Gaussian gauge blocking: exact covariance, decay and the nonlinear threshold
For the specified linearized four-dimensional \(SU(3)\) block average, hard conditioning gives an exact Gaussian covariance and the explicit tube estimate \(-\Delta|_{\ker W}\ge4/9\). A distinct soft constraint has gap \(2/9\) for coefficient \(\alpha\ge130\) and the explicit decay criterion below. These are Gaussian statements. No normalized nonlinear polymer expansion or physical continuum gap is proved by them.
The coarse variable already averages two-link paths. Its angle is \(\theta_c=W\theta\), without a second factor of two. Likewise, an \(\ell^\infty\) kernel bound requires the corresponding lattice shell sum, rather than the smaller product sum for \(\ell^1\) distance. These corrections change the former normalization and numerical threshold estimates. The surviving nonlinear estimate is a conditional scaling diagnosis, not a certified Kotecký–Preiss region or a theorem excluding other methods.
Conventions and gauge-compatible averaging
On \(\mathbb Z^4\) use lattice units \(a=1\), Hermitian generators with \(\operatorname{tr}T^aT^b=\delta^{ab}/2\), link angles \(\theta=aA\) and \(U_\ell=e^{i\theta_\ell}\). The dimensionless coupling obeys \(g^2=\hbar g_{\rm cl}^2\) when a classical normalization is supplied. Then \[\frac{S_W}{\hbar}=\frac2{g^2}\sum_p(N-\operatorname{Re}\operatorname{tr}U_p) =\frac1{2g^2}\|d\theta\|^2+O(\theta^3).\] The cubic/quartic terms, Haar correction and nonlinear averaging belong to the interacting remainder. The action-cost treatment distinguishes an action lower bound from normalized interacting rarity.
For blocks \(B\) of side two define \[(W\theta)_{B,\mu}=\frac1{16}\sum_{x\in B} (\theta_{x,\mu}+\theta_{x+\hat\mu,\mu}),\qquad (W_0\lambda)_B=\frac1{16}\sum_{x\in B}\lambda_x.\] The path telescopes, giving the exact intertwining \[Wd=\bar dW_0,\qquad (Wd\lambda)_{B,\mu}=(W_0\lambda)_{B+2\hat\mu}-(W_0\lambda)_B.\] Thus a fine linearized gauge transformation induces a coarse one. This is a linearized identity, not a construction of the nonlinear group-valued map.
The tube for \((B,\mu)\) has three \(\mu\)-layers of eight sites, weighted \((1,2,1)\). Its total weight is 32; the normalization \(1/16\) makes a constant fine angle map to twice that angle, as a coarse link of length \(2a\) requires. Every fine link lies in one or two tubes of its direction.
Exact conditional covariance and the tube gap
Adding \(\|d^*\theta\|^2/(2g^2)\) gives 32 scalar Feynman-gauge problems. Use the infinite-lattice Green function in four dimensions, or fix the constant modes in a finite-volume version before writing inverses. Put \(G=(-\Delta)^{-1}\) and \(K=\ker W\). The minimizer with \(Wf=\bar f\) is \[H\bar f=GW^T(WGW^T)^{-1}\bar f.\] Variational orthogonality gives \[\langle f,-\Delta f\rangle =\langle H\bar f,-\Delta H\bar f\rangle+\langle\xi,-\Delta\xi\rangle, \quad \xi=f-H\bar f\in K.\] Consequently \[C_{\rm fl}=g^2[G-GW^T(WGW^T)^{-1}WG] =g^2(P(-\Delta)P)^{-1}|_K,\] \[\bar S_2(\bar f)=\frac1{2g^2} \langle\bar f,(WGW^T)^{-1}\bar f\rangle,\] where \(P\) projects onto \(K\). Linearized gauge-invariant observables have the same law in Feynman and Landau gauges: their difference is a pure gauge, and \(Wd=\bar dW_0\) makes its coarse image pure gauge as well.
Tube Poincaré bound. For every \(f\in K\), \[\langle f,-\Delta f\rangle\ge\frac49\|f\|^2.\] To prove it, one tube is a \(2\times2\times2\times3\) graph with first nonzero Laplacian eigenvalue 1. For its plain mean \(m\), \[\|f-m\|_T^2\le\sum_{\text{bonds}\subset T}|\partial f|^2.\] The weighted mean vanishes; with \(\bar w=4/3\) and \(\sum_T(w-\bar w)^2=16/3\), \[m=-\frac1{32}\sum_T(w-\bar w)(f-m),\qquad 24m^2\le\frac18\|f-m\|_T^2.\] Hence \(\|f\|_T^2\le(9/8)\sum_{\text{bonds}\subset T}|\partial f|^2\). Summing tubes covers every site and at most doubles each bond, proving \(\|f\|^2\le(9/4)\|\partial f\|^2\). Alternating layers or transverse alternating modes have energy \(4\|f\|^2\) and satisfy the constraints, so the optimal constant lies in \([4/9,4]\).
In particular, \[\|C_{\rm fl}\|\le\frac94g^2,\qquad (\mathbb E\xi_\ell^2)^{1/2}\le\frac32g.\] For one centered Gaussian colour component this implies \[\Pr(|\xi_\ell|\ge\eta)\le2e^{-2\eta^2/(9g^2)}.\] It bounds that Gaussian fluctuation only; a nonzero background shifts the threshold, and nonlinear reweighting requires its own normalized estimate.
For slowly varying fields, \(Wf\simeq2f\) and coarse differences are \(\bar\partial(Wf)\simeq4\partial f\). The coarse site density is \(1/16\). These factors give the same Maxwell coefficient \(1/(2g^2)\) in four dimensions, with relative momentum corrections of order \(\bar k^2a^2\). The variable is \(\theta_c=\bar f\), not \(2\bar f\). This is the leading Gaussian coefficient, with no nonlinear running claim.
Hard conditioning and soft-constraint decay
For hard conditioning, \(P\) is nonlocal. Along the elongated direction, \[WW^T=\frac{48+8(S+S^{-1})}{256},\qquad \operatorname{spec}(WW^T)\subset[1/8,1/4].\] Conjugating \(P(-\Delta)P+(4/9)(1-P)\) by exponential weights motivates the earlier hard-constraint rate near \(7\times10^{-4}\). The displayed projection calculation used \(\|P_\kappa-P\|\le10\kappa+O(\kappa^2)\) without a finite-\(\kappa\) remainder bound. That number is an estimate, not a certified explicit rate in this treatment. In particular a rate for the following different operator cannot silently certify \(C_{\rm fl}\).
Let \(Q=W\) and \(A_\alpha=-\Delta+\alpha Q^TQ\). Write \(f=f_K+f_\perp\). The tube bound, \(\|{-\Delta}\|\le16\) and \(\|Qf_\perp\|^2\ge\|f_\perp\|^2/8\) give \[\langle f,A_\alpha f\rangle\ge \frac29\|f_K\|^2+(\alpha/8-16)\|f_\perp\|^2 \ge\frac29\|f\|^2\quad(\alpha\ge130).\] The elementary first inequality uses \(\|\partial(f_K+f_\perp)\|^2\ge\frac12\|\partial f_K\|^2 -\|\partial f_\perp\|^2\). The soft operator has range two in the elongated coordinate and one in each other coordinate. Since \(Q^TQ\) has nonnegative entries and row/column sum \(1/4\), conjugation in any one coordinate changes it in norm by at most \((e^{2\kappa}-1)/4\). Therefore \[2\sinh\kappa+\frac\alpha4(e^{2\kappa}-1)\le\frac19 \quad\Longrightarrow\quad |A_\alpha^{-1}(x,y)|\le9e^{-\kappa|x-y|_\infty}.\] Choose the coordinate attaining \(|x-y|_\infty\) for the matrix-element bound. At \(\alpha=130\), \(\kappa=10^{-3}\) is a conservative explicit choice; for larger \(\alpha\) use the displayed criterion with that value. The former \(1.7\times10^{-3}\) and its uniform use for all \(\alpha\ge130\) were not justified by that inequality.
Dimock, Rev. Math. Phys. 25 (2013) 1330010, §2.4 (arXiv passage read in the source work), treats soft Green functions by local Neumann inverses and a random-walk parametrix with sufficiently large cubes. That passage gives parameter-dependent constants, not the missing numerical hard rate. Balaban, CMP 95 (1984) 17, is retained at metadata level.
Folded symbol and the unproved analytic strip
For hard conditioning, coarse momentum \(\bar k\in[-\pi/2,\pi/2]^4\) couples \(k=\bar k+\pi n\), \(n\in\{0,1\}^4\). Up to a common normalization, \[w(k)=\prod_{\nu\ne\mu}\cos^2(k_\nu/2)\,\cos^4(k_\mu/2),\qquad \hat k^2=4\sum_\nu\sin^2(k_\nu/2),\] \[R(\bar k)=\sum_{n\ne0}\frac{w(\bar k+\pi n)}{\hat k^2(\bar k+\pi n)}, \qquad C_{00}(\bar k)\ \propto\ \frac{R(\bar k)}{w(\bar k)+\hat k^2(\bar k)R(\bar k)}.\] The original massless pole cancels. Here \(w\) denotes the analytic extension of the real-momentum squared weight, not an absolute square at complex momentum. In an axial strip, the flipped denominators have positive real part for \(\cosh\kappa<2\), and the unfolded weight has modulus at least \(1/32\). The remaining denominator must still be shown zero-free. An order-one strip is a proposed improvement, not an established decay rate; all folded matrix entries, rather than only this diagonal, need control.
What the remainder estimate actually says
A cubic Wilson vertex has the schematic size \(C_3g\), with \(C_3\) formerly estimated as order 10–30 from \(d\xi\) of size \(6g\), \(\xi^2\) of size \(9g^2/4\) and the colour contraction. Quartic and Haar corrections start at order \(g^2\). These size estimates do not prove a connected expansion. If a valid tree interpolation bounds each edge by a specified covariance sum \(B\), and a rooted plaquette entropy bound \((20e)^{n-1}\) is applicable, the schematic smallness test is \(20e^2C_3gB<1\).
For an assumed bound \(|C(x,y)|\le Ag^2e^{-\kappa|x-y|_\infty}\), the correct unweighted sum is \[B:=\sup_x\sum_y|C(x,y)|/g^2\le A\,\mathcal S_4(\kappa),\] \[\mathcal S_4(\kappa) =1+\sum_{n\ge1}(64n^3+16n)e^{-\kappa n} =1+\frac{64q(1+4q+q^2)}{(1-q)^4}+\frac{16q}{(1-q)^2},\quad q=e^{-\kappa}.\] Indeed the \(\ell^\infty\) shell has \((2n+1)^4-(2n-1)^4=64n^3+16n\) sites. Thus \(\mathcal S_4\sim384\kappa^{-4}\). The earlier \(\coth(\kappa/2)^4\) is the sum for \(\ell^1\) distance and is too small here. A weighted polymer norm also spends part of the available decay rate; \(B\) as defined above is unweighted.
Under those extra expansion hypotheses one obtains the illustrative sufficient test \[g<[20e^2C_3A\mathcal S_4(\kappa)]^{-1},\] whose small-rate scaling is \(g=O(\kappa^4)\). The old numerical table mixed norms and different covariances and is superseded by this formula. Even a correct sufficient estimate far from order-one coupling proves no optimality barrier for other organizations. The interacting numerator, normalization, large fields and iteration remain open in this scheme. The later retained-barrier treatment has its own weak-strip domain and does not fill these gaps by substitution.
Consequence for STATE
The maintained output is exact linearized blocking, the \(4/9\) tube gap and the specified soft-constraint decay criterion. Gaussian scale covariance, nonlinear running, normalized rarity and physical spectral transfer remain different assertions. The nonlinear threshold is a conditional diagnostic; no new research is initiated by correcting its normalization and scope.