The Mass Gap as a Coherence Threshold
The Clay Mathematics Institute’s Millennium problem for quantum Yang–Mills theory asks for a rigorous construction of four-dimensional SU(N) gauge theory satisfying the Osterwalder–Schrader axioms, with a mass gap: the Hamiltonian of the quantized theory must have no spectrum in the interval (0, Δ) for some Δ > 0 above the vacuum. The problem remains open. A number of uncertified claims have circulated in 2026 on preprint servers and blogs; none has passed peer review, and none is treated here as evidence.
This pre-print proposes reframing the mass-gap question as a coherence-threshold problem: the existence of a gap is treated as a threshold phenomenon governed by the competition between the ultraviolet definition of the theory — where asymptotic freedom makes short-distance physics weakly coupled and controllable — and the infrared observable world, where strong coupling produces confinement. The mediating structure is the renormalization-group flow, which depletes regulator dependence scale by scale. The program’s central object is Conjecture-Schema YM: the continuum mass gap is a coherence threshold — regulator-independent uniform exponential clustering of gauge-field correlations implies, via Osterwalder–Schrader reconstruction, a continuum quantum field theory with a spectral gap, and the converse is conjectured but not proved. The threshold is measured by the gap-coherence functional Γ(a) = inf over observable channels of the exponential decay rate at regulator a, made precise as the slowest-decaying channel of a unital algebra 𝔄 of bounded gauge-invariant observables (Wilson loops, bounded functions of smeared field-strength composites).
Banked partial results, each with exact hypotheses stated, include: Lemma 1, a proved transfer from exponential semigroup decay on a dense set to a spectral gap via the spectral theorem; Lemma 2, a proved two-kernel convolution estimate with a corrected N-kernel composition bound; and Lemma 3, the architecture’s hinge — uniform correlator decay over a cyclic observable algebra implies semigroup decay on a dense subspace, hence a gap — with the boundedness requirement made explicit after independent review. The paper assembles the full conditional chain: uniform exponential clustering plus a controlled continuum limit implies limiting clustering, which implies semigroup decay via cyclicity, which implies a spectral gap, while a non-vanishing connected four-point function witnesses non-triviality (an interacting, non-Gaussian theory). Five open problems are posed, each with its exact missing piece named: discharging uniform clustering, controlling the continuum limit, proving spectral visibility (cyclicity, including the electric-magnetic completeness question for ‘t Hooft loops), witnessing non-Gaussianity, and ruling out a decoupled massless sector. Four explicit falsifiability criteria are stated — the schema can lose, and the paper says exactly how.
The manuscript was verified through three clean-room independent reviews with full repair cycles: the first found one blocking and four major issues (including a missing prefactor-uniformity condition in the clustering hypothesis and a false induction remark), the second found three major issues (including a boundedness gap in Lemma 3’s proof and a miscited reference), and all were repaired with the source formalization files synchronized; the third review verified every fix and found no new blocking or major issues (readiness R4 under the program’s internal scale). The complete review trail and the five formalization files are included as supplementary material.
Nothing in this pre-print closes the Millennium problem, and nothing here claims to. Its contribution is location, not closure: a precisely stated conjecture-schema, a concrete functional with proved basic properties, three proved lemmas forming a conditional architecture, a finite list of open problems with their missing pieces named, and explicit conditions under which the schema fails. AV_claim = 0 throughout. Version 1.2.