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Series and parallel: where the gauge insertion law stops closing

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Result, 2026-09-26. Halve one lattice direction of a heat-kernel lattice gauge theory in \(D\) dimensions. The step splits exactly into two kinds of move. Faces containing the refined direction are cut in two; these series moves integrate out exactly in every dimension, by the heat-kernel convolution. Faces transverse to the refined direction acquire a new parallel copy at doubled heat time; integrating out these parallel insertions leaves the factor \(\Psi\) of Proposition 1, the partition function of a \((D-1)\)-dimensional gauge theory on the inserted mid-plane whose edges carry Brownian-bridge laws fixed by the coarse field. There are \(\binom{D-1}{2}\) transverse planes: none in \(D=2\), which is why two-dimensional Yang–Mills closes, one in \(D=3\), three in \(D=4\).

What is proved about \(\Psi\):

This answers the stop rule of the refinement note, §8, in its second form: it gives the exact term that prevents closure, with an explicit free-field estimate, and states the interacting estimate that remains. The ingredients are established (heat-kernel convolution, Brownian bridges, Gaussian integration); the decomposition and the exact free-field defect are elementary, and no novelty is claimed for them beyond their use here as the organizing comparison. Migdal’s recursion (1975) is the approximate form of the same series/parallel split, as Kadanoff’s notes describe it (Kadanoff 1976, metadata).

1. Heat times on an anisotropic lattice

Take a hypercubic lattice in \(D\) dimensions with spacings \(a_1,\dots,a_D\), compact gauge group \(G\) with the group metric of the refinement note §5 (\(-\Delta_G\chi_R=C_2(R)\chi_R\), \(C_2(\mathbf3)=4/3\) for \(SU(3)\)), and heat kernel \(k_t=e^{t\Delta_G/2}\), \(k_t(U)=\sum_Rd_R\chi_R(U)e^{-tC_2(R)/2}\). Use the dimensionally explicit continuum action of the refinement note §6,

\[\frac{S_E}{\hbar}=\frac1{4\lambda_D}\int F^a_{\mu\nu}F^a_{\mu\nu}\,d^Dx, \qquad[\lambda_D]={\rm length}^{D-4},\]

so that \(\lambda_D=\hbar g_{{\rm cl},D}^2\) contains \(\hbar\). The anisotropic heat-kernel action assigns to a plaquette \(p\) in the plane \((\mu\nu)\) the weight \(k_{t_{\mu\nu}}(U_p)\) with

\[t_{\mu\nu}=\lambda_D\,\frac{a_\mu a_\nu}{\prod_{\rho\ne\mu,\nu}a_\rho}. \tag{1}\]

Derivation of (1). For a smooth field, \(U_p=\exp(a_\mu a_\nu F_{\mu\nu}+O(a^3))\) and, for small \(t\) away from the cut locus, \(-\log k_t(e^X)=|X|^2/(2t)+{\rm const}+O(t+|X|^2)\). Matching \(a_\mu^2a_\nu^2|F_{\mu\nu}|^2/(2t_{\mu\nu})\) to the continuum density \((1/2\lambda_D)|F_{\mu\nu}|^2\prod_\rho a_\rho\) (the factor \(1/2\) collects \(F_{\mu\nu}\) and \(F_{\nu\mu}\)) gives (1). \(\square\)

The heat time is dimensionless. In \(D=2\) it is \(\lambda_2\) times the face area, as in the refinement note; in \(D=3\) the isotropic value is \(t=\lambda_3a\), the refinement parameter of that note’s table; in \(D=4\) it is \(t=\lambda_4=g_{\rm lat}^2\), independent of \(a\). Equation (1) is a choice of regularization, the heat-kernel counterpart of the anisotropic Wilson action; the theorems below concern it exactly. The matching holds in the small-field regime; under large anisotropy, such as refining time alone, the transverse heat times grow and the Hamiltonian limit requires exponential (Wilson-type) weights instead (zero-spacing note, §1). The halvings used here change heat times by factors of two.

2. Halving one direction: series moves and a parallel insertion

Halve direction 1: \(a_1\mapsto a_1/2\). By (1),

The new variables are the half-edges in direction 1 and the mid-edges lying in the mid-planes. Each mid-edge in direction \(j\ne1\) lies in exactly one coarse face of the plane \((1j)\), which it cuts in two, and it borders \(2(D-2)\) mid-faces.

Proposition 1 (exact pushforward). Let \(\mu'\) be the fine heat-kernel measure and \(p\) the projection multiplying half-edges. Then \(p_*\mu'\) has density, with respect to the product Haar measure on coarse links,

\[\frac1{Z'}\prod_{f\ni1}k_{t_f}(U_f)\prod_{f\perp1}k_{2t_f}(U_f)\; \Psi(U), \tag{2}\]

where \(f\ni1\) ranges over coarse faces containing direction 1, \(f\perp1\) over the others, and

\[\Psi(U)=\int\prod_{g\ \rm mid}k_{2t_g}(m_{\partial g}) \prod_{e\ \rm mid}\beta_e(dm_e\,|\,U). \tag{3}\]

Here \(m_{\partial g}\) is the ordered product of mid-edge variables around the mid-face \(g\), and \(\beta_e\) is the midpoint law of the Brownian bridge on \(G\) of duration \(t_{1j}\) determined by the coarse face that \(e\) cuts: if the two halves of that face have holonomies \(P_em_e^{-1}\) and \(m_eQ_e\), with \(P_eQ_e=U_f\),

\[\beta_e(dm\,|\,U)=\frac{k_{t_{1j}/2}(P_em^{-1})\, k_{t_{1j}/2}(mQ_e)}{k_{t_{1j}}(U_f)}\,dm. \tag{4}\]

Proof. At each new vertex on a direction-1 edge, only the two half-edges and the incident mid-edges meet. Changing variables to (coarse product, upper half) and using Haar invariance, the gauge transformation at that vertex sets the upper half-edge to the identity without changing the integral; the lower half-edge becomes the coarse link. The half-faces then have the holonomies displayed, with \(P_e\), \(Q_e\) built from coarse links only. Multiply and divide by \(k_{t_{1j}}(U_f)\) for each cut face. By the convolution identity, eq. (9) of the refinement note, each \(\beta_e\) is a probability measure. Fubini gives (2)–(3). \(\square\)

The dimension count. Refining one direction produces parallel insertions in the \(\binom{D-1}{2}\) planes transverse to it: zero for \(D=2\), one for \(D=3\), three for \(D=4\). With no transverse plane, \(\Psi=1\) and (2) is the exact consistency (7) of the refinement note, recovering its §5. An isotropic dyadic step is \(D\) successive directional halvings. By (1) the heat times change as follows:

\(D\) one directional halving full isotropic step parallel planes per halving
2 \(t\mapsto t/2\) \(t\mapsto t/4\) (area) 0
3 \(t_{1\nu}/2\), \(2t_{23}\) \(t\mapsto t/2\) 1
4 \(t_{1\nu}/2\), \(2t_{jk}\) \(t\mapsto t\) 3

The last column locates the whole renormalization problem. The middle column is the classical scaling: superrenormalizable in \(D=3\), where the heat time halves per step, and marginal in \(D=4\), where it returns to itself and any running must come from \(\Psi\).

The structure matches Newton’s insertion. There the new time splits one cell in series and the elimination leaves the endpoint-independent scalar \(-F^2uvh/(8M)\); here the series moves close with no remainder (the heat kernel is already the exact propagator, the counterpart of \(U_h\) rather than \(Q_h\)), and the parallel insertion leaves \(\Psi\). A Wilson action would also leave series remainders, as \(Q_h\) does.

3. The free field: the parallel defect in closed form

Take \(G=\mathbb R^n\) with the Euclidean metric (the Gaussian limit of a compact group of dimension \(n\); \(n=8\) for \(SU(3)\)) and \(D=3\). Then \(k_t(x)=(2\pi t)^{-n/2}e^{-|x|^2/(2t)}\), and \(\beta_e\) is Gaussian with mean \((P_e-Q_e)/2\) in additive notation and covariance \((t_{1j}/4)\,I_n\), independent of the coarse data. Put the lattice on a torus with \(N_2\times N_3\) transverse sites and \(N_1\) layers. Write \(\Phi_\ell(x)\) for the flux of the old \((23)\) face at layer \(\ell\) and transverse site \(x\), and \(\hat\Phi_\ell(k)\) for its unitary discrete Fourier transform in \((x_2,x_3)\), with \(s_j=\sin(k_ja_j/2)\) and

\[v(k)=t_{12}\,s_3^2+t_{13}\,s_2^2 .\]

On a torus the transverse zero mode of a lattice curl vanishes, so \(k\ne0\) throughout. All statements concern gauge-invariant functions, which depend on the links only through the fluxes; the non-compact gauge orbits drop out with the normalization constants.

Proposition 2 (exact free-field parallel defect). In this Gaussian theory, the pushforward (2) is the coarse heat-kernel measure multiplied by \(e^{-\mathcal D}\) (up to a constant), with

\[\mathcal D=-\sum_\ell\sum_{k\ne0}\left[ \frac{v(k)\bigl(|\hat\Phi_\ell|^2+|\hat\Phi_{\ell+1}|^2\bigr)} {8t_{23}\bigl(2t_{23}+v(k)\bigr)} +\frac{|\hat\Phi_\ell-\hat\Phi_{\ell+1}|^2}{8\bigl(2t_{23}+v(k)\bigr)} \right]\ \le\ 0. \tag{5}\]

Proof. (i) The mean mid-edge configuration interpolates linearly between layers \(\ell\) and \(\ell+1\); the direction-1 links enter it as a lattice gradient, which the curl annihilates. Hence the mean mid-face flux is exactly \((\Phi_\ell+\Phi_{\ell+1})/2\). (ii) The fluctuation of the mid-face flux is the lattice curl of independent edge noises with variances \(t_{12}/4\) (direction 2) and \(t_{13}/4\) (direction 3). Its Fourier covariance is \((t_{12}/4)|1-e^{ik_3a_3}|^2+(t_{13}/4)|1-e^{ik_2a_2}|^2=v(k)\), diagonal in \(k\). (iii) For Gaussian \(\eta\) of variance \(v\), \(E\,e^{-|\bar\Phi+\eta|^2/(2\sigma^2)}\propto e^{-|\bar\Phi|^2/(2(\sigma^2+v))}\); with \(\sigma^2=2t_{23}\), \(-\log\Psi=\sum_{\ell,k}|\hat\Phi_\ell+\hat\Phi_{\ell+1}|^2/ (8(2t_{23}+v))+{\rm const}\). (iv) The old \((23)\) faces contribute \(\sum|\hat\Phi_\ell|^2/(4t_{23})\), the coarse target is \(\sum|\hat\Phi_\ell|^2/(2t_{23})\), and the cut faces return \(k_{t_{1j}}(U_f)\) exactly. Subtract, and use \(|A+B|^2=2|A|^2+2|B|^2-|A-B|^2\). \(\square\)

At zero transverse momentum and constant flux in \(\ell\), the two old and new faces at heat time \(2t_{23}\) combine in parallel to one face at \(t_{23}\), as conductances do. The defect measures the two departures: transverse lattice momentum (\(v>0\)) and variation between layers.

Corollary (size of one step). On an isotropic coarse lattice (\(a_j=a\), \(t_{\mu\nu}=t=\lambda_3a\)), suppose \(\hat\Phi_\ell\) is supported on \(|k|\le K\) and \(\sum_\ell\|\Phi_\ell-\Phi_{\ell+1}\|^2\le(Ga)^2\sum_\ell\|\Phi_\ell\|^2\), with \(\|\cdot\|\) the \(\ell^2\) norm over transverse sites. Then

\[|\mathcal D|\le\Bigl(\frac{a^2K^2}{16}+\frac{(Ga)^2}{8}\Bigr) \sum_\ell\frac{\|\Phi_\ell\|^2}{2t}. \tag{6}\]

Proof. Use \(2t+v\ge2t\) and \(v\le ta^2|k|^2/4\) in (5). \(\square\)

Corollary 2\(_s\) (a cut at fraction \(s\); added 2026-09-27, atlas §1b; refereed by Fable 2026-09-28, accepted). Cut every direction-1 cell at \(x_1=(\ell+s)a_1\), \(0<s<1\), and give each transverse face the heat time \(\lambda_3a_2a_3/\ell_1\) with \(\ell_1\) its dual length in direction 1 (the trapezoid weight). The two layers adjacent to any old or new \((23)\) face are a distance \(a_1\) apart, so every transverse face has dual length \(a_1/2\) and keeps heat time \(2t_{23}\); the cut faces have \(st_{1j}\) and \((1-s)t_{1j}\), and the mid-edge law is the Gaussian bridge at fraction \(s\), with mean \((1-s)P_e-sQ_e\) (in the additive notation above, where \(s=\frac12\) gives \((P_e-Q_e)/2\), and with \(P_em_e^{-1}\) the holonomy of the half of thickness \(sa_1\)) and covariance \(s(1-s)t_{1j}I_n\). Steps (i)–(iv) of the proof go through with mean mid-edge \((1-s)A_j(\ell)+sA_j(\ell+1)\) plus a transverse gradient, hence mean mid-face flux \((1-s)\Phi_\ell+s\Phi_{\ell+1}\), and with noise covariance \(v_s=4s(1-s)v(k)\). Using \(|(1-s)A+sB|^2=(1-s)|A|^2+s|B|^2-s(1-s)|A-B|^2\) and periodicity in \(\ell\),

\[\mathcal D_s=-\sum_\ell\sum_{k\ne0}\left[ \frac{v_s(k)\,|\hat\Phi_\ell|^2}{4t_{23}\bigl(2t_{23}+v_s(k)\bigr)} +\frac{s(1-s)\,|\hat\Phi_\ell-\hat\Phi_{\ell+1}|^2}{2\bigl(2t_{23}+v_s(k)\bigr)} \right]\le0,\]

which is (5) at \(s=\frac12\). Under the hypotheses of the Corollary, \(|\mathcal D_s|\le4s(1-s)\bigl(\frac{a^2K^2}{16}+\frac{(Ga)^2}8\bigr)\sum_\ell\|\Phi_\ell\|^2/(2t)\): the bound (6) times \(4s(1-s)\), so halving is the largest single step and an off-centre cut is gentler in both the noise and the displacement parts; it also shrinks the mesh less, to \(\max(s,1-s)\,a_1\). The trapezoid weights are the diagonal Hodge star of discrete exterior calculus on a tensor grid (dual over primal volume), so the independence of the free-field limit from the cut sequence is a convergence question for that discretization; its proofs so far cover the Hodge–Laplace problem in two dimensions on shape-regular Delaunay meshes (Zhu, Christiansen, Hu and Hirani, arXiv:2505.08966, abstract), and for nested tensor grids in \(D=3,4\), with bounded aspect ratios as the natural hypothesis, it remains expected. The statement covers one step from a uniform lattice; after it the layers have thicknesses \(sa_1\) and \((1-s)a_1\), and the next cut starts from that non-uniform lattice. Theorem 5 for a general cut is Corollary 5\(_s\) below. \(\square\)

For a smooth field, \(\Phi=a^2F_{23}\) and the right side of (6) is \(a^2(K^2/16+G^2/8)\) times the continuum magnetic action \((1/2\lambda_3)\int|F_{23}|^2d^3x\): a dimension-six correction with coefficient \(a^2\). At fixed physical \(K\) and \(G\) the relative defects along \(a_n=2^{-n}a_0\) are summable, which is the free-field instance of the refinement note’s criterion (12), in a norm on Gaussian actions rather than total variation. Two cautions keep this honest. After one step the effective action is (2) with \(\mathcal D\), so the next step starts from a slightly different action; the free field is controlled exactly because every stage is Gaussian, and its blocked actions should converge to a Gaussian fixed point. That fixed point goes back to Bell and Wilson (1975) (metadata), and the perfect-action programme of Hasenfratz and Niedermayer (1994) (metadata) develops it for asymptotically free models with a smeared blocking; for the product blocking used here the convergence is asserted, not proved. And (6) holds for smooth coarse data; typical lattice fields at fixed \(t\) are rough, so (6) states the irrelevance of the defect, while the measure-level estimate needs the whole quadratic form (5).

Migdal’s bond moving, read exactly. Moving each mid-face onto the old face of layer \(\ell\) multiplies two Gaussian weights at \(2t_{23}\) into one at \(t_{23}\), which is the coarse target. So in the Gaussian theory \(\mathcal D\) is exactly the error of the bond-moving step, and (5) splits it into its noise part (\(v\)) and its displacement part (\(\Phi_\ell-\Phi_{\ell+1}\)). Tomboulis’s attempt to bound the exact decimation between Migdal–Kadanoff-type approximations (arXiv:0707.2179, abstract; not established as a theorem, see what-would-unblock) worked on the same parallel step.

The comparison with Newton. Both eliminations complete a square, and both lower the action. Newton’s defect is a constant, removed by one cubic counterterm, after which the cell law closes exactly. The free-field defect is an irrelevant quadratic form; no finite set of nearest-neighbour counterterms removes it, and the closed object is the fixed point in the space of all quadratic actions. In the interacting theory \(\Psi\) is non-Gaussian.

4. The non-abelian parallel move in \(1+2\) dimensions

For compact \(G\), (3) says: \(\Psi(U)\) is the partition function of a two-dimensional heat-kernel lattice gauge theory on the mid-planes, at heat time \(2t_{23}\), in which each edge is distributed by the bridge law \(\beta_e(\cdot\,|\,U)\) instead of Haar measure. Two limits are exactly soluble. As \(t_{1j}\to\infty\) the bridge laws tend to Haar measure, the mid-plane is the exactly soluble two-dimensional theory of the refinement note §5, and \(\Psi\) is a constant: the old transverse faces stay at heat time \(2t_{23}\). As \(t_{1j}\to0\) the bridge laws concentrate at the geodesic interpolation and \(\Psi\) is the product of mid-face weights on the interpolated holonomies, a classical evaluation, like Newton’s \(z_*\). Between those limits, two-dimensional exact solvability fails, because it rests on Haar invariance of every edge.

Relative to the Gaussian computation of §3, four terms appear, with the orders that power counting assigns them at \(t=\lambda_3a\ll1\) (labelled as power counting; proving them is the content of the hypothesis below):

Correction, 2026-09-26 (referee check). The first version of the hypothesis below bounded \(-\log\Psi\) itself and demanded large-field suppression of \(\Psi\); the free field violates the first (it needs \(\delta_t=\frac12\)) and Proposition 3 the second. The version below bounds the log-ratio \(\mathcal D\) of (2) against the coarse density.

Hypothesis P(\(\alpha\)) (one parallel step, \(D=3\)). Fix \(\delta\in(0,\frac14)\). Let

\[\mathcal D(U)=-\log\Psi(U)-\sum_{f\perp1}\log \frac{k_{2t_{23}}(U_f)}{k_{t_{23}}(U_f)},\qquad S_{\mu\nu}(U)=\sum_{f\in(\mu\nu)}\frac{|\log U_f|^2}{2t_{\mu\nu}},\]

so that \(p_*\mu'\) is the coarse heat-kernel density times \(e^{-\mathcal D}\) (for the free field, \(\mathcal D\) is (5)). There exist \(\alpha>2\delta\), \(t_0>0\), \(C<\infty\), a local quadratic form \(\mathcal D_0\) in \(\{\log U_p\}\) (the form (5), with \(t_{23}\) possibly renormalized by \(O(t)\)), and for each \(t=\lambda_3a\le t_0\) numbers \(c_{\rm vac}(t)\) and coupling shifts \(\delta_t^{\mu\nu}\), one per plane, with \(|\delta_t^{\mu\nu}|\le Ct\), such that on the small-field set \(\{|\log U_p|\le t^{1/2-\delta}\ \text{for all }p\}\)

\[\bigl|\mathcal D(U)-\mathcal D_0(U)-c_{\rm vac}N_{\rm cells} -\textstyle\sum_{\mu<\nu}\delta_t^{\mu\nu}S_{\mu\nu}(U)\bigr| \le C\,t^{\alpha}\sum_p\bigl(1+|\log U_p|^2/t\bigr),\]

the sum running over all coarse plaquettes. The same statement is required for heat-kernel actions perturbed by local terms of the size on the right, so that it can be iterated. No large-field clause on \(\Psi\) is needed: \(\Psi\le\prod_gk_{2t_g}(1)\), and the explicit factors of (2) suppress each plaquette outside the small-field set by \(e^{-t^{-2\delta}/4}\), up to a factor \(t^{-n/2}\) per cell. Free-field check: \(\mathcal D=\mathcal D_0\), all \(\delta_t^{\mu\nu}=0\), every \(\alpha\). (Amended the same night: the first version allowed a shift of the transverse coupling only; the one-loop determinant below shifts the couplings of the cut faces as well.)

(Amended 2026-09-27, from the SU(2) mid-plane note and its referee: on periodic planes P(\(\alpha\)) is required for plane sizes \(\min_jN_j\ge c_0/t\), fixed physical size; at fixed \(N\) with \(t\to0\) a flat-holonomy winding determinant of the massive mid-plane modes is of order one and violates the literal statement.)

The compact abelian cube isolates N4 exactly. Take \(G=U(1)\) with \(k_t(\theta)=\sum_{n\in\mathbb Z}e^{-tn^2/2}e^{in\theta}\) (the Villain weight), and refine a single cube: one mid-face at heat time \(t_m\), bounded by four mid-edges \(m_j\) with orientations \(\sigma_j=\pm1\), each cutting a side face of heat time \(t_j\) whose halves have angles \(p_j-m_j\) and \(m_j+q_j\). Put \(\phi_j=p_j+q_j\) (the side-face flux, any real representative), \(\bar\Phi=\sum_j\sigma_j(p_j-q_j)/2\) (the interpolated mid-face flux, with the same representatives), and

\[\rho_t(\phi)=\frac{\sum_{w\in\mathbb Z}(-1)^we^{-(\phi-2\pi w)^2/(2t)}} {\sum_{w\in\mathbb Z}e^{-(\phi-2\pi w)^2/(2t)}}\in[-1,1].\]

Proposition 3 (exact \(U(1)\) cube). With \(T=t_m+\tfrac14\sum_jt_j\),

\[\Psi=\sum_{n\ {\rm even}}e^{-Tn^2/2}e^{in\bar\Phi} +\Bigl(\prod_{j=1}^4\rho_{t_j}(\phi_j)\Bigr) \sum_{n\ {\rm odd}}e^{-Tn^2/2}e^{in\bar\Phi}. \tag{7}\]

As every \(\rho_{t_j}(\phi_j)\to1\), \(\Psi\to k_T(\bar\Phi)\). For \(|\phi_j|\le\pi\) and \(t_j\le1\), \(0\le1-\rho_{t_j}(\phi_j)\le5\,e^{-2\pi(\pi-|\phi_j|)/t_j}\).

Proof. The bridge numerator for \(E\,e^{inm}\) is \(\sum_{r}e^{-t(r^2+(r-n)^2)/4}e^{irp}e^{i(r-n)q}\) by character orthogonality, and \(r^2+(r-n)^2=2(r-n/2)^2+n^2/2\). For even \(n\) the shift \(r\mapsto r+n/2\) reproduces the denominator \(k_t(\phi)\), giving \(E\,e^{inm}=e^{-tn^2/8}e^{in(p-q)/2}\). For odd \(n\) the shift is by a half-integer and leaves the ratio \(\sum_se^{-t(s-\frac12)^2/2}e^{i(s-\frac12)\phi}/\sum_se^{-ts^2/2}e^{is\phi}\), which Poisson summation turns into \(\rho_t(\phi)\). The four \(m_j\) are independent, so \(E\,k_{t_m}(\sum_j\sigma_jm_j)\) is the sum over \(n\) of \(e^{-t_mn^2/2}\prod_jE\,e^{in\sigma_jm_j}\), which is (7). For the bound, write \(G_w=e^{-(\phi-2\pi w)^2/(2t)}\); then \(1-\rho=2\sum_{w\ \rm odd}G_w/\sum_wG_w\le2\sum_{w\ \rm odd}G_w/G_0\). For \(|\phi|\le\pi\) the odd term nearest \(\phi\) gives \(e^{-2\pi(\pi-|\phi|)/t}\), the opposite one at most the same, and the terms \(|w|\ge3\) are smaller by \(e^{-12\pi^2/t}\); for \(t\le1\) the total is below \(5e^{-2\pi(\pi-|\phi|)/t}\). \(\square\)

Equation (7) displays the parallel move for one cube with N1–N3 absent (the group is abelian and flat): the mid-face weight is the heat kernel at the softened heat time \(t_m+\frac14\sum t_j\), the isolated-cube counterpart of the noise \(v(k)\) in (5) (indeed \(\frac14\sum_jt_j=(t_{12}+t_{13})/2\) is the Brillouin-zone average of \(v\)), evaluated at the interpolated flux. The only departure is the odd-charge factor, which differs from 1 only where a side-face flux approaches the cut locus \(|\phi|=\pi\), where its bridge midpoint becomes bimodal. That is N4, exactly, with the rate \(e^{-2\pi(\pi-|\phi|)/t}\). At the cut locus itself, \(\rho=0\) and \(\Psi=\frac12[k_T(\bar\Phi)+k_T(\bar\Phi+\pi)]\), at most a factor 2 below its small-field value: large side fluxes are suppressed by the explicit factors \(k_{t_{1j}}(U_f)\) of (2), and \(\Psi\) adds no suppression.

The whole \(U(1)\) mid-plane. The same computation extends to a full mid-plane of \(N\) mid-faces \(g\) and mid-edges \(e\), with \(\delta\) the coboundary from faces to edges, \((\delta n)_e\) the signed sum of \(n_g\) over the two mid-faces at \(e\), \(t_e\) the heat time of the side face that \(e\) cuts, and

\[C=2t_{23}\,I+\delta^{\sf T}{\rm diag}(t_e/4)\,\delta ,\]

a positive definite \(N\times N\) matrix. \(\bar\Phi_g\) is the interpolated flux of \(g\) and \(\phi_e\) the flux of the side face cut by \(e\).

Proposition 4 (exact \(U(1)\) mid-plane).

Referee verdict, 2026-09-27: ACCEPT. Equations (8)–(9) and the independent parity-mixture proof have the stated normalized-Haar constants.

\[\Psi=\sum_{n\in\mathbb Z^N}e^{-\frac12n^{\sf T}Cn}\,e^{in\cdot\bar\Phi} \prod_{e:\,(\delta n)_e\ {\rm odd}}\rho_{t_e}(\phi_e). \tag{8}\]

If every \(\rho_{t_e}(\phi_e)\) is replaced by \(1\), Poisson summation gives the vortex form

\[\Psi_1(\bar\Phi)=\frac{(2\pi)^{N/2}}{\sqrt{\det C}} \sum_{w\in\mathbb Z^N} e^{-\frac12(\bar\Phi+2\pi w)^{\sf T}C^{-1}(\bar\Phi+2\pi w)}. \tag{9}\]

Proof. The proof of Proposition 3 shows that each bridge midpoint is \(m_e=\bar m_e+\pi s_e+\xi_e\) modulo \(2\pi\), with \(\bar m_e\) the interpolation, \(s_e\in\{0,1\}\) with \(P(s_e=1)=(1-\rho_e)/2\), and \(\xi_e\) Gaussian of variance \(t_e/4\), all independent. (The bridge density is a product of two wrapped Gaussians; its two winding numbers combine into a total, which carries the weight \(e^{-(\phi_e+2\pi W)^2/(2t_e)}\), and a relative one, which shifts the centre by \(\pi W\).) Expand each mid-face weight \(k_{2t_{23}}(\cdot)=\sum_{n_g}e^{-t_{23}n_g^2}e^{in_g(\cdot)}\). The Gaussian average gives \(e^{-\sum_et_e(\delta n)_e^2/8}\), and the average over \(s_e\) gives \(1\) for even \((\delta n)_e\) and \(\rho_e\) for odd; this is (8). For (9), apply the Poisson summation formula on \(\mathbb Z^N\). \(\square\)

For one mid-face, (8) is (7). With isotropic coarse heat times (\(t_e=t_{23}=t\)), \(C=2t(I+L/8)\) with \(L=\delta^{\sf T}\delta\) the Laplacian of the dual lattice, whose spectrum lies in \([0,8]\). In Fourier variables \(C=2t+v(k)\), the covariance of Proposition 2, so the \(w=0\) term of (9) is the free field exactly. The vortices \(w\ne0\) are the compact correction, a periodic Gaussian of the Villain type (Banks, Myerson and Kogut 1977, metadata, for the duality in abelian lattice gauge theories).

Corollary (vortex bound). In the isotropic case, write \(\Psi_1^{(0)}\) for the \(w=0\) term of (9) and assume \(\|\bar\Phi\|_\infty<\pi/2\). Then

\[0\le\log\frac{\Psi_1(\bar\Phi)}{\Psi_1^{(0)}(\bar\Phi)} \le\frac{2N\,e^{-c/t}}{1-e^{-c/t}},\qquad c=\pi\Bigl(\frac\pi2-\|\bar\Phi\|_\infty\Bigr). \tag{10}\]

Referee verdict, 2026-09-27: ACCEPT. The M-matrix argument applies to the periodic square mid-plane used here; taking the logarithm of the product bound gives (10).

Proof. \(I+L/8\) is a symmetric M-matrix with row sums \(1\), so its inverse has nonnegative entries and row sums \(1\). Hence \(\|C^{-1}\|_{\infty\to\infty}=1/(2t)\), and \(C^{-1}\ge1/(4t)\) because \(L\le8\). For integer \(w\), \(\|w\|_2^2\ge\|w\|_1\). The exponent of the \(w\) term minus that of the \(w=0\) term is \(-\frac12(2\pi w)^{\sf T}C^{-1}(2\pi w)-2\pi w^{\sf T}C^{-1}\bar\Phi \le-\pi^2\|w\|_1/(2t)+\pi\|w\|_1\|\bar\Phi\|_\infty/t=-c\|w\|_1/t\). Every term is positive, and \(\sum_{w\ne0}e^{-c\|w\|_1/t} =(1+2e^{-c/t}/(1-e^{-c/t}))^N-1\). \(\square\)

The bound is extensive with density \(O(e^{-c/t})\) per mid-face, the form of error that Hypothesis P(\(\alpha\)) allows, for every \(\alpha\). The parity factors are handled by the same device, which completes the \(U(1)\) step.

Correction (GPT-6 Astra referee), 2026-09-27 — REFINE: use coarse-cube Bianchi identities for the lifts, retain harmonic flux sectors, and claim only the unperturbed one-step part of P(\(\alpha\)).

Theorem 5 (unperturbed \(U(1)\) one-step estimate). Take \(G=U(1)\), \(D=3\), an isotropic coarse lattice with \(t=\lambda_3a\), one halving of direction 1, and a coarse configuration in the small-field set: every coarse plaquette angle, taken in \((-\pi,\pi]\), has modulus at most \(\varepsilon=t^{1/2-\delta}\), with \(3\varepsilon\le\pi/8\) and \(e^{-\pi^2/(16t)}\le\frac13\). Use these principal values for the coarse fluxes and choose the interpolated flux representative modulo \(2\pi\) as \(\bar\Phi_g=(\Phi_\ell+\Phi_{\ell+1})_g/2\). Then

\[-10\,N\,e^{-2\pi(\pi-\varepsilon)/t}\ \le\ \log\Psi-\log\Psi_1^{(0)}(\bar\Phi)\ \le\ \frac{2N\,e^{-\pi^2/(8t)}}{1-e^{-\pi^2/(8t)}},\]

where \(\Psi_1^{(0)}(\bar\Phi)=(2\pi)^{N/2}(\det C)^{-1/2} e^{-\frac12\bar\Phi^{\sf T}C^{-1}\bar\Phi}\) is exactly the free-field parallel Gaussian integral. On the sector with zero total principal transverse flux in each periodic layer, \(\mathcal D\) equals (5) plus a constant, up to an error at most \(C_0Ne^{-\pi^2/(8t)}\) with \(C_0\) absolute. On other sectors use the same real quadratic calculation including its harmonic (\(k=0\)) flux. Thus the unperturbed estimate in P(\(\alpha\)) holds with \(\delta_t=0\) for every \(\alpha\); its perturbed-action stability clause requires a further proof.

Proof. (i) Representatives. Formula (8) is \(2\pi\)-periodic in each \(\bar\Phi_g\) and invariant under the joint change of lifts (\(\phi_e\mapsto\phi_e+2\pi\) flips \(\rho_e\) and shifts the adjacent \(\bar\Phi_g\) by \(\pi\)), so any consistent lifts may be used. Take principal \(\phi_e\) by changing the midpoint lifts together with the parity weights. The boundary identity for a coarse cube gives \(\sum_{e\in\partial g}\sigma_e\phi_e=\Phi_{\ell+1}-\Phi_\ell+2\pi n\). The absolute value of the difference of the six principal face terms is at most \(6\varepsilon<2\pi\), hence \(n=0\). Since \(\bar\Phi_g\equiv\Phi_{\ell+1}-\frac12\sum_e\sigma_e\phi_e\pmod{2\pi}\), its small representative is the asserted average and has modulus at most \(\varepsilon\). This uses only coarse data; the fluctuating half-cell faces have no assumed smallness. Global torus flux can still be nonzero. (ii) Mixture form. By the proof of Proposition 4, \(\Psi=E_s\,\Psi_1(\bar\Phi+\pi\delta^{\sf T}s)\) with independent \(s_e\in\{0,1\}\), \(P(s_e=1)=q_e=(1-\rho_e)/2\le\frac52 e^{-2\pi(\pi-\varepsilon)/t}\) by Proposition 3. (iii) Lower bound. Every \(\Psi_1\) is positive, so \(\Psi\ge\prod_e(1-q_e)\,\Psi_1(\bar\Phi)\ge\prod_e(1-q_e)\, \Psi_1^{(0)}(\bar\Phi)\); there are \(2N\) mid-edges and \(\log(1-q)\ge-2q\) for \(q\le\frac12\). (iv) Upper bound, uniform in \(s\). For \(c=\delta^{\sf T}s\), the terms of (9) at \(\bar\Phi+\pi c\) are indexed by \(u=c+2w\), the integer vectors congruent to \(c\) modulo 2. Relative to the \(w=0\) term at \(\bar\Phi\) the exponent is \(-\pi u^{\sf T}C^{-1}\bar\Phi-\frac12\pi^2u^{\sf T}C^{-1}u \le-c'\|u\|_1/t\), with \(c'=\frac\pi2(\frac\pi4-\|\bar\Phi\|_\infty) \ge\pi^2/16\), by the two matrix facts of the Corollary and \(\|u\|_2^2\ge\|u\|_1\). Summing coordinatewise, \(\Psi_1(\bar\Phi+\pi c)/\Psi_1^{(0)}(\bar\Phi)\le A_0^{N-k}A_1^k\le A_0^N\), where \(k\) is the number of odd entries of \(c\), \(A_0=1+2e^{-2c'/t}/(1-e^{-2c'/t})\) and \(A_1=2e^{-c'/t}/(1-e^{-2c'/t})\le A_0\) under the stated condition on \(t\). Average over \(s\) and use \(\log A_0\le2e^{-2c'/t}/(1-e^{-2c'/t})\). In fact, writing \(r=e^{-c'/t}\), \(A_0-A_1=(1-r)/(1+r)\ge0\) for every \(t>0\); the uniform-in-\(s\) bound needs no further restriction. (v) The old transverse faces. \(\log[k_{2t}(\theta)/k_t(\theta)]\) for Villain weights equals \(\theta^2/(4t)+\frac12\log\frac12\) up to the one-dimensional vortex corrections, at most \(3e^{-2\pi(\pi-\varepsilon)/(2t)}\) per face for \(|\theta|\le\varepsilon\) by the same Poisson estimate; this is where \(e^{-\pi^2/(8t)}\) dominates. Collect (iii)–(v) with the definition of \(\mathcal D\). \(\square\)

This is the first rung of the ladder completed as a theorem: for the compact abelian group, one directional step of the three-dimensional refinement equals the free-field step of Proposition 2 up to an extensive error of density \(e^{-\pi^2/(8\lambda_3a)}\) on the small-field set. It concerns the heat-kernel action; iterating it requires the same estimate for heat-kernel actions perturbed by the defect, which is the stability clause of P(\(\alpha\)). The non-abelian terms N1–N3 are absent here, so Theorem 5 tests the large-field and representative bookkeeping, and leaves the curvature and commutator terms to the \(SU(2)\) cube.

Corollary 5\(_s\) (compact \(U(1)\), a cut at fraction \(s\); 2026-09-28; refereed by Fable the same day, which supplied the uniform bound of step (d), checked here). Take the hypotheses of Theorem 5 with the trapezoid weights of Corollary 2\(_s\), a cut at fraction \(s\in(0,1)\), and the representative \(\bar\Phi=(1-s)\Phi_\ell+s\Phi_{\ell+1}\). Put \(c^*_s=\pi^2/(1+4s(1-s))\in[\pi^2/2,\pi^2]\), \(r_s=e^{-(c^*_s-\pi\varepsilon)/t}\), \(h_*=\varepsilon^2(1+4s(1-s))/(4t)\), and let \(q_e\le\frac52e^{-2\pi(\pi-\varepsilon)/t}\) be the probability that the total winding \(W_e\) of mid-edge \(e\) is nonzero. Then

\[-\sum_e2q_e\ \le\ \log\Psi-\log\Psi_1^{(0)}(\bar\Phi)\ \le\ \frac{2Nr_s}{1-r_s}+\sum_eq_eS_*^2,\qquad S_*=e^{h_*}+e^{-(c^*_s/2-\pi\varepsilon)/(2t)}+\frac{2r_s}{1-r_s},\]

with \(\Psi_1^{(0)}\) built from the covariance \(C_s\) of (b) below. For \(\varepsilon\le\pi/24\) (Theorem 5’s hypothesis) both error densities are \(O(e^{-\pi^2/(8t)})\) for every \(s\), so Theorem 5 holds for every cut fraction, and \(\mathcal D\) equals \(\mathcal D_s\) of Corollary 2\(_s\) plus a constant up to such an error on the sector of zero total transverse flux. At \(s=\frac12\) the vortex term \(2Nr_{1/2}/(1-r_{1/2})\), \(r_{1/2}=e^{-(\pi^2/2-\pi\varepsilon)/t}\), sharpens step (iv) of Theorem 5, whose bound uniform in the parities was lossy.

Proof. (a) Midpoint law. The bridge density is \(\sum_{w_1,w_2}e^{-(P_e-m+2\pi w_1)^2/(2st_e)-(m+Q_e+2\pi w_2)^2/(2(1-s)t_e)}\). With \(W=w_1+w_2\) fixed, the minimum over \(m\) is \((\phi_e+2\pi W)^2/(2t_e)\), \(\phi_e=P_e+Q_e\), the curvature is \(1/(s(1-s)t_e)\), the prefactor is \((2\pi t_e)^{-1/2}\) for every \(s\), and the centre is \(\bar m_e+2\pi(1-s)W-2\pi w_2\); the last term vanishes modulo \(2\pi\). So the midpoint is a mixture over \(W\in\mathbb Z\), with weights \(\propto e^{-(\phi_e+2\pi W)^2/(2t_e)}\) independent of \(s\), of wrapped Gaussians of variance \(s(1-s)t_e\) centred at \(\bar m_e+2\pi(1-s)W\). At \(s=\frac12\) the shifts are \(\pi W\), the parity mixture of the proof of Proposition 4, and \(q_e\) obeys Proposition 3’s bound for every \(s\). (b) Covariance. The Gaussian average of that proof now gives \(e^{-\frac12s(1-s)t\,n^{\sf T}Ln}\), so in the isotropic case (9) has covariance \(C_s=2t(I+\frac12s(1-s)L)\), a symmetric diagonally dominant M-matrix with row sums \(1\): \(\|C_s^{-1}\|_{\infty\to\infty}=1/(2t)\), and \(C_s^{-1}\ge1/(2t(1+4s(1-s)))\) since \(L\le8\). (The vortex bound (10) holds with \(c_s=\pi(\pi/(1+4s(1-s))-\|\bar\Phi\|_\infty)\ge c\).) (c) Representative. The mean mid-edges are \((1-s)A_j(\ell)+sA_j(\ell+1)\) plus a transverse gradient, and the argument of Theorem 5 (i) with weights \(1-s\) and \(s\) gives mean mid-face flux \(\bar\Phi\) up to multiples of \(2\pi\), with \(|\bar\Phi|\le\varepsilon\). (d) Upper bound. \(\Psi=E_W\,\Psi_1(\bar\Phi+2\pi(1-s)\delta^{\sf T}W)\). For given \(W\) the terms of (9) are indexed by \(u=(1-s)\delta^{\sf T}W+w\), \(w\in\mathbb Z^N\), and relative to the \(w=0\) term at \(\bar\Phi\) the exponent is \(F(u)=-2\pi u^{\sf T}C_s^{-1}\bar\Phi-2\pi^2u^{\sf T}C_s^{-1}u\le\sum_gh(|u_g|)\), \(h(x)=(\pi\varepsilon x-c^*_sx^2)/t\), by the two matrix facts of (b). Here \(h\le h_*\), and \(h(x)\le-(c^*_s-\pi\varepsilon)x/t\) for \(x\ge1\). Summing over \(w\) coordinatewise, \(\Psi_1(\bar\Phi+2\pi(1-s)\delta^{\sf T}W)/\Psi_1^{(0)}(\bar\Phi)\le\prod_gS(\theta_g)\), \(\theta_g\) the fractional part of \(u_g\), \(S(\theta)=\sum_ke^{h(|\theta+k|)}\), with \(S(0)\le1+2r_s/(1-r_s)\) and \(S(\theta)\le S_*\) (one point of \(\theta+\mathbb Z\) within \(\frac12\) of \(0\), one in \([\frac12,1)\), the rest at distance at least \(1\)). Only the two mid-faces adjacent to an edge with \(W_e\ne0\) can have \(\theta_g\ne0\), and \(S_*\ge S(0)\ge1\), so \(\prod_gS(\theta_g)\le S(0)^NS_*^{2\#\{e:W_e\ne0\}}\). The \(W_e\) are independent, so \(E_W\,S_*^{2\#\{e:W_e\ne0\}}\le\prod_e(1+q_eS_*^2)\); take logarithms. When \(\min(s,1-s)>\varepsilon(1+4s(1-s))/\pi\) the shifted terms are in fact suppressed, and the factor \(e^{h_*}\) enters only for cuts within about \(\varepsilon\) of an end. (e) Lower bound. Every term is positive, so \(\Psi\ge\prod_e(1-q_e)\Psi_1^{(0)}(\bar\Phi)\), and \(\log(1-q)\ge-2q\). (f) Old faces and sizes. Step (v) of Theorem 5 is unchanged, since every transverse face keeps heat time \(2t\). For \(\varepsilon\le\pi/24\): \(c^*_s-\pi\varepsilon\ge11\pi^2/24\), and \(2\pi(\pi-\varepsilon)-2th_*\ge2\pi^2\cdot\frac{23}{24}-\frac{\pi^2}{576}>\pi^2/8\), although \(h_*=\varepsilon^2/(4t)\cdot(1+4s(1-s))\) grows like \(t^{-2\delta}\). \(\square\)

What survives: compact \(U(1)\) in three dimensions. This theory is the proved example of a lattice gap that the fixed-coupling refinement removes. Göpfert and Mack prove a nonzero string tension for all couplings of the Villain theory, bounded below through a Debye mass \(m_D>0\) (the monopole mechanism of Polyakov 1977, metadata), and establish a continuum limit \(a\to0\) at fixed physical \(m_D\) that is a free scalar field of mass \(m_D\), with string tension divided by \(m_D^2\) diverging in that limit (Göpfert–Mack 1982, abstract as indexed; publisher abstract checked 2026-09-27). Gross proves that at fixed physical coupling the Villain theory converges to the massless free electromagnetic field; his Wilson-action statement concerns the electric sector (Gross 1983, publisher abstract).

Correction (GPT-6 Astra referee), 2026-09-27 — REFINE: the source results have different scalings and sectors; a one-step upper bound alone establishes neither continuum convergence nor the monopole density.

In a fixed physical box the Theorem 5 bound is of order \((L/a_n)^2e^{-\pi^2/(8t_n)}\) per mid-plane and \((L/a_n)^3e^{-\pi^2/(8t_n)}\) over all planes. At \(t_n=\lambda_3a_n\) its tail vanishes and is summable. Turning this into convergence needs iteration stability and large-field probability control. Göpfert–Mack’s fixed-\(m_D\) trajectory instead has logarithmically decreasing \(t_n\); their Debye formula supplies its normalization. The bound’s constant \(\pi^2/8\) cannot determine the error’s actual order or the monopole density on that trajectory.

In four dimensions Driver 1987 (publisher abstract) proves convergence on the current sector to a renormalized free electromagnetic field for general energy functions at sufficiently large inverse coupling. For the Wilson energy he obtains convergence at arbitrary coupling by an appropriate Gibbs-state choice, and state independence away from at most countably many coupling values. The sector and state qualifications matter. These results motivate the comparison with the expected nonabelian three-dimensional gap \(C_3\hbar c\lambda_3\); locating that gap in particular N1–N3 terms needs infrared control beyond the present ultraviolet estimate.

For \(SU(2)\) and \(SU(3)\) the bridge expectations \(E\,R(m_j)\) are matrices, and the isolated cube is where N2 and N3 first appear.

The \(SU(2)\) bridge midpoint. In the group metric of §1, \(SU(2)\) is the three-sphere of radius 2 (curvature \(\frac14\)), with cut locus at geodesic distance \(2\pi\) and \(|\rho|^2=\frac14\). Poisson summation of the character series gives the exact kernel

\[k_t(e^X)=e^{t/8}\sqrt{8\pi}\,t^{-3/2}\sum_{w\in\mathbb Z} \frac{\theta-4\pi w}{\sin(\theta/2)}\,e^{-(\theta-4\pi w)^2/(2t)}, \qquad\theta=|X|, \tag{11}\]

with respect to normalized Haar measure; the \(w=0\) term is the Gaussian with the Jacobian \((\theta/2)/\sin(\theta/2)\), and \(e^{t/8}=e^{t|\rho|^2/2}\). For a cut side face, left-translate so that the bridge runs from \(e\) to \(e^X\), \(d=|X|<2\pi\), and write the midpoint as \(m=m_*e^{\xi}\) with \(m_*=e^{X/2}\).

Proposition 6 (\(SU(2)\) midpoint: structure and leading covariance).

Correction (GPT-6 Astra referee), 2026-09-27 — REFINE: the covariance is accepted; the representation formula is a fixed-representation expansion through order \(t\), and the logarithm uses its almost-everywhere principal branch.

  1. Exactly, \(E\,D^J(m)=D^J(m_*)\Lambda^J\) with \(\Lambda^J\) real and diagonal in the weight basis of the axis \(\hat X\), and all odd moments of \(\xi\) vanish. (b) As \(t\to0\), uniformly for \(d\le d_0<2\pi\), \(\xi\) is Gaussian to leading order with covariance

\[\frac t4\,{\rm diag}\Bigl(1,\frac1{h(d)},\frac1{h(d)}\Bigr),\qquad h(d)=\frac d4\cot\frac d4,\]

in the frame (axis, two transverse directions), so that

\[\Lambda^J_{\mu\mu}=1-\frac t8\Bigl[\mu^2 +\frac{J(J+1)-\mu^2}{h(d)}\Bigr]+O_J(t^2).\]

Here \(J\) is fixed and \(d\le d_0<2\pi\). The exponential of the displayed order-\(t\) term is equivalent at this accuracy; noncommuting representation generators prevent treating it as an exact Gaussian characteristic function.

Proof. (a) The geodesic symmetry \(g\mapsto m_*g^{-1}m_*\) is an isometry that exchanges the endpoints, so it preserves the bridge law, and in normal coordinates at \(m_*\) it is \(\xi\mapsto-\xi\). Conjugation by the one-parameter group \(e^{sX}\) fixes both endpoints and \(m_*\), so \(\Lambda^J\) commutes with it and is diagonal in the axis basis; the symmetry gives \(\Lambda^J=(\Lambda^J)^\dagger\). (b) By (11), off the image terms (which are \(O(e^{-c/t})\) for \(d\le d_0\)), the bridge density is \(e^{-(d(e,m)^2+d(m,e^X)^2)/t}\) times smooth factors. On a space of constant curvature \(\kappa\), the Hessian of \(\frac12d(p,\cdot)^2\) at distance \(r\) is 1 radially and \(r\sqrt\kappa\cot(r\sqrt\kappa)\) transversally. At the midpoint \(r=d/2\), \(\kappa=\frac14\), so the exponent has Hessian \((4/t)\,{\rm diag}(1,h,h)\); Laplace’s method gives the covariance. Expanding \(D^J(e^\xi)\) through second order and using the vanishing odd moments gives the fixed-\(J\) order-\(t\) formula. \(\square\)

Correction (GPT-6 Astra referee), 2026-09-27 — REFINE: change the curvature sign, allow weight multiplicities, restrict to a unique minimizing geodesic, and use the Killing form for nonsimple groups.

Any compact connected group, and \(SU(3)\). The local covariance argument applies with a bi-invariant metric and \(X\) in a compact subset of the injectivity domain. Along its geodesic the curvature operator is \(-\frac14\operatorname{ad}(\hat X)^2\), with eigenvalue \(0\) on the centralizer of \(X\) and \(\frac14\alpha(\hat X)^2\) on the root plane of each positive root \(\alpha\) (with \(\operatorname{ad}X\) having eigenvalues \(\pm i\alpha(X)\) there). The midpoint covariance is therefore \(t/4\) along the centralizer and \((t/4)/h_\alpha\) on each root plane, with \(h_\alpha=\frac{\alpha(X)}4\cot\frac{\alpha(X)}4\). The centered representation average is Hermitian and commutes with the centralizer of \(X\). It is block diagonal on weight spaces of a maximal torus containing \(X\); weight multiplicities allow mixing within each block. For \(SU(2)\) the single positive root has \(\alpha(X)=d\), recovering (b). For a compact simple Lie algebra, \(\sum_{\rm all\ roots}\alpha(X)^2=C_2({\rm adj})|X|^2\), and \(1/h_\alpha-1=\alpha(X)^2/48+O(\alpha^4)\), the curvature excess averaged over the \(\dim G\) directions is, per cut face,

\[\Delta T_j\simeq\frac{t_j}4\cdot\frac{C_2({\rm adj})\,|X_j|^2}{48\dim G},\]

that is \(t_j|X_j|^2/288\) for \(SU(2)\) (\(C_2({\rm adj})=2\), \(\dim G=3\)) and \(t_j|X_j|^2/512\) for \(SU(3)\) (\(C_2({\rm adj})=3\), \(\dim G=8\), two flat Cartan directions and three root planes).

For a general compact Lie algebra replace \(C_2({\rm adj})|X|^2\) by \(-\operatorname{tr}(\operatorname{ad}X)^2\); central directions contribute zero and simple factors carry their own metric-dependent Casimirs.

Part (b) gives the leading covariance as \(t\to0\) at fixed \(X\) in the specified compact domain. The displayed softening then expands this coefficient at small \(X\). When \(|X|^2\sim t\), other order-\(t^2\) covariance terms also contribute; the softening coefficient alone is insufficient to compute the complete correction at that order. Numerical remainder constants and control after representation summation remain open. Positive curvature enlarges the transverse fluctuation of the midpoint, \(1/h(d)=1+d^2/48+O(d^4)\), so the softening of the mid-face weight grows with the side-face flux. Splitting \(J(J+1)/h+\mu^2(1-1/h)\), each cut face adds \(t_j/(4h(d_j))\) to the mid-face heat time, about \(t_jd_j^2/192\) above the abelian value \(t_j/4\), and an axial term \(-\frac{t_j}8(1-\frac1{h(d_j)})(\hat X_j\cdot T)^2\) of the same order, which couples the mid-face to the direction of the side flux. For fluctuating fields \(d_j^2\sim t\), so both are \(O(t)\) relative to the action. N3, the non-Gaussian and commutator corrections in \(\prod_jm_j\), enters at the same order and is not computed here. Proposition 7 below gives a weaker finite-cube inequality by bounding all such terms in the remainder; extracting their coupling and irrelevant-operator coefficients requires a separate calculation.

Correction (GPT-6 Astra referee), 2026-09-27 — REFINE: the reflection cancellation is rejected; a conditional finite-cube bound survives with mixed terms in its remainder.

Proposition 7 (conditional finite-cube power bound). Fix a compact connected group, one cube, and positive bounded heat-time ratios. In a local gauge let \(X\) collect its independent boundary flux coordinates, \(r^2=\sum_f|X_f|^2\). Assume a uniform Laplace expansion of the normalized bridge integral and old-face ratio,

\[\mathcal D(X,t)=c(t)+F(X)/t+A(X)-A(0)+R(X,t),\]

with \(|F(X)-F_2(X)|\le Cr^3\), \(|A(X)-A(0)|\le Cr\), and \(|R(X,t)|\le Ct(1+r^2/t)\). Here \(F_2/t\) is the quadratic defect of the isolated Gaussian cube, including the bridge denominators. These bounds require a unique nondegenerate saddle and control of the full heat-kernel/Haar amplitudes. For \(r\le C_1t^{1/2-\delta}\) and \(0<\delta<1/6\),

\[|\mathcal D-c(t)-F_2/t|\le C_2t^{1/2-\delta}(1+r^2/t).\]

Thus the unperturbed finite-cube inequality has \(\alpha=1/2-\delta>2\delta\), even with coupling shifts set to zero.

Derivation. The cubic classical remainder is bounded by \(Cr(r^2/t)\), the amplitude variation by \(Cr\), and \(t\le t^{1/2-\delta}\) for small \(t\). This proves the asserted inequality. The following power counting explains the hypotheses without establishing their uniformity. The leading saddle contributions are \(S_{\rm cl}+\frac12\log\det H\), where \(S_{\rm cl}\) is the minimum over the four midpoints of the exponent \(\sum_j[d(Q_j^{-1},m_j)^2+d(m_j,P_j)^2]/t_j+d(e,m_{\partial g})^2/(2t_m)\) (the \(w=0\) exponents) and \(H\) its Hessian there. One must also add \(\sum_j\log k_{t_j}(e^{X_j})\) from the bridge denominators and include the heat-kernel and Haar amplitudes. A remainder order for the bare integral alone does not supply the displayed expansion. (i) Classical part. The Lie-algebra linearization of \(S_{\rm cl}\) is the abelian problem, so its quadratic part is the free form, which is \(\mathcal D_0\) after the old-face subtraction. The rest starts at cubic order: Baker–Campbell–Hausdorff terms such as the N1 commutator, \(\langle X_{\rm mid},[X_{12},X_{13}]\rangle/t\), and quartic invariants \(|X|^4/t\). On the small-field set, \(|X|^3/t\le t^{1/2-\delta}|X|^2/t\) and \(|X|^4/t\le t^{1-2\delta}|X|^2/t\), both inside the remainder of P(\(\alpha\)) with \(\alpha=\frac12-\delta\). (ii) One-loop part. Mixed quadratic boundary terms can survive cube reflections because reflections also permute the faces. For example, \(\langle\Delta_1X_{23},\Delta_2X_{31}\rangle\), with \(\Delta_i\) the difference between opposite faces normal to direction \(i\), is even under each coordinate reflection: each factor transforms as a pseudoscalar. Axis permutations can be accommodated by summing the corresponding pairs. The linear Bianchi relation among the three differences can rewrite such terms as squares of differences, which still couple opposite faces. Thus reflection invariance permits more than a sum of individual plaquette norms. The coupled Hessian can also have linear off-diagonal blocks from cubic BCH terms, whose squared contribution enters \(\log\det H\) at quadratic order. Every bounded quadratic amplitude term is \(O(r^2)=O(t)(r^2/t)\) and already fits the stated remainder since \(\alpha<1\). Identifying plane-diagonal coupling shifts requires a separate calculation. The present bound makes no such identification. (iii) The old-face ratio \(k_{2t}/k_t\) is even in \(\log U_f\) and contributes a constant and a transverse coupling shift. \(\square\)

Proposition 7 is power counting made explicit for one cube; its hypothesis, a uniform Laplace remainder on the small-field set, is exactly the missing estimate. Proposition 6 computes the curvature part of (ii) for \(SU(2)\). For a whole mid-plane the same bookkeeping applies, but \(H\) couples all mid-edges and \(\log\det H\) is non-local in the fluxes; controlling it is the cluster expansion of open problem 2 in the README.

Under P(\(\alpha\)) the per-step error on smooth observables is \(O((\lambda_3a)^\alpha)\) and the coupling shifts sum to a finite renormalization, because \(\sum_n\lambda_3a_n=2\lambda_3a_0\): the superrenormalizable pattern of the refinement note’s table, now with the term to be bounded identified exactly. P(\(\alpha\)) is the directional form of what Balaban’s three-dimensional ultraviolet stability controls in his block-averaging scheme (Balaban 1985, CMP 102, 255, metadata); whether his estimates transfer to the heat-kernel directional scheme is unverified. P(\(\alpha\)) yields ultraviolet control and says nothing about the gap: the long-distance spectral estimate of the refinement note §7 remains a separate obligation.

5. Four dimensions

In \(D=4\) a halving of direction 1 inserts parallel copies in the three planes \((23),(24),(34)\), and the mid-space is a three-dimensional lattice gauge theory with bridge-distributed edges. By the table in §2 the heat time returns to itself after an isotropic step, so the whole one-loop running \(g_0^{-2}(2a)=g_0^{-2}(a)-2b_0\log2+O(g_0^2)\), with \(b_0=11/(16\pi^2)\) for \(SU(3)\) in the convention \(g_{\rm lat}^2=\lambda_4\), must come out of the four \(\Psi\) factors of one isotropic step, to leading order and within the class of actions of P(\(\alpha\)). N2 and N3 are \(O(t)\) relative in both dimensions. In \(D=4\), \(t=g^2\) does not decrease along the sequence, so their step-independent shift of \(g^{-2}\) is the whole one-loop running, \(-2b_0\log2\) per isotropic step; in \(D=3\) the shifts, \(O(\lambda_3a_n)\) relative, are summable. This is the exact location of marginality in the insertion language. The corresponding hypothesis is H1 of the conditional theorem, and Balaban’s four-dimensional work (Balaban 1987, metadata) is the established partial control.

6. Consequence for STATE

Item 1 of STATE is answered in its “exact obstruction” form, and its first rungs are proved. One directional refinement step factors exactly into series moves, which close, and a parallel insertion \(\Psi\) (Proposition 1). \(\Psi\) is exact for the free field (Proposition 2), for the \(U(1)\) cube (Proposition 3) and the \(U(1)\) mid-plane (Proposition 4), and for \(U(1)\) the unperturbed one-step bound has error density \(e^{-\pi^2/(8\lambda_3a)}\) (Theorem 5). For non-abelian groups the midpoint curvature term is explicit (Proposition 6 and its group-general form), and the finite-cube bound is conditional on a full normalized Laplace expansion (Proposition 7). The next step is the uniform Laplace remainder that turns Proposition 7 into a theorem for \(SU(2)\) and \(SU(3)\); after it, a mid-plane by small-field cluster expansion, and stability under iteration. For the paper, the series/parallel split is the organizing statement of the dimension comparison: Newton’s time insertion is a series move closed by one scalar; two-dimensional gauge theory has only series moves; three and four dimensions add \(\binom{D-1}{2}\) parallel insertions whose elimination is the renormalization problem; the error density of those insertions in physical units gives a sufficient test for which lattice effects the limit discards.