Three-dimensional Yang–Mills: conditional scaling and the physical gap limit
For a constructed physical pure Yang–Mills theory in three spacetime (two space) dimensions, with no additional parameters and a fixed square-torus shape, dimensional covariance gives
\[\Delta(L)=E_3 f(x),\qquad E_3=\hbar^2c g_{\rm cl}^2, \qquad x=\hbar g_{\rm cl}^2L.\]
The positive finite infinite-volume gap would require \(\lim_{x\to\infty}f(x)=C\in(0,\infty)\), together with physical spectral convergence and reconstruction. Dimensional analysis alone proves none of these existence or positivity statements. The constant-mode matrix model and a volume-uniform strong-coupling lattice estimate control different regimes; neither transports itself to that limit. The current-variable kinetic scale \(N/(2\pi)\) of Karabali–Nair is not an established full physical gap constant. New research remains suspended; this note maintains the separate continuum obligation and the exact normalization tests.
Physical units and the one-variable form
Use a connection-valued field with \(S=(4g_{\rm cl}^2)^{-1}\int F^a_{\mu\nu}F^a_{\mu\nu}\,d^3x\) and weight \(e^{-S_E/\hbar}\). Then \([1/g_{\rm cl}^2]=\mathrm{action}\,\mathrm{length}\). Put \(\lambda_3=\hbar g_{\rm cl}^2=1/\ell_g\), which is an inverse length, and \(E_3=\hbar c\lambda_3\). The classical coupling by itself is not a mass when physical \(\hbar,c\) remain explicit. The mass unit is \(\mu=E_3/c^2=\hbar^2g_{\rm cl}^2/c\).
On \(T_L^2\times\mathbb R\), if the gap depends only on those constants and \(L\), the dimensional theorem makes it \(E_3\) times a function of \(x=L/\ell_g\). Vacuum-energy shifts leave a defined excitation gap unchanged; that observation does not establish ultraviolet construction or spectral convergence. The low-dimensional treatment owns the field normalization and solved matrix/circle comparisons.
Constant modes: an exact truncation and a conditional small-volume use
For \(SU(2)\) in temporal gauge, take constant spatial components \(\vec a_1,\vec a_2\). With \(F_{0i}=c^{-1}\dot a_i\) and \(F_{12}=\vec a_1\times\vec a_2\), their Lagrangian and Hamiltonian are
\[L_0=\frac{L^2}{2g_{\rm cl}^2c}|\dot{\vec a}|^2 -\frac{L^2c}{2g_{\rm cl}^2}|\vec a_1\times\vec a_2|^2,\]
\[H_0=\frac{g_{\rm cl}^2c}{2L^2}\sum_{i=1}^2|\vec p_i|^2 +\frac{L^2c}{2g_{\rm cl}^2}|\vec a_1\times\vec a_2|^2, \qquad[a_i^a,p_j^b]=i\hbar\delta_{ij}\delta^{ab}.\]
This is C133 with \(D=2\), \(m_{\rm eff}=L^2/(g_{\rm cl}^2c)\) and \(g_B^2=L^2c/g_{\rm cl}^2\). Dilation gives the energy unit
\[\hbar^{4/3}g_B^{2/3}m_{\rm eff}^{-2/3} =\hbar^{4/3}c(g_{\rm cl}^2)^{1/3}L^{-2/3} =\frac{\hbar c}{L}x^{1/3}=E_3x^{-2/3}.\]
The truncated Gauss-invariant gap is therefore \(\Delta_{0,\rm inv}(L)=\delta_{\rm inv}^{(2)}E_3x^{-2/3}\), with \(\delta_{\rm inv}^{(2)}\ge\delta_1^{(2)}>0\) from the invariant-sector theorem. The unrestricted scalar or matrix first gap cannot silently replace this physical-sector coefficient. This cross-product proof is \(SU(2)\); general \(SU(N)\) constant-mode conclusions need their corresponding matrix theorem.
Transport to a full physical finite-volume theory would give \(f(x)=\delta_{\rm inv}^{(2)}x^{-2/3}[1+o(1)]\) if the nonzero-mode coupling, renormalization and sector/domain controls make the relative error vanish. The earlier wording treated that error as already controlled merely by analogy with Lüscher’s four-dimensional small-volume method. It is not proved here. The natural scaling parameter \(x^{1/3}\) is dimensional bookkeeping, not a remainder estimate. The Feshbach argument states the distinct complement, Schur and effective-Hamiltonian obligations; a finite-matrix gap alone does not satisfy them.
The lattice route and its strong-coupling end
On the periodic square lattice with \(N_s\) sites per side, use
\[H_{a,L}=\frac{\hbar c}{a}\left[ \frac{g_{\rm lat}^2}{2}\sum_\ell(-\Delta_\ell) +\frac{2}{g_{\rm lat}^2}\sum_p (N-\operatorname{Re}\operatorname{tr}U_p)\right], \qquad g_{\rm lat}^2=\lambda_3a.\]
Here \(N\) is colour rank, \(N_s=L/a\) is lattice size and \(g_{\rm lat}\) is dimensionless. The finite-lattice theorem gives a unique physical ground state and \(\Delta_{a,L}=(\hbar c/a)\delta(g_{\rm lat};N_s,G)>0\). The strong-coupling theorem with spatial \(\nu=2\), \(\Lambda_0=\{0,1\}^2\), two links and one plaquette per site gives
\[\beta=\frac{8N}{g_{\rm lat}^4C_2} =\frac{16N^2}{g_{\rm lat}^4(N^2-1)},\qquad g_0^{(2)}(N)=\left[ \frac{16N^2}{(N^2-1)\beta_*(2,\{0,1\}^2)}\right]^{1/4},\]
\[\Delta_{a,L}\ge\gamma_2\frac{g_{\rm lat}^2}{2}C_2\frac{\hbar c}{a}, \qquad g_{\rm lat}\ge g_0^{(2)}(N),\]
uniformly in \(N_s\). The unspecified theorem constants \(\beta_*,\gamma_2\) retain their source hypotheses; they are not numerical continuum estimates.
At fixed \(\lambda_3,L\), taking \(a\to0\) sends \(g_{\rm lat}\to0\) with \(x=g_{\rm lat}^2N_s\) fixed and \(\Delta_{a,L}/E_3=\delta(g_{\rm lat};N_s,G)/g_{\rm lat}^2\). A finite-volume continuum spectral theorem would need
\[\frac{\delta(g_{\rm lat};x/g_{\rm lat}^2,G)}{g_{\rm lat}^2}\longrightarrow f(x).\]
For the volume-first lattice route define \(\delta_\infty(g_{\rm lat})=\liminf_{N_s\to\infty}\delta(g_{\rm lat};N_s,G)\) and \(c_3(g_{\rm lat})=\delta_\infty(g_{\rm lat})/g_{\rm lat}^2\). At strong coupling \(c_3\ge\gamma_2C_2/2\). A finite positive limit \(c_3\to C\) near zero, plus construction, matching of limit orders and surviving physical spectral weight, is the continuum target. Positivity and a common lower bound at every intermediate bare coupling form a sufficient optional strengthening, not a necessary all-coupling theorem for the continuum problem. The strong-coupling lower bound does not control the required weak-coupling end.
For compact \(U(1)\) the all-coupling Villain gap can coexist with a massless fixed-\(\lambda_3\) continuum limit. The Villain proof retains normalized measure bounds, its essential monopole suppression and the distinct massive Gaussian scaling trajectory. The former Debye prefactor with dimensionless \(\beta/a^3\) is not a mass-squared formula in the present physical units and is withdrawn here. These Euclidean source results are not automatically a theorem for the Kogut–Susskind Hamiltonian just defined.
In four dimensions the usual scaling target instead has a transmutation factor: up to its specified higher-order running factors, \(\delta_\infty(g)/g^2\) must vanish like \(g^{-2}e^{-1/(2b_0g^2)}\). This is the required trajectory shape if a physical gap survives, not an estimate proved from the strong-coupling end. The three-dimensional energy unit is simpler, but a nonzero spectral coefficient is still an independent dynamical question.
Construction boundary and consequence for STATE
Chandra–Chevyrev–Hairer–Shen construct a renormalized local evolution in auxiliary stochastic time, with possible blow-up; the cited result supplies neither global survival, an invariant measure nor OS reconstruction (2024 introduction, passage reading recorded in refinement). Karabali–Nair’s \(N/(2\pi)\) is a current-variable kinetic scale (1996 equations (28)–(29), passage), not a proved full gap. The cited 2026 outline likewise leaves that proof open; its source boundary is kept in the refinement treatment.
Conditional on the physical construction and limit, the question is whether \(f(x)\) has a positive finite limit. A lower bound \(f(x)\ge f_->0\) for \(x\ge x_0\) would supply the positive side only after that construction; one still needs finiteness and an appropriate spectral limit. The matrix proof, conditional small-volume transport, strong lattice estimate and continuum limit remain separate. This note stays distinct from the solved low-dimensional models because it owns the explicit \(\nu=2\) lattice thresholds, normalization and unresolved field-theoretic limit, not another matrix-gap proof.