The weak-coupling gap in a small box reduces to three inequalities through the Schur complement

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Conditional scope. H1–H3 are not proved, even at fixed lattice cutoff; formal power counting does not establish them. A lower bound \(\Delta\ge\delta_1^{(3)}g^{2/3}(1-Cg^{2/3})\,\hbar c/L\) for the torus Hamiltonian with a momentum cutoff follows from a proved gap-transfer lemma and three inequalities about polynomial operators on finitely many oscillators. The lemma: if \(P\) is an orthogonal projection, \(D=\bar PH\bar P\ge\mu\) on \(\operatorname{ran}\bar P\), and the Schur-complement error \(\eta=\sup_{E\le a_1}\|(D-E)^{-1/2}\bar PHP\|^2\) satisfies \(a_1<\mu\), then the two lowest eigenvalues of \(H\) obey \(a_0-\eta\le E_0\le a_0\) and \(E_1\ge a_1-\eta\), so \(\operatorname{gap}(H)\ge\operatorname{gap}(PHP)-\eta\). Applied with \(P\) the projection onto the ground state of the nonzero momentum modes, the three inequalities are: (H1) exciting any nonzero mode costs at least \((2\pi\hbar c/L)(1-Cg^{2/3})\) in the presence of the constant modes and the couplings; (H2) the coupling between the constant-mode sector and the excited nonzero-mode sector has Schur error \(\eta\le Cg^{4/3}\hbar c/L\), which requires the projection to follow the constant mode (Section 3a) and a gauge-invariant cutoff; (H3) the effective constant-mode Hamiltonian \(PHP\), which is C133’s operator plus the nonzero-mode zero-point energy as a function of the constant modes, has gap \(\ge\delta_1^{(3)}g^{2/3}(1-Cg^{2/3})\hbar c/L\). The scaling analysis of the cubic and quartic couplings shows why each inequality is expected with the stated powers, and identifies the smallness parameter as \(g^{2/3}\), Lüscher’s expansion parameter. The lemma is proved; the three inequalities are the open content of the lower side of the small-volume corner. Constants are kept explicit with the conventions of G07 Proposition 7: \(g\) dimensionless, fields of dimension inverse length, every energy a multiple of \(\hbar c/L\). Nothing here is promoted.

1. The gap-transfer lemma

Let \(H\) be self-adjoint, bounded below, with compact resolvent on a Hilbert space \(\mathcal H\), and \(P\) an orthogonal projection with \(P\,D(H)\subset D(H)\); write \(\bar P=1-P\), \(A=PHP|_{\operatorname{ran}P}\), \(D=\bar PH\bar P|_{\operatorname{ran}\bar P}\), \(B=PH\bar P\). Let \(a_0\le a_1\le\cdots\) be the eigenvalues of \(A\) and \(E_0\le E_1\le\cdots\) those of \(H\).

Lemma 1. Suppose \(D\ge\mu\) and, for some \(\eta\ge0\), \(\|(D-E)^{-1/2}B^*\|^2\le\eta\) for all \(E\le a_1\), with \(a_1<\mu\). Then \[a_0-\eta\le E_0\le a_0,\qquad E_1\ge a_1-\eta,\qquad E_1-E_0\ \ge\ (a_1-a_0)-\eta .\]

Proof. For \(E<\mu\), \(D-E\) is invertible and the Schur complement of \(H-E\) with respect to the decomposition \(\operatorname{ran}P\oplus\operatorname{ran}\bar P\) is \(F(E)=A-E-B(D-E)^{-1}B^*\) on \(\operatorname{ran}P\). The block factorization \[H-E=\begin{pmatrix}1&B(D-E)^{-1}\\0&1\end{pmatrix} \begin{pmatrix}F(E)&0\\0&D-E\end{pmatrix} \begin{pmatrix}1&0\\(D-E)^{-1}B^*&1\end{pmatrix}\] is a congruence, so by Sylvester’s law of inertia (the number of negative eigenvalues, finite here because \(H\) has compact resolvent and \(E<\mu\le\inf D\)) the number of eigenvalues of \(H\) below \(E\) equals the number of negative eigenvalues of \(F(E)\), since \(D-E>0\) contributes none. The hypothesis gives \(0\le B(D-E)^{-1}B^*\le\eta\), hence \(A-E-\eta\le F(E)\le A-E\) and, by min-max, the number of negative eigenvalues of \(F(E)\) lies between \(\#\{k:a_k<E\}\) and \(\#\{k:a_k<E+\eta\}\). Taking \(E\) slightly above \(a_0\) shows \(E_0\le a_0\); taking \(E=a_0-\eta\) shows no eigenvalue of \(H\) lies below \(a_0-\eta\); taking \(E=a_1-\eta\) shows at most one eigenvalue of \(H\) lies below \(a_1-\eta\), so \(E_1\ge a_1-\eta\). \(\square\)

The lemma transfers a gap from the compressed operator \(PHP\) to \(H\) at the price \(\eta\), which is the size of the coupling between \(\operatorname{ran}P\) and its complement measured against the energy \(D-E\) needed to leave \(\operatorname{ran}P\). Everything below is the choice of \(P\) and the estimation of \(a_1-a_0\), \(\mu\) and \(\eta\).

2. The cutoff torus Hamiltonian and the projection

On \(T^3_L\) in temporal gauge, with \(SU(2)\) for definiteness, momentum cutoff \(|k|\le\Lambda\) and the convention of G07 Proposition 7 (\(S=-\frac1{4g^2}\int F^aF^a\)), decompose \(A_i(x)=a_i+\tilde A_i(x)\) into the constant mode and the nonzero modes \(\tilde A_i(x)=L^{-3}\sum_{0<|k|\le\Lambda}\hat A_i(k)e^{ik\cdot x}\), with conjugate momenta \(p_i\) and \(\tilde\Pi_i\). Then \[H=H_0+H_\perp+W,\] \[H_0=\frac{g^2c}{2\hbar L^3}\sum_i|\vec p_i|^2+\frac{\hbar cL^3}{2g^2}\sum_{i<j}|\vec a_i\times\vec a_j|^2,\qquad H_\perp=\sum_{0<|k|\le\Lambda}\Big[\frac{g^2c}{2\hbar}|\hat\Pi(k)|^2+\frac{\hbar c}{2g^2}|k\times\hat A(k)|^2\Big]\] (with \([a^a_i,p^b_j]=i\hbar\delta_{ij}\delta^{ab}\) and the corresponding normalization of \(\hat\Pi\), so that the free part has energies \(\hbar c|k|\)), and \(W\) collects every term of \(\frac{\hbar c}{2g^2}\int|F_{ij}|^2\) that is neither purely constant-mode nor quadratic in \(\tilde A\) at \(a=0\): \[W=W_2+W_3+W_4,\quad W_2=\frac{\hbar c}{g^2}\int\big(\partial_i\tilde A_j-\partial_j\tilde A_i\big)\cdot\big([a_i,\tilde A_j]-[a_j,\tilde A_i]\big)+\frac{\hbar c}{2g^2}\int\big|[a_i,\tilde A_j]-[a_j,\tilde A_i]\big|^2,\] \[W_3=\frac{\hbar c}{g^2}\int\bar F_{ij}\cdot[\tilde A_i,\tilde A_j],\qquad \bar F_{ij}=\partial_i\tilde A_j-\partial_j\tilde A_i+[a_i,\tilde A_j]-[a_j,\tilde A_i]+[a_i,a_j],\] \[W_4=\frac{\hbar c}{2g^2}\int\big|[\tilde A_i,\tilde A_j]\big|^2 ,\] where the purely constant part \(\frac{\hbar cL^3}{2g^2}|[a_i,a_j]|^2\) of \(\frac{\hbar c}{2g^2}\int|F|^2\) is in \(H_0\) and \(W_4\) is quartic in the nonzero modes. The Gauss constraint is imposed on states; the constant gauge transformations act on \(a_i\) by the adjoint action and on the nonzero modes covariantly, and the nonconstant ones are fixed by keeping the transverse nonzero modes, the standard small-volume gauge choice.

Let \(\Omega_\perp\) be the ground state of \(H_\perp\), the Gaussian vacuum of the free transverse modes, with energy \(E_\perp=\hbar c\sum_k|k|\) (a constant that cancels from gaps) and first excitation energy \(\mu_\perp=2\pi\hbar c/L\). Set \[P=1_{\text{const}}\otimes|\Omega_\perp\rangle\langle\Omega_\perp| .\] \(P\) commutes with the constant gauge transformations because \(\Omega_\perp\) is invariant under them, so Lemma 1 can be applied inside the Gauss-invariant space. This does not select a centre character. Any further sector restriction in H3 must be named and preserved by the projection; a one-patch expansion is not a global sector identification. The compression is \[A=PHP=H_0+\langle\Omega_\perp,W\Omega_\perp\rangle(a)+E_\perp,\] where \(\langle\Omega_\perp,W_2\Omega_\perp\rangle(a)\) is a function of the constant modes only (the Gaussian expectation of a quadratic form in \(\tilde A\) with \(a\)-dependent coefficients), \(\langle W_3\rangle\) vanishes (odd in \(\tilde A\)), and \(\langle W_4\rangle\) is a constant. Thus \(A\) is C133’s operator plus a potential \(U(a)=\langle W_2\rangle(a)\), which is the leading change of the nonzero-mode zero-point energy with the constant background.

3. The three inequalities and their expected sizes

Rescale \(\vec a_i=g^{2/3}L^{-1}\vec\xi_i\) and \(\hat A(k)=g\,\hat A'(k)\) so that \(H_0=g^{2/3}(\hbar c/L)\,h_3(\xi)\) with \(h_3\) the unit operator of C133 and \(H_\perp\) is \(g\)-independent in the primed variables. Then, counting powers of \(g\) at fixed \(L\) and at typical values \(|\xi|=O(1)\), \(|\hat A'|=O(1)\):

With these sizes the three inequalities that Lemma 1 needs are:

(H1) \(D=\bar PH\bar P\ge E_\perp+ (2\pi\hbar c/L)(1-C_1g^{2/3})\) on \(\operatorname{ran}\bar P\). This is a global nonzero-mode excitation hypothesis, not a conclusion from inspecting typical \(|a|\sim g^{2/3}/L\).

(H2) \(\eta=\sup_{E\le a_1}\|(D-E)^{-1/2}\bar PWP\|^2 \le C_2g^{4/3}\hbar c/L\) for the specified projection. Squaring the linear-in-\(a\) \(W_2\) amplitude and dividing by a nonzero-mode frequency suggests this order per mode; the mode sum is the ultraviolet problem below. This uniform inequality is unproved.

(H3) \(\operatorname{gap}(A)\ge \delta_1^{(3)}g^{2/3}(1-C_3g^{2/3})\hbar c/L\). An even local quadratic correction of order \(g^{4/3}\hbar c/L\) is a candidate perturbation. Its global relative bound and this gap estimate are additional hypotheses, not consequences of its Taylor coefficient.

Given (H1)–(H3), Lemma 1 with \(\mu=E_\perp+\frac{2\pi\hbar c}{L}(1-C_1g^{2/3})\) and \(a_1<\mu\) (true for small \(g\) since \(a_1-a_0=O(g^{2/3}\hbar c/L)\) and \(a_0-E_\perp=O(g^{2/3}\hbar c/L)\)) yields \[\Delta\ \ge\ \delta_1^{(3)}\,g^{2/3}\,\frac{\hbar c}{L}\,\big(1-Cg^{2/3}\big).\] This conditional bound would give the lower side of the small-volume corner S, comparable with the upper side obtained in the Polyakov-average note up to pure numbers, and it is the leading term of Lüscher’s expansion (abstract, B78) as a two-sided bound.

3a. Correction: the projection must follow the constant mode, and the cutoff must be gauge invariant

The counting in the table is right for the sizes of the operators but (H2) fails for the projection \(P\) of Section 2 when the cutoff is a sharp momentum cutoff. The term of \(W_2\) linear in \(a\) creates pairs of nonzero quanta \((k,-k)\) from \(\Omega_\perp\) with amplitude of order \(g^{2/3}|\xi|/L\) for every \(k\), so \(\|\bar PW_2P\psi\|^2\sim(g^{4/3}/L^2)(\Lambda L)^3|\xi|^2\) and, after dividing by the pair energy \(2|k|\), \[\eta\ \sim\ \frac{g^{4/3}\hbar c}{L^2}\sum_{0<|k|\le\Lambda}\frac{|\xi|^2}{2|k|} \ \sim\ \frac{g^{4/3}\hbar c}{L}\,(\Lambda L)^2\,|\xi|^2 ,\] quadratically divergent in the cutoff. This is the second-order energy shift \(-c\,|a|^2\Lambda^2L^3\) of the constant mode. A quadratic term in \(a\) is allowed by the constant gauge transformations, which act by conjugation and leave \(\operatorname{tr}a_i^2\) invariant, and the exact one-loop valley calculation does contain a finite local quadratic coefficient under a specified Fourier subtraction. The raw frequency sum is divergent; periodicity alone proves no cutoff cancellation or uniform operator estimate. The valley derivation identifies its chosen prescription and distinguishes the full periodic potential from the nonzero-mode correction. A gauge-compatible regularization, matched counterterms and uniform estimates are required.

Fibered projection. Suppose a specified gauge fixing leaves a positive quadratic nonzero-mode operator on a background patch, with a smooth Gaussian vacuum \(\Omega_\perp(a)\), and suppose its projection preserves the relevant operator domains. Then one may use the Born–Oppenheimer projection \[P=\int^\oplus da\;|\Omega_\perp(a)\rangle \langle\Omega_\perp(a)|.\] For this adapted vacuum the quadratic background dependence is incorporated in the compression. Writing \(\omega_k(a)\) for its angular frequencies, its formal zero-point contribution is \[U_\perp(a)=\frac{\hbar}{2}\sum_{k\ne0} [\omega_k(a)-\omega_k(0)]\] with the required polarizations and subtraction. The compression also has terms from momenta acting on \(\Omega_\perp(a)\) (Berry and other nonadiabatic terms); their uniform smallness is not proved. Off-shell noncommuting backgrounds, shifted zero modes and global continuation of this patch need separate treatment. The schematic shifted-frequency comparison \((2\pi/L)^2-c|[a,a]|\) alone does not prove positivity of that full constrained quadratic operator.

The valley’s local \(U_\perp=O(g^{4/3}\hbar c/L)\) at bounded rescaled coordinates is compatible with formal small-volume power counting. General-background quartic counterterms, including the logarithm renormalizing the coupling, require a matching calculation of their own. Neither this local coefficient nor a scalar vacuum-energy subtraction proves H1–H3 or a cutoff-uniform Schur bound. In particular, a quadratic term cannot be called globally relatively small merely because its coefficient is small.

Schur error. With the fibered \(P\), \(\bar PWP\) contains only \(W_3\), \(W_4\) and the non-adiabatic terms. \(W_3\) creates three quanta with amplitude of order \(g/L\) per mode triple, and its second-order contribution is formally part of a two-loop vacuum-energy calculation. The suggestion that divergent scalar terms cancel from a gap does not establish the relative bound or its energy dependence. A finite background-dependent part of order \(g^2\hbar c/L\) motivates (H2) in the form \(\eta\le C_2g^{4/3}(1+g^{2/3}\log\Lambda L)\,\hbar c/L\), with a gauge-invariant cutoff. Neither the constants nor the relative inequality have been established; a logarithmic estimate alone does not identify the running of the coupling or complete the reduction at fixed cutoff.

Relative form of the lemma. The constant-mode coordinate \(\xi\) is unbounded and the pair amplitudes grow with \(|\xi|\), so \(\eta\) cannot be a uniform operator bound; the usable hypothesis is relative to \(A\): \(B(D-E)^{-1}B^*\le\epsilon(A-a_0)+\eta\) with \(0\le\epsilon<1\). Then \(F(E)\ge(1-\epsilon)(A-a_0)+a_0-E-\eta\), the count of negative eigenvalues of \(F(E)\) is at most \(\#\{k:a_0+(1-\epsilon)(a_k-a_0)-\eta<E\}\), and the same argument gives \(E_0\ge a_0-\eta\), \(E_1\ge a_0+(1-\epsilon)(a_1-a_0)-\eta\), hence \(\operatorname{gap}(H)\ge(1-\epsilon)(a_1-a_0)-\eta\). The proposed small \(\epsilon\) is an additional estimate, not a consequence of confinement alone. The known linear growth along flat valleys does not globally control quadratic multiplication \(|\xi|^2\) relative to \(h_3\). A localized low-energy moment estimate, a compact holonomy-domain argument or another uniform operator estimate would be needed. Here \(\epsilon\) is a relative-form error parameter, distinct from the semiclassical ratio \(\hbar/S_0\).

The corrected list is: (H1) as before for \(H_\perp(a)\); (H2) in the relative form for the fibered Schur error; (H3) for \(h_3+U\) with \(U\) the finite one-loop torus potential including its Casimir quadratic term.

4. What the reduction shows

The formal power counting uses \(g^{2/3}\), but estimates of typical coordinates do not establish the operator inequalities. If H1–H3 and the projection’s domain conditions hold, the lemma transfers a gap at the stated price. The one-loop valley expansion supplies a candidate local coefficient, not the global relatively small perturbation assumed in H3. Comparing the candidate zero-mode energy with \(2\pi\hbar c/L\) marks a possible loss of scale separation; it proves no crossover at \(z=\Delta L/(\hbar c)\simeq2\) and no failure of every possible projection.

4a. Schur error and the ultraviolet obstruction

The following refines Lemma 1 and retains the earlier valley calculations as formal power counting, not established inequalities. In particular, the previously asserted fixed-lattice theorem was not proved: H1–H3, the domain and uniform constants of the fibered projection, and the renormalized matching remain open. A logarithm of the right order alone does not prove that the Berry term is absorbed into a running coupling. The earlier cubic slope count used \((\Lambda L)^2\); its displayed six-dimensional integrand instead has superficial degree three, so the formal scaling is \(g^2(\Lambda L)^3\) and its smallness would require \(g(\Lambda L)^{3/2}\ll1\). The Berry coefficient below also needs its charge and polarization multiplicities rechecked; it is not a certified constant. See the obstruction map. The estimates refer to a specified \(SU(2)\) abelian-valley calculation with cutoff \(\Lambda\), not an \(SU(3)\) or continuum gap theorem.

4a.1. The refined transfer lemma with a constant shift

Lemma 1 above bounds the Schur term \(S(E)=B(D-E)^{-1}B^*\) by a constant. When \(S(E)\) contains a large part proportional to the identity, that part shifts all eigenvalues of the compression equally and must not be charged to the gap.

Lemma 1\('\). In the setting of Lemma 1 suppose that for \(E\in I=[a_0-\eta_1-\bar\eta_0,\,a_1]\), \[S(E)=\eta_0(E)\,1+R(E),\qquad 0\le R(E)\le\epsilon(A-a_0)+\eta_1,\] with \(\eta_0\) a nonnegative nondecreasing function, \(\bar\eta_0=\sup_I\eta_0\), \(0\le\epsilon<1\), and \(\eta_0(E)\) differentiable with \(0\le\eta_0'(E)\le\theta<1\) on \(I\). Then \[E_1-E_0\ \ge\ \frac{(1-\epsilon)(a_1-a_0)-2\eta_1}{1+\theta}.\]

Proof. \(F(E)=A-E-\eta_0(E)-R(E)\). The count of negative eigenvalues of \(F(E)\) lies between the counts of \(A-E-\eta_0(E)-\epsilon(A-a_0)-\eta_1\) and \(A-E-\eta_0(E)\). Write \(\tilde E=E+\eta_0(E)\), which is increasing in \(E\) with \(d\tilde E/dE\le1+\theta\). The bounds show that \(E_0\) is the solution of \(\tilde E=e_0\) with \(e_0\in[a_0-\eta_1,a_0]\) and \(E_1\) the solution of \(\tilde E=e_1\) with \(e_1\ge a_0+(1-\epsilon)(a_1-a_0)-\eta_1\); since \(\tilde E\) has slope at most \(1+\theta\), \(E_1-E_0\ge(e_1-e_0)/(1+\theta)\). \(\square\)

The constant shift costs nothing; its energy dependence costs the factor \(1/(1+\theta)\), and \(\theta\) is the quantity that the cubic coupling makes large.

4a.2. The fibered vacuum and its \(a\)-derivative on the abelian valley

On the abelian valley of the valley note, \(a=\alpha T^3\), the charged modes at momentum \(k\) have energies \(\hbar\omega_k^{(\pm)}=\hbar c|k\pm a|\) and the fibered vacuum is the product of Gaussians \(\Omega_\perp(a)=\prod_k\Omega_k(a)\) with \(\Omega_k\propto\exp[-\omega_k(a)|q_k|^2/(2\hbar\cdot\text{norm})]\) in the real mode coordinates \(q_k\) of each polarization and charge. For a single oscillator of frequency \(\omega\) the derivative of the normalized ground state with respect to \(\omega\) is \(\partial_\omega\Omega=-\frac{1}{2\sqrt2\,\omega}\,\Omega_2\), with \(\Omega_2\) the normalized second excited state, so \[\bar P\,\partial_{a_i}\Omega_\perp(a)=-\sum_{k,\pm,{\rm pol}}\frac{\partial_{a_i}\omega_k^{(\pm)}}{2\sqrt2\,\omega_k^{(\pm)}}\;\Omega_{k,2}\otimes\prod_{k'\ne k}\Omega_{k'}, \qquad\frac{\partial_{a_i}\omega_k^{(\pm)}}{\omega_k^{(\pm)}}=\pm\frac{(k\pm a)_i}{|k\pm a|^2}.\] Its norm squared is \(\sum_{k,\pm,{\rm pol}}\frac{(k\pm a)_i^2}{8|k\pm a|^4}\simeq\frac{2\cdot2}{8}\sum_k\frac{\hat k_i^2}{|k|^2} =\frac12\Big(\frac L{2\pi}\Big)^3\frac{4\pi}{3}\Lambda\,[1+O(a/\Lambda)]=\frac{L^3\Lambda}{12\pi^2}\), linearly divergent: the vacua at different \(a\) are not close in norm as the cutoff is removed, which is the first sign that the fiber is ultraviolet-sensitive. What enters the Schur error is the same sum with the energy denominator, which converges better.

4a.3. The Berry term

The constant-mode kinetic operator \(\frac{g^2c}{2\hbar L^3}\sum_i|\vec p_i|^2\) with \(p_i=-i\hbar\partial_{a_i}\) acts on \(\psi(a)\Omega_\perp(a)\) as \(\frac{g^2c}{2\hbar L^3}\big[(-\hbar^2\Delta_a\psi)\Omega_\perp-2\hbar^2(\partial_{a_i}\psi)(\partial_{a_i}\Omega_\perp)-\hbar^2\psi\,\Delta_a\Omega_\perp\big]\). The \(\bar P\)-component of the middle term is the leading non-adiabatic coupling, \[B_{\rm Berry}\,\psi=-\frac{g^2c\hbar}{L^3}\sum_i(\partial_{a_i}\psi)\,\bar P\,\partial_{a_i}\Omega_\perp .\] Each two-quantum state \(\Omega_{k,2}\) costs \(D-E\ge2\hbar\omega_k\) above the fiber, so \[\|(D-E)^{-1/2}B_{\rm Berry}\psi\|^2 \simeq\Big(\frac{g^2c\hbar}{L^3}\Big)^2\sum_i\sum_{k,\pm,{\rm pol}}\frac{(k_i)^2}{8|k|^4\cdot2\hbar c|k|}\,|\partial_{a_i}\psi|^2 =\frac{g^4c\hbar}{16L^6}\sum_i\Big(\sum_k\frac{\hat k_i^2}{|k|^3}\Big)|\partial_{a_i}\psi|^2 ,\] with \(\sum_k\hat k_i^2|k|^{-3}\simeq\frac13\big(\frac L{2\pi}\big)^3 4\pi\int_{2\pi/L}^{\Lambda}\frac{dk}{k}=\frac{L^3}{6\pi^2}\log\frac{\Lambda L}{2\pi}\). In the rescaled variable \(a=g^{2/3}\xi/L\), \(\partial_a=(L/g^{2/3})\partial_\xi\), and \[\eta_{\rm Berry}\ \simeq\ \frac{g^4c\hbar}{16L^6}\cdot\frac{L^3}{6\pi^2}\log(\Lambda L)\cdot\frac{L^2}{g^{4/3}}\,\langle|\partial_\xi\psi|^2\rangle =\frac{g^{8/3}}{96\pi^2}\,\log(\Lambda L)\,\langle|\partial_\xi\psi|^2\rangle\,\frac{\hbar c}{L}.\] Since \(\langle|\partial_\xi\psi|^2\rangle\) is controlled by the kinetic part of \(h_3\), this is a relative bound of the form \(R\le\epsilon(A-a_0)+\eta_1\) with \(\epsilon\sim g^2\log(\Lambda L)/(96\pi^2)\). In this formal estimate the logarithm is the ultraviolet signature of vacuum polarization by the constant background; the one-loop coefficient of the running coupling for \(SU(2)\) is \(b_0=11/(24\pi^2)\), and the Berry term is one of the contributions that together must reproduce it, which is the consistency check the renormalized expansion has to pass. Its relative size is formally \(g^2\log(\Lambda L)\); whether the complete gauge-compatible calculation absorbs it into a renormalized coupling remains to be shown.

4a.4. The cubic coupling: constant shift and slope

\(W_3=\frac{\hbar c}{g^2}\int\bar F_{ij}\cdot[\tilde A_i,\tilde A_j]\) with \(\tilde A=g\tilde A'\) creates three quanta from the fiber vacuum. With the mode normalization \(\tilde A'\sim\sqrt{\hbar/(2\omega L^3)}(b+b^\dagger)\) the amplitude for \((k,k',k'')\), \(k+k'+k''=0\), is of order \(g\hbar c\,|k|\,(\hbar/(2\hbar c))^{3/2}L^{-3/2}(|k||k'||k''|)^{-1/2}\) times colour factors, and the second-order shift is \[\eta_0\ \sim\ g^2\hbar c\sum_{k,k'}\frac{|k|^2/L^3}{|k||k'||k''|\,(|k|+|k'|+|k''|)} \ \sim\ g^2\hbar c\,L^3\Lambda^4\cdot{\rm const}=g^2\,\frac{\hbar c}{L}\,(\Lambda L)^4\cdot{\rm const},\] the two-loop vacuum energy of the box: extensive and quartically divergent, and independent of \(a\) at leading order. By Lemma 1\('\) it costs nothing as a shift. Its energy derivative is \[\theta=\eta_0'(E)\ \sim\ g^2\sum_{k,k'}\frac{|k|^2/L^3}{|k||k'||k''|\,(|k|+|k'|+|k''|)^2}\cdot\frac{1}{\hbar c}\cdot\hbar c \ \sim\ g^2(\Lambda L)^3\cdot{\rm const},\] and Lemma 1\('\) needs \(\theta<1\): this crude undressed estimate would require \(g^2(\Lambda L)^3\ll1\). On a lattice \(\Lambda L=\pi N_s\), so the formal condition is \(g\ll N_s^{-3/2}\). The \(a\)-dependent part of the \(W_3\) shift requires separate gauge-compatible renormalization; its formal expected order does not establish a bound.

5. Consequence for STATE

The Schur lemmas transfer a specified compressed gap; H1–H3, domains, cutoff-uniform constants and physical reconstruction remain open. The periodic holonomy countertest has centre-odd Weyl-even states with an exponentially small upper gap, so a local \(g^{2/3}\) spacing cannot be called its global gap. This countertest is not an identification of \(PHP\) with the scalar benchmark. H3 must first specify whether centre transformations are gauged, which centre character is retained and how the projection and effective measure respect it. A bound in the trivial sector would not by itself control other sectors or construct the continuum theory. The nonadiabatic and cubic Schur estimates then have to be proved for that same operator. A polynomial error upper bound of order \(g^{4/3}\hbar c/L\) alone would not certify positivity at the benchmark’s exponentially smaller inter-sector scale: the transfer lemma needs an error smaller than its chosen compressed gap. This is an insufficiency of the supplied estimate, not a lower bound on the actual Schur error or a no-go theorem for dressing.