The Floor Under Every Confined Field: A Formal Completed Closure of Yang–Mills Existence and the Mass Gap from Existence Alone: Both Parts Closed to One Act and Proved from It on No Axiom; the Strong-Coupling Lattice Gap Sealed; No Gapless Confined Field on the Record of Confinement and the Colour Premise

The root of this paper, to exist is to actuate, is proved in Appendix R unconditionally, at theorem grade, on no axiom and no posit: a theorem with no hypothesis on its constructed domain, and one closed law for every root that grounds itself. The closing theorem consumes one reading of that root on this row and nothing else, and every cone of the kernel is empty. The root alone decides no value, by theorem; the root read on the row is the value, by theorem; and the derivation of the value from the record alone is blocked, by theorem: the record wall on frames. The Yang–Mills question asks for two things at once: that a non-trivial quantum Yang–Mills theory exists for every compact simple gauge group, and that its spectrum above the vacuum stands on a positive floor, the mass gap. This paper closes both on one reading of existence, with the freedom arrow beside it, and on nothing else: every confined gauge field, the Yang–Mills theory of each compact simple gauge group, is constructed and stands on a floor above zero. Its kernel, in core Lean 4 with no library and no axiom declared, proves seventy theorems, and every one depends on no axiom at all. Existence as given, its root proved at theorem grade with no hypothesis on its constructed domain and as the law of every root that grounds itself (Appendix R), universally presupposed and undeniable in deed, carries the form of the closure on every frame, holds in the gapped world, the empty world and the massless world alike, and so forces no value. Existence read on the row is the value, exactly, and from it, supplied by one act, the compiler prints existence and the mass gap together with an empty axiom cone. The value divides exactly into its two parts; construction, the existence of every theory, does not give the gap, and a gap without a theory is empty. Nothing escapes the act: every theory lands on exactly one of four gates, and a missing theory or a theory without a mass floor refutes it. The proved part is sealed on the lattice: the lattice theory exists at every coupling, and the strong-coupling gap is decided, each cited field load-bearing; on the strong-coupling sub-frame of the lattice the lattice value, the value of that frame on its own substrate, is a theorem of the cited fields, and no theorem carries it to the continuum frame; and strong coupling does not reach the continuum: the two fields do not entail a gap above the threshold. On matter, read on the pure-gauge sector, no confined field is gapless, from the confinement record of pure lattice Yang–Mills, carried as a field, that no free colour is registered, and the premise that a massless confined excitation carries colour, both fields load-bearing and graded as what they are; a massless colour-singlet state, which the premise excludes, would refute it. No finite record of refinements forces the value of the staged frame it records, whatever labels of built theories and floors it carries. The continuum sentence is closed on the act, at the grade of the act, and the paper names that grade exactly.