The Scalar Glueball and the Trace Anomaly: Phenomenological Outlook and Open Lattice Challenges for a Dilaton Eective Model of Pure SU(3)

A dilaton effective model of pure SU(3) gauge theory, whose logarithmic potential is fixed by trace-anomaly matching rather than assumed, is confronted with continuum lattice data. Inverting the anomaly relation gives the dilaton vacuum expectation value χ0 = 304 ± 30 MeV, the stiffness B = 7.14, and a quenched gluon condensate ⟨(αs/π)G2⟩SU(3) = 0.044 ± 0.008 GeV4, with no phenomenological input. We state precisely what is and is not being claimed: the potential carries two parameters and was given two lattice numbers, so its reproduction of the scalar glueball mass MS and of the matrix element s = ⟨Ω|θ|S⟩ is an identity of the parametrisation. The falsifiable content is the condensate, and only against an independent determination of the same operator in the same scheme. We then argue that the apparent agreement between our number and the action density reported by gradient-flow and cooling studies of the topological substructure of the vacuum must not be read as confirmation: the operators are distinct, the flow reparametrises rather than removes the O(Λ4) renormalon ambiguity, and normalisation conventions differ by an order of magnitude. We formulate instead a four-point protocol under which the comparison would become a genuine test. Finally we identify the single measurement that would decide the model: the continuum-extrapolated scalar glueball matrix element s r03 in pure SU(2) gauge theory, which tests the predicted N-dependence s2/MS2 = (11N/6)⟨(αs/π)G2⟩SU(N) in a regime where the model was not fitted. An internal 1.7σ tension between two lattice determinations of MS/√σ is recorded rather than averaged away. All algebra is machine-checked in Lean 4 with Mathlib, with the axiom footprint audited to the Mathlib baseline.