Existence
A quantum field theory is constructed on four-dimensional spacetime rather than assumed through formal path-integral notation.
The result
A constructive existence and mass-gap theorem for Yang–Mills theory.
The paper claims the full four-dimensional pure Yang–Mills existence and mass-gap result—not a conditional reduction or a finite-lattice analogue.
Unconditional proof claimed in Volumes I–III. Independent expert review ongoing.
Theorem
Polzer · 2026
For every compact connected non-abelian gauge group G with simple Lie algebra, including each finite central quotient as a distinct global form, there exists a nontrivial pure Yang–Mills Wightman quantum field theory on ℝ⁴ with a unique invariant vacuum Ω and a number Δ > 0 such that
spec(H) ∩ (0, Δ) = ∅.
Equivalently, the Hamiltonian is bounded below by Δ on the orthogonal complement of the vacuum.
What it establishes
Yang–Mills identity
The continuum theory is identified with pure Yang–Mills for the chosen compact simple gauge group and its global form.
Mass gap
Above the unique vacuum, the Hamiltonian spectrum begins at a strictly positive energy Δ.
Claim architecture
Built against the Clay specification.
The Jaffe–Witten problem description was used as the guideline for the theorem’s quantifiers and claim structure: any compact simple gauge group, a nontrivial quantum Yang–Mills theory on ℝ⁴, axiomatic control, and a strictly positive Hamiltonian mass gap.
The proof guide also tracks the surrounding specification: local gauge-invariant operators, short-distance asymptotic freedom, continuum reconstruction, and the consequences of the gap.
In plain language
Why the gap matters
The classical Yang–Mills equations contain no particle mass parameter. Quantum Yang–Mills theory nevertheless behaves as though its lightest physical excitation has positive mass. The gap is the mathematical form of that statement.
Proving it requires both constructing the continuum quantum theory and showing that its vacuum is separated from every excited state by a fixed, positive amount of energy.
Unconditional proof claimed in Volumes I–III. Independent expert review ongoing.
Lean formalisation of selected proof interfaces is in progress.
Glueball spectrum project is in development.