In three dimensions the mass gap is the positivity of one function at infinity

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Qualification, 2026-09-26. The continuum spectral formulas below are conditional on constructing the physical theory. The cited stochastic-quantization result constructs local auxiliary-time dynamics, not an invariant measure or OS reconstruction. Small physical volume, strong lattice coupling and the infinite-volume continuum limit are separate regimes. The refinement note gives the corrected source boundary and the corresponding construction target.

For pure Yang–Mills in \(d=3\) (two space dimensions) the coupling \(g^2\) is itself a mass, so the finite-volume gap of the continuum theory on a periodic square torus of side \(L\), when that theory exists, has the form \[\Delta(L)=g^2\hbar^2c\,f(x),\qquad x=g^2\hbar L\ \text{(dimensionless)},\] by dimensional analysis alone, and the Jaffe–Witten conjecture for \(d=3\) is the single statement \[\lim_{x\to\infty}f(x)=C\in(0,\infty),\qquad m=C\,g^2\hbar^2/c .\] The small-\(x\) end of \(f\) is known from the constant-mode sector: with two spatial components the zero-mode Hamiltonian is C133 with \(D=2\), \(m_{\rm eff}=L^2/(g^2c)\) and \(g_B^2=L^2c/g^2\), giving \(\Delta_0(L)=\delta_1^{(2)}\,g^2\hbar^2c\,x^{-2/3}\), that is \(f(x)=\delta_1^{(2)}x^{-2/3}[1+o(1)]\) as \(x\to0\). On the lattice, the strong-coupling theorem of the previous note with \(\nu=2\) gives a gap \(\ge\gamma_2(g_{\rm lat}^2/2)C_2\hbar c/a\) uniform in the lattice size, and the continuum limit is the statement that \(\delta_\infty(g_{\rm lat})/g_{\rm lat}^2\) stays bounded below as \(g_{\rm lat}^2=g^2\hbar a\to0\), together with construction and a finite positive limiting ratio. The strong-coupling lower bound supplies only one end of this lattice question. The small-volume result concerns a different order of limits. Contrast \(d=4\), where the ratio must pass from a power law \(\propto g^2\) at strong coupling to the essential singularity \(e^{-1/(2b_0g^2)}\) at weak coupling. The three-dimensional problem is therefore a natural target: it has a dimensionful coupling and a single scaling variable. Karabali–Nair’s \(N/(2\pi)\) is a current-variable kinetic scale, not an established value of the full physical gap (1996 paper, eqs. (28)–(29), passage). The constant \(C\) remains to be determined. Nothing here is promoted.

1. Units and the one-variable form

With \(S=\frac1{4g^2}\int F^a_{\mu\nu}F^{a\mu\nu}\,d^3x\) and the quantization \(e^{iS/\hbar}\), the dimensional census of G07 §4 gives \([1/g^2]=\) action\(\cdot\)length in \(d=3\), so \(g^2\hbar=1/\ell_g\) is an inverse length, \(\mu:=g^2\hbar^2/c=\hbar/(c\ell_g)\) is a mass, and \(\varepsilon_3:=\mu c^2=g^2\hbar^2c\) is the unique energy formed from \((\hbar,c,g)\). The continuum theory on the periodic torus \(T^2_L\times\mathbb R\) has the fixed constants \(\hbar\), \(c\), \(g\) and the box size \(L\), and a gap \(\Delta(L)\), when the finite-volume theory exists, depends on them only. By the floor/unit theorem C131 applied to the observable “energy”, \(\Delta(L)\) is \(\varepsilon_3\) times a function of the single dimensionless combination \(x=L/\ell_g=g^2\hbar L\), because \((\hbar,c,g)\) span exactly one energy and \(L\) adds exactly one dimensionless ratio. Setting \(f(x)=\Delta(L)/\varepsilon_3\): \[\Delta(L)=\varepsilon_3\,f(x),\qquad\varepsilon_3=g^2\hbar^2c,\qquad x=g^2\hbar L .\] The infinite-volume gap, if it exists, is \(m c^2=\varepsilon_3\lim_{x\to\infty}f(x)\), and Jaffe–Witten’s \(0<m<\infty\) is \(0<\lim f<\infty\). Vacuum-energy renormalization, the only ultraviolet divergence of the superrenormalizable theory relevant here, shifts \(E_0\) and \(E_1\) equally and leaves \(f\) untouched.

2. The small-\(x\) end from the constant modes

Temporal gauge on \(T^2_L\) leaves two constant spatial components \(a_1,a_2\) in the Lie algebra. With \(S=\frac1{4g^2}\int F^a_{\mu\nu}F^{a\mu\nu}d^2x\,c\,dt\), \(F_{0i}=c^{-1}\dot a_i\) and \(F_{12}=\vec a_1\times\vec a_2\), the constant-mode Lagrangian is \(\frac{L^2}{2g^2c}|\dot{\vec a}|^2-\frac{L^2c}{2g^2}|\vec a_1\times\vec a_2|^2\) and \[H_0=\frac{g^2c}{2L^2}\sum_{i=1}^2|\vec p_i|^2+\frac{L^2c}{2g^2}\,|\vec a_1\times\vec a_2|^2, \qquad[a_i^a,p_j^b]=i\hbar\delta_{ij}\delta^{ab},\] which is C133 with \(D=2\), \(m_{\rm eff}=L^2/(g^2c)\) and \(g_B^2=L^2c/g^2\); \(\hbar\) enters only through \(g^2\) and the commutator. Its energy unit is \[\varepsilon=\hbar^{4/3}g_B^{2/3}m_{\rm eff}^{-2/3} =\hbar^{4/3}\Big(\frac{L^2c}{g^2}\Big)^{1/3}\Big(\frac{L^2}{g^2c}\Big)^{-2/3} =\hbar^{4/3}c\,g^{2/3}L^{-2/3} =\frac{\hbar c}{L}\Big(\frac{L}{\ell_g}\Big)^{1/3} =\varepsilon_3\,x^{-2/3},\] using \(g^{2/3}\hbar^{1/3}=\ell_g^{-1/3}\) and \(\hbar c/L=\varepsilon_3/x\). So the constant-mode gap is \(\Delta_0(L)=\delta_1^{(2)}\varepsilon_3x^{-2/3}\) with \(\delta_1^{(2)}>0\) the pure number of C133 for \(D=2\), and \[f(x)=\delta_1^{(2)}\,x^{-2/3}\,[1+o(1)],\qquad x\to0,\] where \(o(1)\) stands for the coupling to the nonzero momentum modes, controlled perturbatively by the small-volume expansion in the same way as in \(d=4\) (Lüscher’s method; the \(d=3\) expansion parameter is \(x^{1/3}\) rather than \(g^{2/3}\), since \(x\) is the only parameter). The claim \(f\to\infty\) as \(x\to0\) is the statement that the small box is gapped with a gap growing like \(L^{-2/3}\), slower than the \(1/L\) of a free particle in a box: the zero-point mechanism of G08 gives a softer confinement than a hard wall.

3. The lattice route in \(d=3\) and the strong-coupling end

On the periodic square lattice \((a\mathbb Z/N_sa\mathbb Z)^2\) the Kogut–Susskind Hamiltonian is \[H=\frac{\hbar c}{a}\Big[\frac{g_{\rm lat}^2}{2}\sum_\ell(-\Delta_\ell) +\frac{2}{g_{\rm lat}^2}\sum_p(N-\operatorname{Re}\operatorname{tr}U_p)\Big], \qquad g_{\rm lat}^2=g^2\hbar a=\frac a{\ell_g},\] with \(g_{\rm lat}\) dimensionless. Theorem 1 of the obligations map holds verbatim in two dimensions: \(\Delta_{a,L}=(\hbar c/a)\,\delta(g_{\rm lat};N_s,G)>0\) with a unique physical ground state. The strong-coupling theorem holds with \(\nu=2\), \(\Lambda_0=\{0,1\}^2\), two links and one plaquette per site: \[\beta=\frac{4}{g_{\rm lat}^4C_2}\cdot2N=\frac{16N^2}{g_{\rm lat}^4(N^2-1)},\qquad \Delta_{a,L}\ge\gamma_2\,\frac{g_{\rm lat}^2}{2}\,C_2\,\frac{\hbar c}{a},\] for \(g_{\rm lat}\ge g_0^{(2)}(N)=\big(16N^2/[(N^2-1)\beta_*(2,\{0,1\}^2)]\big)^{1/4}\), uniformly in \(N_s\), by the same hypothesis check.

The continuum limit is \(a\to0\) at fixed \(g^2\), \(L\), that is \(g_{\rm lat}^2\to0\) with \(N_s=L/a\) and \(x=g^2\hbar L=g_{\rm lat}^2N_s\) fixed. The lattice gap in units of \(\varepsilon_3\) is \[\frac{\Delta_{a,L}}{\varepsilon_3}=\frac{\hbar c}{a\,g^2\hbar^2c}\,\delta =\frac{\delta(g_{\rm lat};N_s,G)}{g_{\rm lat}^2},\] so, conditional on the physical continuum construction and convergence of its spectral threshold, its finite-volume scaling requires \(\delta(g_{\rm lat};x/g_{\rm lat}^2,G)/g_{\rm lat}^2\to f(x)\) as \(g_{\rm lat}\to0\), and the infinite-volume mass is \[\frac{mc^2}{\varepsilon_3}=\lim_{g_{\rm lat}\to0}\ \frac{\delta_\infty(g_{\rm lat})}{g_{\rm lat}^2}, \qquad\delta_\infty(g_{\rm lat})=\liminf_{N_s\to\infty}\delta(g_{\rm lat};N_s,G).\] At strong coupling the same ratio is bounded below by \(\gamma_2C_2/2\). The \(d=3\) conjecture on the lattice route is therefore:

T3\(_{(3)}\). The function \(c(g_{\rm lat})=\delta_\infty(g_{\rm lat})/g_{\rm lat}^2\), which satisfies \(c\ge\gamma_2C_2/2\) for \(g_{\rm lat}\ge g_0^{(2)}\), stays bounded below by a positive constant for all \(g_{\rm lat}>0\) and has a finite limit \(C\) as \(g_{\rm lat}\to0\).

This is a sufficient all-coupling version; the continuum target needs the positive finite limit near \(g_{\rm lat}=0\), with construction, rather than positivity at every intermediate lattice coupling.

The abelian contrast in \(d=3\). For compact \(U(1)\) in three dimensions Göpfert and Mack (abstract, B78) prove a gap at every coupling, so T2\('\) holds for the abelian group here, and their Debye mass \(m_D^2=(2\beta/a^3)e^{-\beta v(0)/2}\), \(\beta=4\pi^2/e^2_{\rm lat}\), makes \(c(g_{\rm lat})=\delta_\infty/g_{\rm lat}^2\) vanish with an essential singularity as \(g_{\rm lat}\to0\): the continuum photon is massless. In \(d=3\) the abelian/non-abelian distinction is therefore the limit of the ratio, zero against a positive \(C\), with positivity at each coupling shared by both. Everything in this note up to here is a collection of known facts in one normalization; the only statement of this programme’s own is the reduction to \(f\) and its two ends.

No exponential enters. In \(d=4\) the corresponding ratio \(\delta_\infty(g)/g^2\) is bounded below at strong coupling by the same theorem and must vanish like \(g^{-2}e^{-1/(2b_0g^2)}\) at weak coupling; the two ends of the \(d=4\) problem have different functional forms and the lattice gap must interpolate between them, which is what makes T3 in \(d=4\) a transmutation statement. In \(d=3\) both ends are linear in \(g_{\rm lat}^2\).

4. What is in hand and what is not

Chandra, Chevyrev, Hairer and Shen explicitly leave global stochastic survival and an invariant measure open (2024 paper, introduction, passage). The earlier table entry claiming finite-volume existence by stochastic quantization was too strong.

Open in \(d=3\): (i) the required finite- and infinite-volume physical construction; (ii) a lower bound on \(f\) beyond the small-volume regime; (iii) the identification of \(f\)’s small-\(x\) correction with the stochastic construction’s estimates. Item (ii) is the conjecture itself, and the one-variable form means that a single inequality \(f(x)\ge f_->0\) for \(x\ge x_0\), proved for the continuum finite-volume theory, together with the existence of \(\lim f\), would settle the \(d=3\) mass gap.

5. Consequence for STATE

The \(d=3\) spectral target, conditional on construction, concerns one function of one variable: \(f(x)\sim\delta_1^{(2)}x^{-2/3}\) in the small-volume approximation at \(x\to0\) and \(f\to C\) conjectured at \(x\to\infty\), with the lattice ratio \(\delta_\infty/g_{\rm lat}^2\) bounded below at strong coupling. It replaces the \(d=3\) line of STATE. The next theorem-sized step in \(d=3\) is a lower bound \(f(x)\ge f_->0\) on an interval \(x\in[x_0,x_1]\) beyond the small-volume expansion, for which the candidate tool is a variational or operator-inequality argument on the torus Hamiltonian that keeps the zero-point mechanism of the constant modes while controlling the nonzero modes by their Gaussian part, which is what the weak-coupling lattice bound of STATE step 3 is meant to prepare.