 2's string tensions and inheriting — with caption-level disclosure — their documented ≈10% overestimate, which biases this ratio low. The series-internal headline is the construction in which that bias cancels: m₀₊₊/T_c = (m₀₊₊/√σ)/(T_c/√σ) with both factors from the series' own data, giving 5.33(34) at β = 2.45 — consistent with the continuum expectation ≈ 5.4 derived from the literature values [page verification at deposit] and comparable to the verified SU(3) determination 5.73(9) — and 4.62(23) at β = 2.50. The two couplings' ratios differ at 2.2σ: the pre-registered scaling cut returns INDETERMINATE, the tension is reported per coupling rather than averaged away, and its leading candidate systematics (the window asymmetry between the two extractions foremost) are stated with the pre-registered repair path deferred to a successor. Nothing in this paper bears on the existence or construction half of the Clay problem.