Gauge flow: exact conjugation, Jacobian locality and spectral truncation

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Finite-lattice Wilson flow is a diffeomorphism and has a unitary pullback. It preserves spectra and transformed multiplication norms. Its first derivative has a local path-expansion bound, but that estimate alone controls no relative Hamiltonian truncation error.

The continuum auxiliary covariant heat equation below has an absolute Gaussian kernel-tail bound under a full-interval curvature hypothesis. The exact variation of the usual gauge-modified flow contains additional linear terms. A stationary chromomagnetic background has an aligned mode growing at \(gB-k_\parallel^2\), after the Landau cost is included. These are different operators and domains. Their proofs identify what a proposed spectral reduction would still have to control; they do not supply a physical mass gap.

Finite-lattice invariances

On \(\mathcal M=G^{\mathcal E}\), the Wilson flow \[\dot V_s(x,\mu)=-g^2\{\partial_{x,\mu}S_w(V_s)\}V_s(x,\mu), \qquad V_0=U,\] has a smooth all-time flow on the compact finite-dimensional manifold; \(\Phi_t^{-1}=\Phi_{-t}\). Lüscher, arXiv:1006.4518v3, (1.4) and the following sentence (passage-level reading in the local companion), supplies the finite-lattice flow and action monotonicity. Let \(\rho_t=|\det D\Phi_t|\) relative to Haar measure and define \[(W_t\psi)(U)=\rho_t(U)^{1/2}\psi(\Phi_t(U)).\] Change of variables proves \(W_t\) is unitary. Hence \(W_tHW_t^*\) has exactly the spectrum, multiplicities and gap of \(H\). Also \[W_tM_fW_t^*=M_{f\circ\Phi_t},\qquad \|f\circ\Phi_t\|_\infty=\|f\|_\infty,\] because \(\Phi_t\) is onto. Thus any criterion using only the spectrum of \(H_0\) and the norm of \(\phi\) is unchanged under the same unitary conjugation of the pair. This does not show invariance under an unrelated new splitting or under projection: flow can mix links and change the supports and hypotheses of a local theorem.

For the Kogut–Susskind Hamiltonian \[H=\frac{\hbar c}{a}\left[\frac{g^2}{2}K+\frac2{g^2}V\right],\quad K=\sum_\ell(-\Delta_\ell),\quad V=\sum_p(N-\operatorname{Re}\operatorname{tr}U_p),\] flow gives \(0\le V\circ\Phi_t\le V\) for \(t\ge0\), while its supremum is unchanged. It follows that \(\langle\psi,(V\circ\Phi_t)\psi\rangle\le\langle\psi,V\psi\rangle\) for a fixed state \(\psi\); if both state and observable are transformed consistently, their expectation is unchanged. In particular this is no proof that the physical ground-state measure has uniform local large-field tails. The corrected Agmon analysis controls integrated global excess only; the magnetic identities show that its simple Gibbs-comparison repair does not supply the missing local estimate.

Kinetic metric, free cancellation and lattice derivatives

The electric operator becomes a second-order operator whose coefficients are determined by the pullback metric and the Jacobian. Flow mixes links: the transformed operator is generally nonlocal. The scale \(\sqrt{8t}\) is a smoothing radius, not a strict finite interaction range. Bounds on the flow Jacobian do not by themselves prove relative operator-error bounds for a truncated Hamiltonian; the one-step distinction is explained below.

In a finite-volume regularized free abelian model in Coulomb gauge, a nonzero mode obeys \(A_k\mapsto e^{-tk^2}A_k\) and the Jacobian is a constant. With a canonical momentum convention consistent with the Hamiltonian, the transformed quadratic coefficients acquire reciprocal factors \(e^{-2tk^2}\) and \(e^{+2tk^2}\), so their product and oscillator frequency \(\omega_k=c|k|\) are unchanged. This illustrates the exact unitary result. Treating the ratio of those coefficients as a Yarotsky smallness parameter is not justified: Fourier modes are not the local site decomposition required by that theorem and a free mode’s coordinate and momentum are unbounded. The former assertion that an arbitrary local criterion can always be improved without limit by conjugation is withdrawn.

One lattice step: coupling cancellation and local derivatives

For the stated Wilson normalization the \(g^2\) in the flow equation cancels the \(1/g^2\) in the action. At fixed dimensionless time, the finite-lattice vector field is therefore independent of \(g\). Choosing \(\sqrt{8t}=2a\) gives \(t/a^2=1/2\). The link-local vector field has bounded derivatives at fixed compact group and bounded lattice incidence. More explicitly, write its linearization in link coordinates as \(\dot J=A(s)J\) with \(A_{xy}=0\) beyond a fixed interaction distance and row sums bounded by \(B\), uniformly in the lattice size. Expanding the integral equation, a matrix element at graph distance \(d\) can receive a contribution only after at least \(d\) local factors, so \[\|J_{xy}(t)\|\le\sum_{n\ge d}\frac{(B|t|)^n}{n!}.\] Here time and \(B\) are in lattice units, and any finite-range edge in \(A\) is one step of this graph. This elementary bound is uniform in volume and coupling under the stated derivative and frame bounds; its constants depend on the group and local geometry. It controls the first flow derivative, not automatically the inverse derivative, Jacobian potential, unbounded kinetic form or full Hamiltonian truncation. The former claim \(Ce^{-cK}\) for the relative operator error and the assertion that \(K=3\) suffices are withdrawn: no constants or relative-form proof were provided. Compactness also does not itself prove normalized large-field rarity or multiscale stability.

Exact continuum variation and its gauge convention

Take Lüscher’s gauge-modified continuum flow \[\partial_sB_\mu=D_\nu G_{\nu\mu}+D_\mu q, \qquad q=\partial_\nu B_\nu.\] The definition is from arXiv:1006.4518v3, equations (1.1)–(1.2) and (2.2), passage level via the local companion. Repeated spatial indices are summed. Write \(u=\delta B\).

Proposition 1 (exact variation and auxiliary gauge choice). The Jacobian of this gauge-modified flow obeys \[\partial_su_\mu=D^2u_\mu+2[G_{\mu\nu},u_\nu] +D_\mu(\partial_\nu u_\nu-D_\nu u_\nu)+[u_\mu,q].\] The ungauged flow instead has linearization \[\partial_su_\mu=D^2u_\mu-D_\mu D_\nu u_\nu +2[G_{\mu\nu},u_\nu].\] Adding the linear background-covariant gauge term \(D_\mu D_\nu u_\nu\) to the latter defines the auxiliary evolution used in the auxiliary bounds below. Identifying it with another flow Jacobian requires an additional gauge comparison, not supplied here.

Proof. Since \(\delta D_\nu=[u_\nu,\cdot]\) and \(\delta G_{\nu\mu}=D_\nu u_\mu-D_\mu u_\nu\), \[\delta(D_\nu G_{\nu\mu}) =D^2u_\mu-D_\nu D_\mu u_\nu+[u_\nu,G_{\nu\mu}] =D^2u_\mu-D_\mu D_\nu u_\nu+2[G_{\mu\nu},u_\nu].\] Here \([D_\nu,D_\mu]u_\nu=[G_{\nu\mu},u_\nu]\). The gauge term has the exact variation \[\delta(D_\mu q)=D_\mu\partial_\nu u_\nu+[u_\mu,q],\] not \(D_\mu D_\nu u_\nu\). Adding these formulas proves the first identity; omitting the gauge term or adding the specified linear one proves the other assertions. \(\square\)

Thus the original cancellation “to the same order” was unjustified: the omitted terms are themselves linear in \(u\). The propositions below are about \(J\), not about the full \(D\Phi_t\) in Lüscher’s convention. At \(B=0\) both equations reduce to the free heat equation. At a stationary background, background-transverse modes of the ungauged linearization also agree with the auxiliary generator; no general preservation of that transverse condition is asserted.

Auxiliary Gaussian bound

Equip the compact Lie algebra with an invariant positive inner product, and its vector fields with \(|u|^2=\sum_\mu|u_\mu|^2\). The connection is metric-compatible because \(\operatorname{ad}(B_\mu)\) is skew-adjoint. Assume \(B(s,x)\) is smooth on \([0,t]\times\mathbb R^3\), with bounded coefficients and spatial derivatives sufficient for the covariant parabolic propagators and kernels below, and define \[(\mathcal C(s,x)u)_\mu=\sum_\nu[G_{\mu\nu}(s,x),u_\nu],\qquad \Gamma(s)=\sup_x\|\mathcal C(s,x)\|_{\rm op},\qquad A_t=2\int_0^t\Gamma(s)\,ds.\] Assume \(\Gamma\in L^1([0,t])\). These are hypotheses, not estimates derived from action monotonicity.

Proposition 2. Under these assumptions the auxiliary propagator satisfies \[\|J(t,0;x,y)\|_{\rm op}\le e^{A_t}K_t^{\rm free}(x-y), \qquad K_t^{\rm free}(z)=(4\pi t)^{-3/2}e^{-|z|^2/(4t)}.\] In particular, if \(\Gamma(s)\le\Gamma_*\) then \(A_t\le2t\Gamma_*\).

Proof. First consider \(\partial_s v=D(s)^2v\). Metric compatibility gives \[(\partial_s-\Delta)|v|^2=-2\sum_\alpha|D_\alpha v|^2.\] Apply the chain rule to \(h_\epsilon=(|v|^2+\epsilon^2)^{1/2}\) and use \(|\partial_\alpha|v|^2|\le2|v|\,|D_\alpha v|\). It follows that \((\partial_s-\Delta)h_\epsilon\le0\). Scalar comparison followed by \(\epsilon\downarrow0\) yields, for the time-dependent covariant propagator \(U(t,r)\), \[|U(t,r)f|(x)\le(e^{(t-r)\Delta}|f|)(x).\] With the assumed kernels this also gives \(\|U(t,r;x,y)\|_{\rm op}\le K_{t-r}^{\rm free}(x-y)\), by testing localized initial data. This proves the needed Kato domination here; no bibliographic sharpness claim is used.

Set \(M(s)=2\mathcal C(s)\). Duhamel uses the two-time propagator, \[J(t,0)=U(t,0)+\int_0^t U(t,s)M(s)J(s,0)\,ds.\] In the term with \(n\) insertions, each multiplication factor has norm at most \(2\Gamma(s_i)\) and scalar heat kernels convolve to \(K_t^{\rm free}(x-y)\). The ordered time integral is bounded by \(A_t^n/n!\). Summing this absolutely bounded series proves the result. \(\square\)

Norm convention. If instead \(g_*=\sup_{s,x,\mu,\nu}\|\operatorname{ad}(G_{\mu\nu}(s,x))\|_{\rm op}\), then \(\Gamma_*\le2g_*\) in three dimensions, or \(\Gamma_*\le(d-1)g_*\) in \(d\) dimensions. Indeed \[|(\mathcal Cu)_\mu|\le g_*\sum_{\nu\ne\mu}|u_\nu|,\] and Cauchy–Schwarz, followed by summation over \(\mu\), gives \(|\mathcal Cu|^2\le(d-1)^2g_*^2|u|^2\). The former unspecified “adjoint operator norm” cannot silently omit this index factor.

At \(B=0\), \(J=K_t^{\rm free}I\) and the pointwise bound is attained. This free-case equality does not establish optimality of the curvature exponent for nonzero backgrounds.

Absolute tails and the missing metric comparison

Define \(J_R\) by cutting its kernel to \(|x-y|\le R\), and set \[T_R=J-J_R,\qquad \delta(R,t)=e^{A_t}\int_{|z|>R}K_t^{\rm free}(z)\,d^3z.\] Proposition 2 bounds both row and column integrals of \(\|T_R(x,y)\|_{\rm op}\) by \(\delta(R,t)\). Consequently the Schur test (or the scalar convolution majorant) gives the absolute bound \[\|T_R\|_{L^2\to L^2}\le\delta(R,t).\] It is not normalized by \(\|J\|\) or by a kinetic quadratic form.

Corollary 3 (kernel-tail budget). Put \(q=R/(2\sqrt t)\). Then \[\delta(R,t)=e^{A_t} \left[\operatorname{erfc}(q)+\frac{2q}{\sqrt\pi}e^{-q^2}\right] \le e^{A_t-q^2}\left(1+\frac{2q}{\sqrt\pi}\right).\] Thus at \(R=\kappa\sqrt{8t}\) a sufficient condition for \(\|J-J_R\|_{2\to2}\le\epsilon\) is \[2\kappa^2\ge A_t+ \log\left(1+\frac{2\sqrt2\kappa}{\sqrt\pi}\right) +\log(1/\epsilon),\qquad 0<\epsilon<1.\] For every finite \(A_t\) this is achieved at sufficiently large \(\kappa\). It is a sufficient condition only.

Proof. Radial integration and \(r=2\sqrt t\,v\) give \[\int_{|z|>R}K_t^{\rm free}(z)\,d^3z =\frac4{\sqrt\pi}\int_q^\infty v^2e^{-v^2}dv =\operatorname{erfc}(q)+\frac{2q}{\sqrt\pi}e^{-q^2}.\] For \(q\ge0\), \(\operatorname{erfc}(q)\le e^{-q^2}\): their difference is zero at zero, has derivative \(2e^{-q^2}(1/\sqrt\pi-q)\), and tends to zero at infinity. The stated inequality and logarithmic budget follow. \(\square\)

At \(\kappa=1\), even the exact free discarded mass is \(\operatorname{erfc}(\sqrt2)+(2\sqrt2/\sqrt\pi)e^{-2}\). This is a fixed positive number, not a parameter tending to zero on small fields. A growing upper bound for \(A_t\) implies neither a large actual tail nor a lower bound on a truncation error.

The Hamiltonian conversion remains open. The kinetic metric in the finite-lattice kinetic construction above, involves quadratic Jacobian combinations and inverse metrics; the transformed operator also has density derivatives and lower-order terms. Already for bounded operators \(F\) and \(F_R\), with \(E=F-F_R\), \[\|F^*F-F_R^*F_R\| \le2\|F\|\,\|E\|+\|E\|^2.\] This follows by expanding \(F_R=F-E\). It is an absolute estimate; a relative comparison additionally needs a positive lower bound on the relevant metric or a bound on its inverse. The free continuum heat multiplier \(e^{-t|k|^2}\) has no bounded inverse on \(L^2\). The finite lattice has an invertible flow Jacobian, but uniform inverse, configuration-derivative and relative form estimates still need proof. The auxiliary \(J\) above is not identified with that lattice Jacobian. Thus the original Hamiltonian interpretation of \(\delta\) remains a heuristic motivation, not a proved relative truncation theorem.

A sufficient curvature budget

Write \(\ell=\sqrt{8t}\). If the block norm satisfies \[\Gamma(s)\le\frac\eta{\ell^2}\quad(0\le s\le t),\] then \(A_t\le\eta/4\) and the kernel-tail corollary yields \[\|J-J_R\|_{2\to2} \le\left(1+\frac{2\sqrt2\kappa}{\sqrt\pi}\right) \exp\left[\frac\eta4-2\kappa^2\right].\] The original \(2\eta\) growth exponent in this substitution was incorrect. In component norm, the hypothesis \(g_*\le\eta/\ell^2\) gives instead \(A_t\le\eta/2\) in three dimensions by the index bound of the auxiliary-bound section.

This budget motivates comparison with small/large-field splitting in constructive renormalization, but neither forces that split nor controls all steps of the finite-lattice construction above. The comparison to Balaban’s series (Commun. Math. Phys. 122 (1989) 355, metadata level) is context, not a derivation of its hypotheses. No unproved decay of a “typical” interacting curvature is used here.

Action monotonicity and curvature norms

For smooth finite-action solutions of the ungauged gradient flow, with decay allowing integration by parts and the gradient normalization, \[\frac{d}{ds}S(B_s)=-\|\nabla S(B_s)\|_2^2\le0.\] The added gauge term is an infinitesimal gauge transformation and does not change this identity under the same boundary assumptions. Since \(S\) is proportional to \(\int|G|^2\), it gives \[\|G(t)\|_2^2\le\|G(0)\|_2^2.\] It supplies no \(L^\infty\) bound by itself, and Proposition 2 requires control over the whole flow interval, not just its endpoint.

Free or abelian smoothing, corrected. Around the zero connection \(J\) is the heat semigroup. In an abelian theory the curvature itself obeys the heat equation, so Cauchy–Schwarz and the Gaussian convolution identity give \[\|e^{t\Delta}f\|_\infty \le\|K_t^{\rm free}\|_2\,\|f\|_2 =(8\pi t)^{-3/4}\|f\|_2.\] For abelian initial curvature this bounds \(\|G(t)\|_\infty\) by \((8\pi t)^{-3/4}\|G(0)\|_2\) in the Euclidean component norm. For fixed nonzero initial norm its ratio to \(1/t\) is proportional to \(t^{1/4}\|G(0)\|_2\): it tends to zero for small \(t\) and grows for large \(t\), the reverse of the former remark. Integrating this estimate over \(0<s<t\) also gives a finite budget proportional to \(t^{1/4}\|G(0)\|_2\), using the finite-dimensional conversion to the block norm. This is an abelian estimate, not an interacting theorem. The coefficient \((4\pi t)^{-3/4}\) formerly displayed is a looser valid bound; the expression above is the exact heat-kernel \(L^2\) norm.

Monotonicity alone does not supply the missing norm. A fixed \(L^2\) curvature norm permits concentration in small regions with arbitrarily large height. For example rescale a smooth compactly supported abelian connection by \(B_\lambda(x)=\lambda^{1/2}B(\lambda x)\); its curvature is \(G_\lambda(x)=\lambda^{3/2}G(\lambda x)\), with constant \(L^2\) norm and unbounded supremum as \(\lambda\to\infty\). This disproves a pointwise estimate from the action bound alone, not parabolic estimates using additional evolution information.

The Hamiltonian formulation also has configuration probabilities, for example its ground-state density; see the transfer note, §§1–2 (full-read). Such measure estimates do not automatically give the signed operator comparisons needed for spectral stability. The distinction emphasized in the operator-inequality note, §3 (full-read), is relevant here; this note does not assume its claimed Hamiltonian truncation estimate or conclude that every Hamiltonian route fails.

Instability of the full background generator

Let \(\mathfrak g\) be a compact matrix Lie algebra with invariant inner product \(\langle X,Y\rangle=-\operatorname{tr}(XY)\). Then \(\operatorname{ad}(Z)\) is antisymmetric: \(\langle[Z,X],Y\rangle=-\langle X,[Z,Y]\rangle\).

Lemma (curvature symmetry). At each point, on \(\mathfrak g\)-valued vectors with inner product \(\sum_\mu\langle u_\mu,v_\mu\rangle\), the operator \((Mu)_\mu=2[G_{\mu\nu},u_\nu]\) is symmetric.

Proof. \[\langle Mu,v\rangle =2\sum_{\mu\nu}\langle[G_{\mu\nu},u_\nu],v_\mu\rangle =-2\sum_{\mu\nu}\langle u_\nu,[G_{\mu\nu},v_\mu]\rangle =2\sum_{\mu\nu}\langle u_\nu,[G_{\nu\mu},v_\mu]\rangle =\langle u,Mv\rangle.\] The two antisymmetries cancel. \(\square\)

A finite-dimensional symmetric operator of norm \(N\) has an eigenvalue of modulus \(N\); symmetry does not determine its sign. Moreover, positive eigenvalues of \(M\) alone do not determine those of \(D^2+M\). For compactly supported fields, integration by parts gives \[\langle u,Lu\rangle=-\sum_\alpha\|D_\alpha u\|_2^2 +\langle u,Mu\rangle.\] The first term must be retained.

The preceding Duhamel bound uses \(\Gamma=\frac12\sup\|M\|\); its exponent is an upper bound only. The following example retains both terms of \(L\).

The charged Landau mode

We work with the Yang–Mills flow linearized at a stationary background \(\bar A\), in background-covariant gauge. This specifies the operator rather than assuming that every gauge-modified flow has the same Jacobian. With \(D=\partial+[\bar A,\cdot]\) and \(D_\nu G_{\nu\mu}=0\), variation of the ungauged flow gives \[\partial_su_\mu=D^2u_\mu-D_\mu D_\nu u_\nu +2[G_{\mu\nu},u_\nu].\] Adding the linear background-gauge term \(D_\mu D_\nu u_\nu\) gives \(L\). Equivalently, on a mode with \(D_\nu u_\nu=0\), the ungauged linearization already agrees with \(L\). This does not repair or identify the full Jacobian in every gauge convention.

Choose an \(SU(2)\) colour sector with \([T^3,E_+]=iE_+\), and absorb the coupling into the connection, so \(G_{12}=bT^3\), \(b=gB>0\). In symmetric gauge on the transverse plane, \[\bar A_1=-\frac b2x_2T^3,\qquad \bar A_2=\frac b2x_1T^3.\] On the \(E_+\) component, \[D_1=\partial_1-\frac{ibx_2}{2},\qquad D_2=\partial_2+\frac{ibx_1}{2}.\] Writing \(Q=D_1-iD_2\), integration by parts and \([D_1,D_2]=ib\) give \(Q^*Q=-(D_1^2+D_2^2)-b\) and \([Q,Q^*]=2b\). Thus the oscillator ladder has transverse magnetic-Laplacian energies \((2n+1)b\), \(n=0,1,\ldots\). To fix the factor without relying on a bibliographic convention, its lowest mode can be checked directly: \[f(x_1,x_2)=e^{-b(x_1^2+x_2^2)/4},\qquad (D_1-iD_2)f=0,\qquad (D_1^2+D_2^2)f=-bf.\] On the two transverse vector components, \[M=\begin{pmatrix}0&2ib\\-2ib&0\end{pmatrix},\qquad M\binom f{-if}=2b\binom f{-if}.\] The charged complex notation represents real Lie-algebra fields by adding the conjugate charge sector. The vector \[u=(f,-if,0,\ldots)e^{ik_\parallel\cdot x_\parallel}\] is background-transverse, since \(D_1f-iD_2f=0\). Consequently \[D^2u=-(b+k_\parallel^2)u,\qquad Lu=(b-k_\parallel^2)u.\] This supplies both the positive eigenvalue of \(M\) and the Laplacian cost on the same vector. The two transverse polarizations at general Landau level have the spectral values \[\omega^2=k_\parallel^2+(2n+1)b\mp2b\] for \(-L\). Here \(k_\parallel^2\) is the sum of the squared momenta in directions outside the magnetic plane: one for spatial three-dimensional flow, two for four-dimensional Euclidean flow. The unstable branch is \[\omega^2=k_\parallel^2-b<0\quad\text{when }k_\parallel^2<b,\] and its gradient-flow growth is \[e^{-\omega^2s}=e^{(b-k_\parallel^2)s}.\] At \(k_\parallel=0\) the rate is \(b\), half the exponent coefficient \(2b\) in the Duhamel majorant. That example therefore does not saturate the majorant. Whether a better universal bound holds for other backgrounds is not settled by this calculation.

On the whole space the longitudinal plane waves are generalized modes; wave packets supported in \(k_\parallel^2<b\) give growing \(L^2\) perturbations, with rates approaching \(b\). Periodic longitudinal boundary conditions admit the zero-momentum mode. The constant field on the transverse plane has infinite total action; the calculation is a stationary-background spectral model, not a finite-action four-dimensional construction or a statement for arbitrary finite boxes.

This is the instability historically associated with Nielsen and Olesen, “An unstable Yang–Mills field mode,” Nucl. Phys. B144 (1978) 376–396 (abstract read; the original spectral passage was not read here). The numerical factors used above are supported by the displayed calculation, not by that abstract or a normalization left implicit.

Action descent and growing perturbations

For the unmodified gradient flow with its gradient-metric normalization, \[\frac{d}{ds}S(A_s)=-\|\nabla S(A_s)\|_2^2\le0.\] A constant abelian curvature has \(D_\nu G_{\nu\mu}=0\) and is stationary. Its transverse unstable modes give negative Hessian directions, \(\operatorname{Hess}S=-L\) on the background-transverse sector in the normalization of the auxiliary-bound section. Nearby trajectories can separate at rate \(b-k_\parallel^2\) while descending the action. For the infinite-action background these are statements about the local equation and quadratic variation on admissible perturbations, rather than differentiation of an infinite total action.

The original comparison with the valley note is also corrected: constant commuting connections in its flat holonomy valley have \(G=0\). They are not the nonzero constant chromomagnetic background considered here. The valley’s zero-point potential is a different calculation and gives no justification for this saddle or its growth rate.

Free-field size is a benchmark, not a supremum bound

From the free-field susceptibility calculation, the flowed magnetic two-point function at coincident points is \[\big\langle b_{t,i}(0)b_{t,i}(0)\big\rangle =\int\!\frac{d^3k}{(2\pi)^3}\sum_i\tilde G_{ii}(k) =\int\!\frac{d^3k}{(2\pi)^3}\,\frac{g^2k}{2}\,e^{-2tk^2}\cdot2 =\frac{g^2}{2\pi^2}\int_0^\infty\!k^3e^{-2tk^2}dk,\] and \(\int_0^\infty k^3e^{-\alpha k^2}dk=1/(2\alpha^2)\) with \(\alpha=2t\) gives \[\big\langle|b_t|^2\big\rangle=\frac{g^2}{2\pi^2}\cdot\frac1{8t^2}=\frac{g^2}{16\pi^2t^2}, \qquad \big\langle|b_t|^2\big\rangle^{1/2}=\frac{g}{4\pi\,t}.\] The flow time is the only scale, so the dimensionless combination is \[t\,\big\langle|b_t|^2\big\rangle^{1/2}=\frac{g}{4\pi},\] independent of \(t\) and of the lattice spacing. In the interacting theory at scale \(\ell=\sqrt{8t}\) the same estimate with the running coupling \(g(\ell)\) is the leading perturbative statement.

The displayed convention absorbs the quantum factor in \(g^2\); writing \(g^2=\hbar g_{\rm cl}^2\) restores physical \(\hbar\) explicitly. Rotational invariance makes each component’s variance \(\sigma^2=g^2/(48\pi^2t^2)\), one third of this trace. Their joint Gaussian law is \(\sigma^2 I_3\). Its radial density is proportional to \(r^2e^{-r^2/(2\sigma^2)}\), so in the large-threshold regime \[\log\mathbb P\left(|b_t|>\frac{\eta_t}{t}\right) =-\frac{24\pi^2\eta_t^2}{g^2} +O\big(\log(\eta_t/g)\big),\qquad \eta_t/g\to\infty.\] The polynomial radial prefactor is not a constant uniform in threshold. Using the trace variance as the variance of each component would instead give the incorrect \(8\pi^2\) coefficient. With \(\ell^2=8t\), a threshold \(\eta_\ell/\ell^2\) means \(\eta_t=\eta_\ell/8\); even this particular free-field logarithmic exponent becomes \(-3\pi^2\eta_\ell^2/(8g^2)\). Neither value is a sharp-block or an interacting constant. A sharp average uses a different covariance; nonabelian field averaging also requires a gauge convention or an invariant block-\(L^2\) observable, with its own covariance.

A single-point Gaussian variance is neither a bound on the field supremum over a continuum region nor on the whole-interval curvature block norm. Substitution of \(g/(4\pi t)\) into a deterministic kernel hypothesis is only a model estimate. Even for a Gaussian vector field, its component variance, radial tail, spatial correlation and continuum supremum require their own constants; a \(V/(8t)^{3/2}\) independent-cell count does not prove such a theorem. Replacing \(g\) by a running coupling is a perturbative suggestion, not a nonperturbative interacting probability bound.

At \(R=\kappa\sqrt{8t}\), the earlier schematic substitution gave \(\exp[g/(2\pi)-2\kappa^2]\) and \(\kappa^2\ge g/(4\pi)+1/2\) for tolerance \(e^{-1}\). The exact tail above additionally has a polynomial prefactor and uses a supremum over all flow times. These numbers therefore remain illustrative scale bookkeeping, not a Hamiltonian truncation guarantee at any coupling. Hamiltonian ground states have configuration probabilities; their transfer identity does not turn an absolute auxiliary kernel bound into the required signed operator comparison.

Effective spectral reductions

Suppose a scale change constructs effective operators \(H^{(n)}\) at \(\ell_n=2^na\), with a specified low-energy subspace, generated interactions and a spectral comparison. Relative Schur bounds can ignore a common scalar shift but must control the state-dependent remainder and the complementary energy denominator; see the Feshbach reduction, including its ultraviolet correction. A candidate truncation must be compared with the exact transformed operator in a suitable domain and relative-form norm, with constants uniform in volume and with enough control across the cutoff trajectory. One-loop running can estimate how many doublings might be needed, but it is not a proof of the effective action or its arrival in a gapped region.

A state-relative bound, for example \[\langle\psi,\phi\psi\rangle\le\epsilon\langle\psi,(H-E_0)\psi\rangle +\eta\|\psi\|^2\] on a specified low-energy domain, could exploit more information than a supremum norm. Such a bound, the relevant probability estimates, and stability after projection are independent obligations. The blocking analysis gives conditional surface-to-gap comparisons; it does not prove that projection blocking is always worse at every coupling.

Consequence for STATE

The canonical flow argument separates exact unitary invariance, local lattice first derivatives, auxiliary continuum tails, background instability and relative spectral comparison. The explicit curvature budget controls the auxiliary kernel only. A projected or blocked Hamiltonian still needs the inverse metric, density derivatives, signed relative-form error and uniform iteration estimates. Action descent and free-field statistics provide none of those hypotheses automatically.