Yang–Mills and the Cost of Decomposition

Read this as a lattice result with an honest continuum gap. The four theorems below are established on the lattice, where the transfer matrix is a finite object you can actually hold. The Clay problem asks for the continuum field, and the bridge to it is stated here as an open conjecture, not a crossed one. The full preprint and its proofs are on Zenodo; judge it there.

The Yang–Mills puzzle is that the equations look massless and the particles are not. The classical field theory has no mass term anywhere in it, yet the quantum theory only ever produces excitations with a strictly positive lowest energy — the mass gap. Explaining how the gap appears, rigorously, from a theory that has no mass written into it, is the Clay Millennium problem.

This preprint reads that gap as a treewidth gap — the same backbone-versus-no-backbone meter the volume points at primes, circuits, and ancient scripts. Two quantities carry the load. Let V be the Wilson loop expectation ⟨W(C)⟩ in the zero-flux sector, and D be the string tension σa² in the nonzero-flux sector.

The claims, stated plainly

• V² + D² = 1, proved by the transfer matrix. Zero-flux and nonzero-flux states partition the lattice Hilbert space into two orthogonal sectors. Their squared norms sum to one by Parseval's theorem — and V and D are those norms. The identity is forced, not fitted.
• The classical field carries treewidth Θ(N^((d−1)/d)). A surface-area law — the backbone of a smooth field grows like the boundary of its volume.
• The quantum field at strong coupling carries treewidth Ω(N). Confinement makes the field globally coupled; no small separator cuts it apart.
• The running coupling interpolates between them. As g(μ) flows, the treewidth moves between the classical surface law and the strong-coupling volume law — and the mass gap Δ>0 bounds the correlation length ξ = 1/Δ, which in turn bounds the treewidth to O(ξ^(d−1)).

Why the gap and the mass are the same fact

The chain is short. A positive mass gap Δ means correlations die over a finite length ξ = 1/Δ. A finite correlation length means the field decomposes into nearly-independent blocks of size ξ — which bounds how large its treewidth can grow. So "the theory has a mass gap" and "the field's backbone stays bounded by its correlation length" are the same statement read at two altitudes. The mass gap is the cost the field pays to be decomposable.

How to read a claim like this

I hold it the way I hold the rest of this project's bolder moves — proven core, honest about the reach. Three things to keep in view:

• The lattice is solid; the continuum is the open question. All four theorems live on the lattice, where the transfer matrix is finite and the sectors are exactly orthogonal. That is real, and it is not yet the Clay problem.
• One conjecture carries the weight. Whether the treewidth gap survives the continuum limit a → 0 is stated as open. Lattice gauge theory has a long history of quantities that behave on the lattice and misbehave in the limit; the paper does not pretend this one is settled.
• The V²+D²=1 step is the strongest part. It is a clean consequence of orthogonal sectors and Parseval, independent of the harder treewidth claims — the same identity proven from first principles in The Equals Sign as a Rotation.

None of that is a dismissal. A lattice proof that organizes confinement, the mass gap, and the running coupling under one structural meter — and isolates the remaining difficulty into a single named limit — is a real contribution. It is the kind of claim that should be published with its open conjecture stated plainly and handed to specialists to attack.

Related: separator growth in one dimension · where V²+D²=1 comes from