A conditional continuum theorem and anisotropic terminal gap bounds for SU(3) lattice Yang–Mills theory -Full proofs, abandoned routes and open problems

We give complete proofs of the results of twelve research notes (R18–R29) on the step that the Hamiltonian strong-coupling results of the G-series leave open: how fixed-spacing lattice bounds could be transferred to the continuum for the SU(3) Yang–Mills mass gap. (1) A conditional continuum theorem: for a Euclidean Wilson lattice with Balaban’s centered block averaging, if (E1) the coarse-graining is controlled together with sources, so that the certified object is the true effective measure, and (E2) the scaling trajectory lands, for every sufficiently fine bare spacing, in a terminal class of coarse measures with two-sided one-step bounds, then subsequential continuum limits exist, are reflection positive and non-trivial, and their mass gap in physical units lies in an explicit interval. The key identity is that coarse observables span an invariant subspace of the bare transfer matrix. (2) Terminal estimates. On isotropic lattices the lower bound holds for a polymer class, while the available upper bound cannot absorb cross-time corrections because the leading one-step correlation is of order 10−10. In the Hamiltonian formulation both bounds are robust at fixed spacing, but the transfer to the continuum is obstructed. On anisotropic lattices with strong temporal coupling, a space-time cluster expansion around independent link Markov chains bounds the neutral gap from both sides; for example 3.440 ≤ m ≤ 3.488 per time step at bt = 2, bs = 10−10, where the exact value at bs = 0 is 3.446; the numbers are verified for explicit witnesses in 50-digit arithmetic. (3) We record two routes that were tried and abandoned and an overestimate that was corrected, with the quantitative reasons. The open core is (E2), the passage through intermediate coupling. No statement about the continuum mass gap is proved.