DRSN XXII: Yang–Mills Mass Gap under Spectral Drift (De Rerum Spectrale Natura, Report XXII, Version 1.0)

DRSN XXII: YANG–MILLS MASS GAP UNDER SPECTRAL DRIFT De Rerum Spectrale Natura series REPORT XXII (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro October 2025 •Yang–Mills mass gap reformulated as a spectral coercivity problem. •Drifted covariant Laplacian preserves gauge geometry and ellipticity. •BCH expansion isolates an emergent zero-order mass sector. •Mass gap localised in the coercivity of the C2spectral component. •Structural parallel with the Navier–Stokes Clay problem. Spectral Localisation of the Yang–Mills Mass Gap J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We apply the drifted spectral framework developed in Clay–Perspective I to the Yang–Mills mass gap problem. Rather than attempting a resolution of the Jaffe–Witten conjecture, we identify the precise spectral sector in which the obstruction to a mass gap must reside. Using bounded spectral drift of the covariant Laplacian, we show that the Baker–Campbell– Hausdorff expansion isolates a zero-order operator acting as an emergent mass-like analytic potential. We prove that spectral coercivity of this sector implies the existence of a uniform mass gap for the effective generator. The construction preserves gauge covariance, ellipticity, and locality, and reveals a structural parallel between the Yang–Mills mass gap and the Navier–Stokes regularity problem. This work localises the mass gap as a concrete spectral coercivity question and opens a clear programme for further analysis. Keywords: Yang–Mills theory; mass gap; spectral drift; BCH expansion; covariant Laplacian; spectral coercivity; Clay Millennium Problems. I. INTRODUCTION AND SCOPE The Yang–Mills mass gap problem, formulated by Jaffe and Witten as a Clay Millennium Problem, asks whether four-dimensional Yang–Mills theory admits a mathematically rigorous quantum formulation with a strictly positive spectral gap between the vacuum and the first excited state. The purpose of the present work is not to resolve this problem. Rather, following the drifted spectral methodology developed in Clay–Perspective I for the Navier–Stokes equations, we aim to localise the mass gap obstruction within a precise operator-theoretic framework. The guiding principle is that classical formulations obscure the structural locus of the obstruction, while a spectral reformulation isolates it in a concrete analytic object. In the Navier–Stokes case, the obstruction was shown to reside in the coercivity of a dissipative operator. Here, we show that the Yang–Mills mass gap can be reformulated as a coercivity problem for an emergent zero-order spectral sector. ∗jp[email protected] 3 This work therefore tests the universality of spectral drift across distinct Clay problems and identifies the precise object in which the Yang–Mills mass gap must reside. Canonical dependency. This work operates within the canonical drifted spectral framework fixed in DSRN XVII and synthesised in DSRN XVIII. It does not introduce new foundational definitions and does not claim resolution of the Clay Millennium Problem considered. II. THE YANG–MILLS MASS GAP: CLASSICAL STATEMENT Consider pure Yang–Mills theory with compact gauge group G in four spacetime dimensions. In the Euclidean formulation, one seeks a probability measure on gauge fields satisfying the Osterwalder– Schrader axioms, or equivalently a Hamiltonian formulation with a self-adjoint energy operator HYM. The mass gap conjecture asserts the existence of a constant m > 0such that Spec(HYM)∩(0, m)=∅, that is, the vacuum is separated from the rest of the spectrum by a strictly positive gap [1]. Equivalently, the conjecture implies exponential decay of gauge-invariant correlation functions and a strictly positive lower bound on the spectrum of the generator of correlations [2,3]. Despite extensive progress in lattice gauge theory and constructive approaches, no proof of the mass gap exists in four dimensions. What remains unclear in classical formulations is not only whether the gap exists, but where, structurally, such a gap should be encoded. III. SPECTRAL OBJECT AND FUNCTIONAL SETUP A natural operator-theoretic object in which to encode the mass gap is a Laplace-type operator associated with the gauge connection A . After gauge fixing, a canonical choice is the covariant Laplacian acting on adjoint-valued fields, ∆A:= −DµDµ,(III.1) where Dµ=∂µ+ [Aµ,·]denotes the gauge-covariant derivative. The operator ∆ A is elliptic, essentially self-adjoint, and admits a well-defined heat kernel expansion. Spectral properties of operators of this type play a central role in both constructive field theory and spectral analysis of gauge theories [4,5]. 4 In this setting, the Yang–Mills mass gap can be formulated as the existence of a uniform positive lower bound in the spectrum of an appropriate generator built from ∆A. IV. REAL SPECTRAL DRIFT FOR YANG–MILLS THEORY Let Xbe a bounded self-adjoint operator acting on the Hilbert space of adjoint-valued squareintegrable fields. We define the real spectral drift of the covariant Laplacian by (∆A)s:= e−sX ∆AesX .(IV.1) Bounded similarity transformations preserve domain, essential self-adjointness, and the principal symbol of the operator. In particular, the drift does not alter the gauge-covariant geometric structure encoded in the highest-order part of ∆A. Proposition 1 (Structural Invariance under Drift).For all s∈R , the operator (∆ A ) s is elliptic, essentially self-adjoint on the same domain as ∆A, and isospectral to ∆A. Proof. The result follows from bounded similarity and standard self-adjointness theory for elliptic operators [5]. A. BCH Expansion and Zero-Order Structure The drifted covariant Laplacian admits a convergent BCH expansion, (∆A)s= ∆A+s C1+s2 2C2+s3 6C3+··· , Cn:= adn X(∆A).(IV.2) Since X is bounded, each Cn is a lower-order operator. In particular, the second-order commutator C2 is of order zero and therefore acts as a pointwise analytic potential compatible with gauge covariance and ellipticity. This observation singles out C2 as the natural candidate for an emergent mass-like spectral sector. B. Interpretation The real spectral drift generates a controlled hierarchy of corrections without introducing dissipation or breaking gauge covariance. At this level, the Yang–Mills mass gap problem is 5 ∆A C1= [∆A, X] C2= [X, [∆A, X]] C3= ad3 X(∆A) . . . Zero-order sector candidate mass term FIG. 1. BCH tower of the drifted covariant Laplacian. The zero-order term C2 emerges as a natural mass-like spectral sector. reformulated as the question of whether the emergent zero-order sector C2 can be shown to be spectrally coercive. This prepares the analysis of spectral coercivity developed in the next section. V. EMERGENT MASS SECTOR AND SPECTRAL COERCIVITY A. The Zero-Order Sector as an Emergent Mass Term The BCH expansion of the drifted covariant Laplacian (∆A)s= ∆A+s C1+s2 2C2+· · · isolates the second-order commutator C2= [X, [∆A, X]] as a zero-order operator. Unlike higher-order corrections, C2 acts as a pointwise analytic potential and does not modify the principal symbol of ∆A. This distinguishes C2 as the unique sector capable of generating a mass gap without altering gauge covariance or ellipticity. Definition 2 (Emergent Mass Sector).The emergent mass sector of drifted Yang–Mills theory is defined as the zero-order operator C2arising in the BCH expansion of (∆A)s. 6 B. Spectral Coercivity and the Mass Gap We now state the minimal spectral result required to localise the Yang–Mills mass gap obstruction. Proposition 3 (Mass-Like Coercivity Implies a Spectral Gap).Assume that there exists m2> 0 such that the emergent mass sector satisfies ⟨ψ, C2ψ⟩ ≥ m2∥ψ∥2for all ψ∈Dom(∆A). Then the spectrum of the drifted covariant Laplacian (∆A)sobeys Spec((∆A)s)∩(0, m2)=∅, i.e. there exists a uniform positive spectral gap of size at least m2. Proof. Since C2 is of order zero, the inequality provides a uniform coercive lower bound on the quadratic form associated with (∆ A ) s . Lower-order perturbations do not affect the principal symbol and therefore cannot close a gap generated by a strictly positive zero-order term. Standard spectral theory for elliptic operators then implies the existence of a uniform gap [5]. Remark 4. This proposition does not assert that C2 is coercive in full Yang–Mills theory. It shows that, within the drifted spectral framework, the mass gap reduces to a coercivity property of a precisely identified zero-order operator. C. Spectral Localisation of the Mass Gap Proposition 3localises the Yang–Mills mass gap in a single spectral sector. The gap is neither a purely perturbative nor a dynamical effect; it is a property of the emergent analytic potential generated by the drift. This mirrors the Navier–Stokes case, where global regularity was shown to depend on the coercivity of a dissipative operator. D. Interpretation Within the drifted spectral framework, the Yang–Mills mass gap problem is reframed as follows: the existence of a gap is equivalent to establishing a uniform lower bound on the emergent mass sector C2. All geometric and gauge-theoretic structures remain intact. This prepares the final comparison with the Navier–Stokes Clay problem and the concluding programme. 7 Yang–Mills theory Covariant Laplacian ∆ASpectral drift (∆A)sZero-order sector C2 Coercive ⇒mass gap Degenerate ⇒no gap Mass gap localised as a spectral coercivity question on C2. FIG. 2. Spectral localisation of the Yang–Mills mass gap under drift. The existence of a gap is controlled by the coercivity of the zero-order sector C2. VI. COMPARISON WITH THE NAVIER–STOKES CLAY PROBLEM Before concluding, it is instructive to compare how two distinct Clay problems are localised within the same drifted spectral framework. Despite their very different physical origins, both problems reduce to a coercivity question for a precisely identified spectral sector. Clay problem Spectral object Obstruction localised as Navier–Stokes (3D) Dissipative operator Ksin Dz=Hs−iKsLoss of spectral coercivity of Ks Yang–Mills mass gap Drifted covariant Laplacian (∆A)sAbsence of a positive lower bound in the zero-order sector C2 Drift mechanism BCH layer Emergent structure Navier–Stokes {Cn}n≥1with compatible dissipation Closed Reynolds cycle (production–mixing–dissipation) Yang–Mills C2(order-zero term) Mass-like spectral potential compatible with gauge geometry TABLE I. Spectral localisation of two Clay problems under drift. Distinct phenomena are reduced to coercivity questions for specific spectral sectors generated by the same operator-theoretic mechanism. This comparison highlights the universality of the drifted spectral approach. The method is not tailored to a particular equation, but provides a systematic way to identify where structural obstructions must reside. VII. CONCLUSIONS AND PROGRAMME We have applied the drifted spectral framework to the Yang–Mills mass gap problem, following the same methodological principles developed for the Navier–Stokes equations in Clay–Perspective I. The main conclusions are: 8 • The Yang–Mills mass gap admits a natural spectral formulation in terms of a drifted covariant Laplacian. • The BCH expansion isolates a unique zero-order sector C2 acting as an emergent mass-like analytic potential. •The existence of a mass gap is equivalent to the spectral coercivity of this sector. •Gauge covariance, ellipticity, and locality are preserved under drift. As in the Navier–Stokes case, the present work does not resolve the Clay problem. It identifies, however, a sharply defined spectral locus where the problem must be addressed. The programme suggested by this localisation is clear: to establish the Yang–Mills mass gap, one must prove a uniform positive lower bound for the emergent zero-order sector C2 in a fully interacting gauge theory. Whether such a bound holds remains an open question. What is gained here is precision. The mass gap is no longer a diffuse dynamical mystery, but a concrete spectral coercivity problem. [1] A. Jaffe and E. Witten, “Quantum Yang–Mills Theory,” Clay Mathematics Institute Millennium Problem Description (2000). [2] K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions,” Commun. Math. Phys. 31 (1973), 83–112. [3] J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Springer (1987). [4] B. Simon, The P(ϕ)2Euclidean (Quantum) Field Theory, Princeton University Press (1974). [5] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, SelfAdjointness, Academic Press (1975).