The Yang-Mills Mass Gap on the Lattice: A Conditional Synthesis

This replacement preserves the published version 2 and appends a page-fixed erratumand an object-provenance sheet. The erratum withdraws the printed derivations ofTheorems 1.1, 3.1, and 4.1 without claiming that their conclusions for the intendedYang-Mills operators are false. It separates defects in the Euclidean-measure toground-state-measure identification, the use of a four-dimensional effective actionon a three-dimensional spatial orbit space, the volume-dependent transfer-operatortrace extraction, the finite-error resolution window, the factorisation-erroraccumulation, and the admissible-volume quantifiers. Three explicit positivetrace-class counterexamples show why ordinary spectral convergence, positivityimproving, first-excited multiplicity control, and total-tail control do not bythemselves imply gap doubling. A sufficient exact-trace repair is stated at thefinite-volume scale: the leading-eigenvalue normalisation error must be little-o ofthe sum of the two actual first-excited contributions, together with subexponentialfirst-excited multiplicities and vanishing relative excited tails. The previous v2text remains included solely for provenance; no unconditional four-dimensional orcontinuum mass-gap result is claimed.